Equivariant Degree Theory Reprint 2012 Jorge Ize Alfonso Vignoli

Equivariant Degree Theory Reprint 2012 Jorge Ize
Alfonso Vignoli download
https://ebookbell.com/product/equivariant-degree-theory-
reprint-2012-jorge-ize-alfonso-vignoli-51010392
Explore and download more ebooks at ebookbell.com

Here are some recommended products that we believe you will be
interested in. You can click the link to download.
Equivariant Degree Theory Jorge Ize Alfonso Vignoli
https://ebookbell.com/product/equivariant-degree-theory-jorge-ize-
alfonso-vignoli-992014
Applied Equivariant Degree Zalmon Balanov Wieslaw Krawcewicz
https://ebookbell.com/product/applied-equivariant-degree-zalmon-
balanov-wieslaw-krawcewicz-11165550
Equivariant Stable Homotopy Theory And The Kervaire Invariant Problem
Michael A Hill Michael J Hopkins Douglas C Ravenel
https://ebookbell.com/product/equivariant-stable-homotopy-theory-and-
the-kervaire-invariant-problem-michael-a-hill-michael-j-hopkins-
douglas-c-ravenel-51663934
Equivariant Poincar Duality On Gmanifolds Equivariant Gysin Morphism
And Equivariant Euler Classes Lecture Notes In Mathematics 1st Ed 2021
Alberto Arabia
https://ebookbell.com/product/equivariant-poincar-duality-on-
gmanifolds-equivariant-gysin-morphism-and-equivariant-euler-classes-
lecture-notes-in-mathematics-1st-ed-2021-alberto-arabia-51748894

Equivariant Dmodules On Rigid Analytic Spaces Konstantin Ardakov
https://ebookbell.com/product/equivariant-dmodules-on-rigid-analytic-
spaces-konstantin-ardakov-52181358
Equivariant Cohomology In Algebraic Geometry David Anderson William
Fulton
https://ebookbell.com/product/equivariant-cohomology-in-algebraic-
geometry-david-anderson-william-fulton-53021578
Equivariant Stable Homotopy Theory And The Kervaire Invariant Problem
Michael Anthony Hill Michael J Hopkins Douglas C Ravenel
https://ebookbell.com/product/equivariant-stable-homotopy-theory-and-
the-kervaire-invariant-problem-michael-anthony-hill-michael-j-hopkins-
douglas-c-ravenel-32780696
Equivariant Topology And Derived Algebra 1st Edition Scott Balchin
https://ebookbell.com/product/equivariant-topology-and-derived-
algebra-1st-edition-scott-balchin-36489610
Equivariant Cohomology Of Configuration Spaces Mod 2 The State Of The
Art Pavle V M Blagojevi
https://ebookbell.com/product/equivariant-cohomology-of-configuration-
spaces-mod-2-the-state-of-the-art-pavle-v-m-blagojevi-44838086

de Gruyter Series in Nonlinear Analysis and Applications 8
Editors
A. Bensoussan (Paris)
R. Conti (Florence)
A. Friedman (Minneapolis)
K.-H. Hoffmann (Munich)
L. Nirenberg (New York)
A. Vignoli (Rome)
Managing Editor
J. Appell (Würzburg)

Jorge Ize
Alfonso Vignoli
Equivariant Degree Theory
w
DE
G_
Walter de Gruyter · Berlin · New York 2003

Authors
Jorge Ize
Instituto de Investigaciones en Matematicas
Aplicadas y en Sistemas
Universidad Nacional Autonoma de Mexico
01000 MEXICO D. F.
MEXICO
Alfonso Vignoli
Department of Mathematics
University of Rome "Tor Vergata'
Via della Ricerca Scientifica
00133 ROMA
ITALY
Mathematics Subject Classification 2000: 58-02; 34C25, 37G40, 47H11, 47J15, 54F45, 55Q91,
55E09
Keywords: equivariant degree, homotopy groups, symmetries, period doubling, symmetry breaking,
twisted orbits, gradients, orthogonal maps, Hopf bifurcation, Hamiltonian systems, bifurcation
© Printed on acid-free paper which falls within the guidelines of the ANSI
to ensure permanence and durability.
Library of Congress Cataloging-in-Publication Data
Ize, Jorge, 1946—
Equivariant degree theory / Jorge Ize, Alfonso Vignoli.
p. cm. - (De Gruyter series in nonlinear analysis and
applications, ISSN 0941-813X ; 8)
Includes bibliographical references and index.
ISBN 3-11-017550-9 (cloth : alk. paper)
1. Topological degree. 2. Homotopy groups. I. Vignoli,
Alfonso, 1940- II. Title. III. Series.
QA612.I94 2003
514'.2-dc21 2003043999
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed biblio-
graphic data is available in the Internet at <http://dnb.ddb.de>.
© Copyright 2003 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany.
All rights reserved, including those of translation into foreign languages. No part of this book may
be reproduced or transmitted in any form or by any means, electronic or mechanical, including
photocopy, recording, or any information storage and retrieval system, without permission in
writing from the publisher.
Printed in Germany.
Cover design: Thomas Bonnie, Hamburg
Typeset using the authors' TgX files: I. Zimmermann, Freiburg
Printing and binding: Hubert & Co. GmbH & Co. Kg, Göttingen
ISBN 3-11-017550-9
Bibliographic information published by Die Deutsche Bibliothek

J. I. expresses his love to his wife, Teresa,
and his sons, Pablo, Felipe and Andres.
Α. V. wishes to dedicate this book to his beloved wife Lucilla, to his
son Gabriel and to Angela, who kept cheering him up in times
of dismay and frustration becoming more frequent at sunset.

Preface
The present book grew out as an attempt to make more accessible to non-specialists
a subject - Equivariant Analysis - that may be easily obscured by technicalities and
(often) scarcely known facts from Equivariant Topology. Quite frequently, the authors
of research papers on Equivariant Analysis tend to assume that the reader is well
acquainted with a hoard of subtle and refined results from Group Representation
Theory, Group Actions, Equivariant Homotopy and Homology Theory (and co-counter
parts, i.e., Cohomotopy and Cohomology) and the like. As an outcome, beautiful
theories and elegant results are poorly understood by those researchers that would
need them mostly: applied mathematicians. This is also a self-criticism.
We felt that an overturn was badly needed. This is what we try to do here. If
you keep in mind these few strokes you most probably will understand our strenuous
efforts in keeping the mathematical background to a minimum. Surprisingly enough,
this is at the same time an easy and very difficult task. Once we took the decision of
expressing a given mathematical fact in as elementary as possible terms, then the easy
part of the game consists in letting ourselves to go down to ever simpler terms. This
way one swiftly enters the realm of stop and go procedures, the difficult part being
when and where to stop. In our case, we felt relatively at ease only when we arrived
at the safe harbor of matrices. Of course, you have to buy a ticket to enter. The fair
price is to become a jingler with them. After all, nothing is given for free.
We have enjoyed (and suffered) with the fact that so many beautiful results can be
obtained with so little mathematics. Our hope is that you will enjoy (and not suffer)
reading this book.
Acknowledgments. We would like to thank our families for their patience and support
during the, longer than expected, process of writing the book. Very special thanks to
Alma Rosa Rodriguez for her competent translation of ugly hieroglyphics to beautiful
MgX. Thanks to our colleagues, Clara Garza, for reading the manuscript, to Arturo
Olvera for devising and running some of the numerical schemes which have given
evidence to some of our results and to Ana Cecilia Pérez for her computational support.
We are grateful to L. Vespucci, Director of the Library at La Sapienza, for her help in
our bibliographical search. Last but not least, let us mention the contributions of our
friend and collaborator Ivar Massabó with whom we started, in 1985, the long journey
through equivariant degree.
During the last two years, the authors had the partial support of the CNR, of
the University of Rome, Tor Vergata, given through the scientific agreement between
IIMAS-UNAM and Tor Vergata, and of several agencies on the Italian side, including

viii Preface
CANE, and from CONACyT (grant G25427-E, Matemáticas Nolineales de la Física
y la Ingeniería, and the agreement KBN-CONACyT) on the Mexican side.
México City and Rome, February 2003 Jorge Ize
Alfonso Vignoli

Contents
Preface vii
Introduction xi
1 Preliminaries 1
1.1 Group actions 1
1.2 The fundamental cell lemma 5
1.3 Equivariant maps 8
1.4 Averaging 12
1.5 Irreducible representations 17
1.6 Extensions of Γ-maps 25
1.7 Orthogonal maps 29
1.8 Equivariant homotopy groups of spheres 35
1.9 Symmetries and differential equations 42
1.10 Bibliographical remarks 57
2 Equivariant Degree 59
2.1 Equivariant degree in finite dimension 59
2.2 Properties of the equivariant degree 61
2.3 Approximation of the Γ-degree 67
2.4 Orthogonal maps 69
2.5 Applications 72
2.6 Operations 77
2.6.1 Symmetry breaking 78
2.6.2 Products 78
2.6.3 Composition 79
2.7 Bibliographical remarks 85
3 Equivariant Homotopy Groups of Spheres 86
3.1 The extension problem 86
3.2 Homotopy groups of Γ-maps 102
3.3 Computation of Γ-classes 108
3.4 Borsuk-Ulam results 119
3.5 The one parameter case 136
3.6 Orthogonal maps 156
3.7 Operations 165
3.7.1 Suspension 165

χ Contents
3.7.2 Symmetry breaking 171
3.7.3 Products 180
3.7.4 Composition 188
3.8 Bibliographical remarks 195
4 Equivariant Degree and Applications 197
4.1 Range of the equivariant degree 197
4.2 Γ-degree of an isolated orbit 211
Example 2.6. Autonomous differential equations 222
Example 2.7. Differential equations with fixed period 232
Example 2.8. Differential equations with first integrals 234
Example 2.9. Time dependent equations 237
Example 2.10. Symmetry breaking for differential equations 238
Example 2.11. Twisted orbits 240
4.3 Γ-Index for an orthogonal map 245
Example 3.4. Bifurcation 255
Example 3.5. Periodic solutions of Hamiltonian systems 256
Example 3.6. Spring-pendulum systems 266
4.4 Γ-Index of a loop of stationary points 288
Example 4.1. The classical Hopf bifurcation 289
Example 4.3. Hopf bifurcation for autonomous differential equations. 301
Example 4.4. Hopf bifurcation for autonomous systems with
symmetries 303
Example 4.5. Hopf bifurcation for time-dependent differential
equations 304
Example 4.6. Hopf bifurcation for autonomous systems with
first integrals 308
Example 4.7. Hopf bifurcation for equations with delays 324
4.5 Bibliographical remarks 325
Appendix A Equivariant Matrices 327
Appendix Β Periodic Solutions of Linear Systems 332
Bibliography 337
Index 359

Introduction
Nonlinearity is everywhere. But few nonlinear problems can be solved analytically.
Nevertheless much qualitative information can be obtained using adequate tools. De-
gree theory is one of the main tools in the study of nonlinear problems. It has been
extensively used to prove existence of solutions to a wide range of equations.
What started as a topological (or combinatorial) curiosity has evolved into a va-
riety of flavors and represents, nowadays, one of the pillars, together with variational
methods, of the qualitative treatment of nonlinear equations.
In the simplest situation, the "classical" degree of a continuous map f(x) from
R" into itself with respect to a bounded open set Ω such that f(x) is non-zero on 3Ω
is an integer, deg(/; Ω), with the following properties:
(a) Existence. If deg(/; Ω) φ 0, then f (x) = 0 has a solution in Ω.
(b) Homotopy invariance. If one deforms continuously f(x), without zeros on
the boundary, then the degree remains constant.
(c) Additivity. If Ω is the union of two disjoint open sets, then deg(/; Ω) is the
sum of the degrees of f(x) with respect to each of the pieces.
If one has in mind studying a set of equations, those properties have a striking
conceptual importance: a single integer gives existence results by loosening the rigidity
of the equations and allowing deformations (and not only small ones). In other words,
one does not need to solve explicitly the equations in order to get this information and
one may obtain it by deforming the equations to a simpler set for which one may easily
compute this integer. Furthermore, one has a certain localization of the solutions or
one may obtain multiplicity results for these solutions.
Thus, in dimension one, the degree is another way to view die Intermediate Value
Theorem of Calculus and, in dimension two, it is nothing else than the winding number
of a vector field, familiar from Complex Analysis.
If, furthermore, one requires the property
(d) Normalization. The degree of the identity with respect to a ball containing the
origin is 1,
then, one may show that the degree is unique.
Now there are many ways to construct the degree. As a consequence of the unique-
ness, they are all equivalent and depend more on the possible application or on the
particular taste of the user. For instance, one may take a combinatorial approach, or

xii Introduction
analytical (through perturbations or integrals), or topological (homotopical, cohomo-
logical) or an approach from fixed point theory.
Classical degree theory, or Brouwer degree, would have remained a simple cu-
riosity if it were not for the extension to infinite dimensional problems, in particu-
lar to non-linear differential equations. This extension has required some compact-
ness, starting from the Leray-Schauder degree with compact (or completely continu-
ous) perturbations of the identity, continuing with k-set contractions, A-proper maps,
0-epi maps (these terms will be defined in Chapter 1) and so on. In most of these exten-
sions the compactness is used to construct a good approximation by finite dimensional
maps. One of the by-products of the construction presented here is to pinpoint a new
way to see where the compactness is used.
Now the subject of this book is also that of symmetry. This is a basic concept in
mathematics and words like symmetry breaking, period doubling or orbits are familiar
even outside our discipline. In fact, many problems have symmetries: in the domains
and in the equations. Very often these symmetries are used in order to reduce the set of
functions to a special subclass: for instance look for odd (or even) solutions, or radial,
or independent of certain variables. They are also used to avoid certain terms in series
expansions or, in connection with degree theory, in order to get some information on
this integer, the so-called Borsuk-Ulam results. However, since any continuous (i.e.,
not necessarily respecting the symmetry) perturbation is allowed, the ordinary degree
will not give a complete topological information. This very important point will be
clearer once the equivariant degree is introduced and computed in many examples.
In this book we shall integrate both concepts, that of a degree and that of symmetry,
by defining a topological invariant for maps which commute with the action of a group
of symmetries and for open sets which are invariant under these symmetries, i.e., for
equivariant maps and invariant sets.
More precisely, a map f{x), from M" to Rm for instance or between two Banach
spaces, is said to be equivariant under the action of Γ (a compact Lie group, for
technical reasons) if
Πγχ) = y fix)
for all y in Γ, where y and γ represent the action of the element y in R" and Rm
respectively. Think of odd maps (y = y = — Id) or even maps (y = — Id, y = Id),
or any matrix y expressed in two bases. The set Ω will be called invariant if, whenever
χ is in Ω, then the whole orbit Γ χ is also in Ω. By looking only at maps with these
properties, including deformations of such maps, one gets an invariant, degr(/; Ω),
which is not an integer anymore (unless m = η, Γ = {e}, in which case one recovers
the Brouwer degree) but with properties (a)-(c) valid and (d) replaced by a universality
property.
Since the construction of this equivariant degree is quite simple, we shall not resist
the temptation to present it now. Let f(x) be an equivariant map, with respect to the
actions of a group Γ, defined in an open bounded invariant set Ω and non-zero on
3Ω. Since Ω is bounded, one may choose a very large ball Β containing it. Then
one constructs an equivariant extension / of / to Β. The new map f(x) may have

Introduction xiii
new zeros outside Ω. One takes an invariant partition of unity φ(χ) with value 0 in Ω
and 1 outside a small neighborhood Ν of Ω, so small that on N\Q the map fix) is
non-zero (it is non-zero on 3Ω). Take now a new variable t in / = [0, 1] and define
f(f,x) = (2f + 2φ{χ) — 1, /(*)).
It is then easy to see that /(?, x) = 0 only if Λ; is in Ω with /( x) = fix) = 0 and, since
φ(χ) = 0, one has t = 1/2. In particular, the map fit, x) is non-zero on 3(7 χ Β)
and defines an element of the abelian group (this group will be studied in Chapter 1)
n£„(Sm)
of all Γ-equivariant deformation (or homotopy) classes of maps from 3(I χ B) into
Km+1 \{0}. We define the Γ-equivariant degree of fix) with respect to Ω as the class
of f{t, χ) in n£„(Sm):
degr(/; Ω) = [/V
This degree turns out to have properties (a)-(c), where having non-zero degree here
means that the class [/]p is not the trivial element of (Sm). Furthermore, by
construction, this degree has the Hopf property, which is that if Ω is a ball and [/]p
is trivial, then f\;>n has a non-zero Γ-equivariant extension to Ω. In other words,
degr(/; Ω) gives a complete classification of Γ-homotopy types of maps on spheres.
This property implies also that degr(/; Ω) is universal in the sense that, if one has
another theory which satisfies (a}-(c) such that, for a map / and a set Ω, one has a
non-trivial element, then degr(/; Ω) will be non-zero.
The simplest example is that of a non-equivariant map from M" into itself. Then
we shall see that [/]r is the Brouwer degree of f with respect to / χ Β. Since f is not
zero on / χ (Β\Ω), this degree is that of / with respect to 7 χ Ω, where / is a product
map. A simple application of the product theorem implies that [/]r = deg(/; Ω),
a result which is, of course, not surprising but which indicates that our approach has
the advantage of a very quick definition, with an immediate extension to the case of
different dimensions, including infinite ones.
A second simple example is that of a ^-action on M" = Rk χ W~k = Mm, where
x = (y, z) and fiy, z) = ifoiy, z), f\(j, z)) with /o even in ζ and f\ odd in z. It
turns out that in this case ΠfZ(S") = Ζ χ Ζ, and that deg% (f; Ω) is given by two
integers: deg(/o(;y, 0); Ω Π M*) and deg(/; Ω). As a consequence of the oddness of
/i, with respect to z, one has f\ (jc, 0) = 0 and it is clear that these two integers are
well defined. The set {λ, 0} is the fixed point subspace of the action of Z2 and it is not
surprising that these two integers are important. What is less intuitive is that if Ω is a
ball then these two integers characterize completely all Z2-maps defined on Ω.
A third example is that of an S1 -action on M.k χ Cm 1 χ · · · x C'"p, where S1 leaves
R* fixed and acts as &χρ(ίη]φ), for j = 1,..., p, on each complex coordinate of CmJ.
This is an important example because if one writes down the autonomous equation
dX k
fiX) = 0, X in R ,
dt

xiv Introduction
for X(t) = Σ xne,n\ that is for 2π-periodic functions, then the fact that f(X) does
not depend on t implies that its component fn(X) on the rc-th mode has the property
that
fn(X(t + <p)) = ein<fifn(X(t)),
i.e., the equation is equivalent to an 5'-equivariant problem (infinite dimensional). It
turns out that, in this case, degsi (/; Ω) is a single integer given by deg(/|K*; Ω|κ*)>
i.e., by the invariant part of /. This is a slightly disappointing result but it can
also be viewed as indicating that points with large orbits, in the sense of positive
dimension, corresponding to the complex coordinates do not count when classifying
the Γ-equivariant classes. This is a general fact which will be true for any group. Thus,
in this particular example, one will have new invariants if the domain has (at least)
one more dimension than the range, i.e., / is a function of a parameter ν and of X. In
the case of differential equations, the extra parameter ν may come from a rescaling of
time and represent the frequency. This occurs when one looks for periodic solutions
of unknown period. In that case, it turns out that
(Sn) = Z2 χ Ζ χ Ζ χ · • •
with one Z, giving an integer for each type of one-dimensional orbits, and Z2, an
orientation, corresponding to the invariant part. It is clear that we now have a much
richer structure, which will lead to a host of applications, ranging from Hopf bifurcation
to period doubling and so on. For instance, one may perturb an autonomous differential
equation by a small time-periodic function. Then one may see what happens to the
invariants in (Sn), where one forgets about the S -action, i.e., in ns«+i (Sn) =
Z2. Of course, one could also break the symmetry by adding a (27r//?)-periodic
perturbation, giving rise to other types of invariants.
A last example would be that of the action of a torus Tn, or of the largest torus in
a general group. If this torus is generated by the phases <p\,..., φη, each in [0, 2π],
one may look at Γ-equivariant maps f(x) which have the additional property of being
orthogonal. This means that
f(x)-AjX= 0, j = l,...,n,
where A¡ is the infinitesimal generator corresponding to <pj. This situation occurs when
one considers gradients of invariant functionals: if f(x) = V<p(x), where φ(γχ) =
φ(χ), then, by differentiating with respect to <pj, one obtains this orthogonality. For
instance, this is the situation for Hamiltonian systems, where one of the orthogonality
relations is the conservation of energy. For such Γ-orthogonal maps one may repeat
the construction of the degree and obtain a new invariant
deg±(/;n) inn^tS"),
a group which is much larger than n£„ (S"). In fact, it is a product of Ζ 's, one for each
orbit type, independent of the dimension of the orbit, as we shall describe below, by

Introduction XV
relating this new degree to "Lagrange multipliers". One may look at zeros of the map
f(x) + ^kjAjX = 0,
where if one takes the scalar product with f(x) one obtains a zero of f(x) and the
relation Σ λ-jAjX — 0. In particular, if, for some x, the Aj χ 's are linearly independent,
this implies that k¡ = 0. Of course, this linear independence depends on x, but the
introduction of these multipliers will enable us to compute completely the group
It is now time to have a closer look at the content of the book. We shall do so by
pointing out the parts which may be of special interest to a given group of readers. As
explained in the Preface, we have tried to write a book as self-contained as possible.
This implies that the first chapter is devoted to a collection of some simple facts
from different fields which are needed in the book. Thus we introduce group actions,
equivariant maps, averaging and irreducible representations, in particular, Schur's
Lemma and its consequences. This is all which will be needed from Representation
Theory.
From the point of view of Topology, one of our main tools will be that of extensions
of equivariant maps. There is a special extension for orthogonal maps. A full proof
is given in Theorem 7.1, using the Gram-Schmidt orthogonalization process. We
give also the definition and some basic properties of equivariant homotopy groups
of spheres, the groups where our degrees live. The last section in the chapter is
a review of some of the results from Analysis, in particular, Ordinary Differential
Equations, which will be needed in the last chapter. Thus we integrate a quick survey
of Bifurcation Theory, Floquet Theory (also expanded in Appendix B), Hamiltonian
systems and the special form of orbits arising in these problems (twisted orbits).
Hence, an expert in any of these fields should only glance at some of these results
in order to get acquainted with our notation, and look at some of the examples. For
a reader who is not familiar with these subjects, we hope that (s)he will find all
the necessary tools and acquire a working knowledge and a good intuition from this
chapter.
In this brief description of the first chapter, we left out the second section on the
fundamental cell. This construction, explained here for abelian groups, is the key to
most of the work on equivariant homotopy groups. It says that one may find a region
in Ε", made of sectorial pieces, such that, if one has any continuous function defined
on the cell with some symmetry properties on its boundary, then one may extend the
map to the whole space, using the action of the group. Think of a map defined on a
half-space and extended as an odd map or of a map defined on a sector in C of angle
(2 π/η).
The second chapter is devoted to the definition and study of the basic properties
of the equivariant degree. Furthermore, we show how this degree may be extended
to infinite dimensions by approximations by finite dimensional maps, à la Leray-
Schauder, and how one may define the orthogonal degree. Next, we present abstract
applications to continuation and bifurcation problems and, finally, we study the usual

XVI Introduction
operators on our degree: symmetry breaking, products and composition, operations
which will be studied more deeply in the next chapter and applied in the last chapter.
Of course, this chapter is the abstract core of the book.
Chapter 3 has a more topological flavor. In it we compute the equivariant homotopy
groups of spheres, in the particular case of abelian groups. The reason for this choice
is that we are able to give explicit constructions of the generators for the groups
with elementary arguments (although sometimes lengthy). Thus, anyone should be
able to follow the proofs. The basic idea is that of obstruction theory, that is, of
extension of maps. The program is to start from an equivariant function which is non-
zero on a sphere 9 Β and see under which conditions one may construct an extension
inside the sphere, first to the fundamental cell where one has either an extension, if
the dimension is low enough, or a first obstruction given by some Brouwer degree,
or secondary obstructions which are not unique but may be completely determined.
Then, one uses the group action to extend the map to the whole ball B. Finally, the
homotopy group structure enables one to subtract a certain number of generators and
write down any map as a sum of multiples of explicit generators. These multiples will
be the essence of the degree.
In order to make this program a reality, we work stage by stage. (Here, we ask
the reader to allow us to use some technical arguments so that we may illustrate the
range of ideas developed in the book.) The first step is to consider a map which is Γ-
equivariant and non-zero on d BH and on the union of all BK, such that Η is a subgroup
of K, and where BH stands for the ball in the subspace fixed by Η. In particular,
all points in BH\ (J BK have the same orbit type Η, and extensions are completely
determined by the behavior on the boundary of the fundamental cell. Hence, if the
map is between the spaces VH and WH, the fundamental cell has dimension equal to
dim V H — dim Γ /Η, and, if this difference is less than dim W H, one always has a non-
zero extension, while if one has equality one obtains a first obstruction: the degree of
the map on the boundary of the fundamental cell. This is the content of Theorem 1.1.
The next step is to give conditions under which this obstruction is independent of the
previous extensions. One obtains a well-defined extension degree.
The next step is to continue this extension process to non-zero Γ-maps defined
from [J dBH with dim Γ/Η = k, which are also non-zero on (J BK for Κ with
dim Γ/ΑΓ < k. For this purpose the concept of complementing maps is quite important.
We show that essentially this set of maps behaves as a direct sum of maps characterized
by the extension degrees. The final step is to go on for all k's which meet the hypothesis.
For instance, if V = Rk χ W, then one proves that
= χ Ζ χ Ζ χ ··· ,
with one Ζ for each orbit type Η with dim Γ/Η = k and concerns only orbits
of dimension lower than k.
The next question is the following. Given a map, how does one compute its
decomposition into the direct sum? This is done in two different ways: either by
approximations by normal maps (a topological substitute to Sard's lemma) or by

Introduction xvii
looking at global Poincaré sections. One may relate the Z-components in the above
decomposition to ordinary degrees (see Corollary 3.1 in Chapter 3).
The fourth section is devoted to Borsuk-Ulam results, that is to the computation
of the ordinary degree of an equivariant map. The purpose of this section is to show
how the extension ideas can be used in this sort of computations.
The next section treats the case of maps from IxW into W, which is particularly
important when one breaks the S '-symmetry, for instance for an autonomous differ-
ential equation with unknown period by perturbing it by a (27r/p)-periodic field. We
compute then Πο, in the above formula, and prove that now there are obstructions
for extensions to the faces of the fundamental cell and to the body of that cell. For
each Η with Γ/Η finite one has a classification of the secondary obstructions in a
group isomorphic to Z2 χ Γ/Η, with explicit generators according to the different
presentations of Γ/Η.
The sixth section deals with the computation of the homotopy group of spheres
for Γ-orthogonal maps, proving that
nr±sy(Sv) = ZxZx
with one Ζ for each orbit type, independent of its dimension. This is done via the
Lagrange multipliers already mentioned, and the reader will guess why the case of
Γ-equivariant maps with parameters, from IR* χ W onto W, is important here.
The last section of Chapter 3 deals with operations: suspension, products, compo-
sition and symmetry breaking. That is, what happens to the explicit generators under
one of these operations.
As we have already said this third chapter is more topologically inclined. A reader
more interested in applications should only look at the statements of the results, which
will be used in the last chapter, and see some of the examples.
However, we would like to make a few points. Our entire construction relies
on a single basic fact: a map from a sphere into a higher dimensional sphere has a
non-zero extension to the ball, while, if the dimensions are equal, one has a unique
"obstruction", an integer, for extension (and other invariants if the dimension of the
range is lower). From this, with "elementary" but explicit arguments, and with no
algebraic machinery, we obtain surprising new results which may be understood by
any non-specialist. Of course, there is a price to be paid: our actions are linear and
the groups are abelian (the non-abelian case may be dealt with in a similar, but less
explicit way). On the other hand, our pedestrian approach stresses some new concepts,
like those of complementing maps, normal maps and global Poincaré sections, which
may be useful in a more abstract context. In short, independently of the reader's
background, we believe that this chapter may be useful and interesting to anyone.
The last chapter is essentially devoted to applications, although the first section
states that any element in n£v (Sw) is the Γ-degree of a map defined on a reasonable
Ω. Now, in order to be useful, a degree should be computable in some simple generic
cases, for instance for an isolated orbit or an isolated loop of orbits. For the case of an
isolated orbit, the natural hypothesis is to assume that 0 is a regular value. (We recall

xviii Introduction
here our introductory remarks: one does not have to consider the nonlinear equation
under study, but a, hopefully, simpler equation where one may look at these generic
situations.) This leads to approximation by the linearization of the map at the orbit.
The simplest case is when one has a stationary solution, or, even better, a family of
such solutions, leading to bifurcation. In this case, the Γ-index is given by the sign of
determinants of the linearization on the fixed point subspace of Γ and on the subspaces
where Γ acts as Z2, giving conditions for period doubling. The next case is when the
isolated orbit has an orbit type which is not the full group. For this sort of solution,
we obtain an abstract result (Theorem 2.4) and the Γ-index is given in terms of the
spectrum of the linearization, à la Leray-Schauder, but with many of these indices.
This abstract result is applied to autonomous differential equations of unknown period
or of fixed period but with an extra parameter, or with a first integral. One may then
perturb this autonomous differential equation with a time-periodic function and obtain
subharmonics or phase locking phenomena. If the autonomous differential equation
has also a geometrical symmetry, then one obtains twisted orbits.
We are phrasing this part of the introduction in a way which will be easily recog-
nizable by a reader familiar with low dimensional dynamical systems. However, each
specific behavior will be explained in that chapter.
A similar situation occurs for orthogonal maps. In that case the orthogonal index
has components which are of the previous type (i.e., leading to period doubling) and
a new type given by a full Morse index, i.e., the number of negative eigenvalues of a
piece of the linearization. This is applied to Hamiltonian systems of different types,
where variational methods give also invariants depending on Morse numbers. In the
present case it is the orthogonality which brings in this invariant.
In order to show how to apply our degree, we give the complete study of two spring-
pendulum systems. We hope that this example makes the point of the usefulness of
the equivariant degree approach and we challenge the reader to guess (a priori) the
type of solution we obtain.
The final section deals with the index of a loop of stationary solutions, with ap-
plications to Hopf bifurcation, systems with first integrals and so on. It is important
to point out that all our examples (except a very simple retarded differential equation)
come from Ordinary Differential Equations. The main reason for this choice is to avoid
technicalities. It should be clear to anyone interested in Partial Differential Equations,
for instance, how to adapt these result to many situations. For example, replace Fourier
series by eigenfunctions expansions or other Galerkin-type approximations. Another
reason for this choice is that the reader may easily see how the degree arguments are
used to obtain information on the solutions of a nonlinear equation in an integrated
way, that is, with the same tool in different situations (and not with ad hoc degrees),
and see what happens if one modifies the conditions of the problem, as in symmetry
breaking. Here, we would like to stress the Hopf property, i.e., that, if the degree is
zero, then it is likely that one may perturb the problem (in the sense of extensions of
maps) so that the new problem has no solutions. This property and the global picture
which enables one to relate two different solutions or two different problems, is one of

Introduction xix
the main conceptual contributions of degree theory. Of course, we are not computing
the actual solutions (nothing is for free), although it would be interesting to adapt the
homotopy numerical continuation methods to equivariant problems.
Each chapter has a final section on bibliographical remarks. We have tried to
indicate some other approaches to the subject matter of this book. However, it is clear
that most of this book is based on the authors' research in the last 15 years. It is also
clear that there is still much to do. For instance, perform similar computations for
actions of non-abelian groups with its endless list of applications. Similarly, there are
more or less straightforward extensions (we have mentioned several times the word
¿-set contraction) or applications to P.D.E.'s (essentially some technical problems)
and many more. We hope that this book will serve as an incentive for the reader to
follow up in that direction.
A last technical point: theorems, lemmas, remarks and examples are listed inde-
pendently. For instance Theorem 5.2 refers to the second theorem in Section 5 of the
chapter. When referring to a result from another chapter, this is done explicitly: for
instance, Theorem 5.2 of Chapter 1. On the other hand, our notations are standard,
but we would like to emphasize a particular one (maybe not too familiar): Η < Κ
means that Η is a subgroup of Κ (and could be Κ itself).

Chapter 1
Preliminaries
As mentioned in the Introduction, the main purpose of this chapter is to collect some of
the most useful definitions and properties of actions of compact Lie groups on Banach
spaces, as well as the elements of homotopy theory and some facts about operators
which will be most frequently used in this book. Thus, the reader will find here almost
all the results needed in this text. The expert will have only to glance at the definitions
in order to get acquainted with our notation.
1.1 Group actions
In the whole book Γ will stand for a compact Lie group (the reader will see below
which properties of a Lie group are used here).
Definition 1.1. A Banach space £ is a Γ-space or a representation of the group Γ,
if there is a homeomorphism ρ of Γ into GL(£), the general linear group of (linear)
isomorphisms over E. In this case, we say that Γ acts linearly on E, via the action
ρ(γ)χ, such that
P(YY') = P(V)P(Y'),
p(e) = Id.
When no confusion is possible, we shall denote the action simply by y.
Example 1.1. Let Ε = Μ" χ Mm and Γ = Z2 = {-/, /} with
p(-I)(X,Y) = (-X,Y). (1.1)
Example 1.2. If E — C and Γ = TLm = {0,1,... ,m — 1} is the additive group of
the integers modulo m, let
p(k)z = eljlikp/mz, where ρ is a fixed integer. (1.2)
Example 1.3. If E = C and Γ = S1 = Μ/2π = [φ e [0, 2ττ)}, then one may have
ρ(φ)ζ = βίηψζ (1.3)
for some integer n.

2 1 Preliminaries
Example 1.4. If E = C and Γ = Τ" χ Zm| χ · · · χ Zm¡ = {{ψ\,... ,<pn,k\,... ,ks)
with ψ] e [Ο, 2π), O < kj < my}, then one may have
η s
ρ(γ)ζ = expi(^nyçpy + 2 π ^¿y/y/my)z, (1.4)
ι ι
where tij and lj are given integers.
Remark 1.1. We shall see below that this is the general case of an irreducible rep-
resentation of any compact abelian Lie group. It is easy to see that if TLm acts on C,
then p(m) = 1 = p(l)m and p() must have the form given in (1.2). Since the same
argument applies to S1 acting on C, then any Γ given by an abelian product as in (1.4),
must act on C as in that formula.
On the other hand if Zm acts non-trivially on R, then m is even and ρ (I) = —1,
while S may act only trivially on R, i.e., ρ(φ) = ρ(φ/Ν)Ν, take Ν so large that the
continuity of ρ implies that ρ(φ/Ν), being close to 1, must be positive. Hence, ρ (φ)
is always a positive number. Since p{2π) = 1 = ρ(2π/Ν)Ν one gets ρ(2π/Ν) = 1
and p{2np/q) — p(2n/q)p = 1 and by denseness of Q in R, one obtains ρ (φ) = 1.
For convenience in the notation, we shall very often use ( 1.4) to denote also the action
of Γ on R, with the convention that, in that case, n¡ = 0, lj is a multiple of my/2 if
rrij is even, or lj = 0 if my is odd.
Example 1.5. Let E = C^ibe the space of continuous, 2π -periodic functions
on R^ with the uniform convergence norm. The group Γ = S1 may act on E as
ρ{φ)Χ{ί) = X{t + ψ)
i.e., as the time shift.
One may also set this action in terms of Fourier series by writing
00
X(t) = ΣΧη*ίη'.
—oo
with Xn e CN, X-„ — Xn (since X(t) e Rw). For the Fourier coefficients Xn one
has the equivalent action:
ρ(φ)Χη=βίη^Χη. (1.5)
Definition 1.2. Let £ be a Γ-space and χ e Ebe given. The isotropy subgroup of Γ
at χ is the set Γχ = [γ e Γ : γχ = x}, which is a closed subgroup of Γ.
Definition 1.3. The action of Γ on £ is said to be free if Γ* = [e] for any χ e £\{0).
The action is semi-free if = {e} or Γ for any χ e E.

1.1 Group actions 3
For instance, in Example 1.1, Γ^ ^ = %% if and only if X = 0 and the action
is semi-free. In Example 1.2, the action is free only if ρ and m are relatively prime
(denoted as (p : m) — 1), while if p/m = q/n with (q : n) = 1, then Γζ = Zm/„ —
{k — sn, s — 0,..., m/n — 1}. In Example 1.3, one has Γζ = Ζ/γ = {ψ = k/N, k
0 V 1}. The case of Example 1.4 will be given below in Lemma 1.1.
Definition 1.4. The element χ e E is called a fixed point of Γ if Γ* = Γ. The
subspace of fixed points of Τ in E is denoted by ER. If Η is a subgroup of Γ then
Eh = {x e Ε : γχ = χ for any γ e Η] is a closed linear subspace of E.
Notation 1.1. If Η is a subgroup of A", we shall write Η < K. Note that if Η < Κ,
then EK c EH.
Definition 1.5. If Η < Γ, the normalizer N(H) of Η is
N(H) = {γ e Γ : γ~ιΗγ c Η}
and the Weyl group W(H) of Η is
W(H) = N(H)/H.
Note that if Γ is abelian, then Ν (Η) = Γ.
Also, if χ e Eh , then γχ e EH for any γ e N(H), since yiyx = γγ2χ = γχ
for some γι and γ2 in Η. Hence γχ is fixed by the action of Η. Furthermore, if
Η = Γ χ for some χ and γχ e EH for some y, then it is easy to see that γ belongs to
N(H), i.e., N(H) is the largest group which leaves EH invariant. Moreover, if Γ is
abelian, then N(H) — Γ and EH is Γ-invariant.
Let us now consider the case of Example 1.4.
Lemma 1.1. Let Γ = Τ" χ Zm, χ · · · χ Zmj act on C via
expi((N, <b)+2n(K,L/M)),
where {Ν,Φ) = J2" nj<Pi and {K, L/M) = ^ kjlj/mj. Iflj/mj = Ij/mj, with lj
and thj relatively prime, let m be the least common multiple of the thj's (I.e.m) and
set |JV| = ΣΪ j\. Then:
(a) IfL φ 0, there is KQ such that (KQ, L/M) Ξ Ì / m, [2π], and any other Κ gives
an action of the form q /m for some q 6 {0,..., m — 1}. In particular, if Ν = 0
and Η is the isotropy subgroup, then W(H) = Z^.
(b) If Ν φ 0, the congruence (Ν, Φ) = 0, \2π], gives | TV | hyperplanes in T". In
particular, if L = 0, then W(H) = Sl = T/Z\N\.
(c) IfL φ 0 and Ν φ 0, then W(H) = S1 = Τ/ΖΛ\Ν\·

4 1 Preliminaries
Proof, (a) If s = 1, then kl/m is an integer if and only if k is a multiple of m and
e2nikl/m gjves ^ distinct roots of unity, hence the result is clear.
If s = 2, from the preceding case, one has kjïj/mj = kj/m¡, with 0 < kj < rhj
and one has to consider k\/m\ + ¿2/^2· Now, m = p\m\ = piñi2, with p\ and
P2 relatively prime by the definition of a l.c.m. Thus, there are integers «ι, «2 such
that a\p\ + ot2P2 = 1, where a\ and ctj have opposite signs. Assume that a¡ > 0.
Divide ai by m\ and get αϊ = ah\ + with αϊ > 0 and 0 < k® < m\. Likewise,
—«2 = (^2 + 1)^2 — kj, with Ü2 > 0 and 0 < k® < m2. Then, p\k® + p2k® =
αϊ pi + ct2P2 + («2 + 1 — a)m, defining Kq in this case. For any other pair (¿1, k.2),
we have k\/m\ + ¿2/^2 = {p\k\ + pi^lm = {p\k\ + P2¿2)(¿?/>"1 + k^/m.2),
proving the result for s = 2.
For the general case, assume the result true for s — 1. Let m be the l.c.m. of
(mi,... ms-1) and m be the l.c.m. of m and ms. We have
s—1
^ kjïj/mj + kl/m s = qo/rh + kï/ms,
1
where qo is given by the induction hypothesis in such a way that
i—1
kjlj/mj ξ \/m and kj = qok'·.
1
One is then reduced to the two "modes" case.
(b) For the action of Tn, one has that {Ν, Φ) spans an interval of length 2π |yV|.
The congruence {Ν, Φ) = 0, [2π], gives |TV| parallel hyperplanes in T". One may
change <Pj to 2π — <pj whenever Nj is negative, defining an isomorphism of Tn for
which all Nj's are positive. Then, (Ν, Φ) = \Ν\φ, with 0 < φ < 2π/\Ν\, will give
that, if L = 0, then Η = Τη~λ χ Z\N\ χ Zmi χ···χΖΛ) with W(H) = S1 = T/Z\N\.
(c) In general, one may write (Ν, Φ) + 2π(Κ, L/M) as + 2nq/m, with
0 < q < m,(p in [0, 2ττ/|Λ^|). The relation + 2nq/m = 2kn will give
φ = {Ν,Φ)/\Ν\ = 2kn/\N\ — 2nq/m\N\ which represents m\N\ different par-
allel hyperplanes in T". Thus, Η = Γ""1 χ ΖΛ\Ν\ and W(H) = 51 = Τ/ΖΑ\Ν\·
•
Definition 1.6. An isotropy subgroup Η is maximal if Η is not contained in a proper
isotropy subgroup of Γ.
Lemma 1.2 (Golubitsky). If Η is a maximal isotropy subgroup of Γ and Er = {0},
then W(H) acts freely on EH\{ 0}.
Proof In fact, if γ χ = χ for some χ φ 0 in EH and some γ e N(H)/H, then
Tx D Η U {y}. Hence, from the maximality of H, one has Γχ = Γ, but then
χ € Er = (0). •

1.2 The fundamental cell lemma 5
Remark 1.2. The groups which act freely on Euclidean spaces have been completely
classified: a reduced number of finite groups, S1 and NCS1) in 53 and S3 (see [Br]
p. 153). For an abelian group with an action given by (1.4), one has H = {e}, i.e.,
W(H) = Γ, only if either η = 0, .<? = 1 and the action of Zm is given by e2jr'^/m,
with ρ and m relatively prime (hence m = m), or η — 1, s = 0, \N\ — 1, with an
action of S1 given by ε'φ (see Lemma 1.1).
Definition 1.7. The orbit of χ under Γ is the set Γ(χ) = {y χ G E : γ e Γ).
It is easy to see that Γ (χ) is homeomorphic to Γ/Γ*, that Γγχ = γΓχγ~ι (in
particular Γγχ = Γ\ if Γ is abelian) and that the orbits form a partition of E. The set
Ε/ Γ is the orbit space of E with respect to Γ.
Definition 1.8. Two points χ and y have the same orbit type H if there are yo and y
such that Η = γ^λΤχγϋ = yf λΤγγ\.
If E is finite dimensional, then it is clear that there are only a finite number of orbit
types.
Definition 1.9. The set of isotropy subgroups for the action of Γ on £ will be denoted
by Iso(£).
1.2 The fundamental cell lemma
In this section we shall assume that one has a finite dimensional representation V of
the abelian group Γ = Tn χ Zm, χ · · · χ Zms in such a way that any X in V is
written as X = Y^Xjej, where x¡ e C if W(Tej) = Zp or Sl, ρ > 2, or x¡ e Κ if
W(rej) = {e} or The action of Γ on the elements of the basis is given by
γ e¡ = expil((Nj, Φ) + 2 π {Κ, Lj/M))ej,
as in (1.4) and Remark 1.1, with
NJ = (n{,...,nJnf and V ¡M = (///mi,..., l¡/ms)T.
Then yX = Σ xjYej and γΧ = X gives yej = e¡ if χ¡ Φ 0. Hence, Γχ = Ρ) Tej,
where the intersection is over those j's for which Xj Φ 0. Thus, W(Ye¡) < ΐν(Γχ).
Lemma 2.1. VT" = {X e V : Ψ(Γχ) < oc}.
Proof. If is finite, then W(rep is a finite group and Γ^ contains T". In this
case, Γχ contains also T", that is, X belongs to VT". Conversely, if X is fixed by T",
then νν(Γχ) is a factor of Zm, χ · · · χ Zm¡ and hence is finite. •
Denote by Hj = and define Hj-i — H\ Π ••• Π //,_), H0 = Γ. Then H}-
acts on the space Vj generated by e¡ (Vj = MorC), with isotropy Hj-i Π Hj = Hj, if

6 1 Preliminaries
Xj φ 0, and Hj-\/Hj acts freely on V/\{0}. Then, from Lemma 1.1, this Weyl group
is isomorphic either to Sl, to {e}, or to Zp, ρ > 2. Let kj be the cardinality of this
group: kj = \Hj-\/Hj\. If the group is S1, then kj = oo, while kj = 1 means that
Hj^i = Hj. If kj = 2 and Vj is complex, then Vj splits into two real representations
of Hj-\/Hj = Z2, while if Vj is real, then kj = 1 or 2.
Consider G = {X 6 V : \x¡\ = 1 for any j}, a torus in V. Let Η = H\ Π
ÍÍ2 Π • · · Π Hm+r be the isotropy type of G, where there are m of the Vj ' s which are
complex and r which are real (hence dim V = 2m + r). Let k be the number of j's
with kj = 00. Let
Δ = {X e G : 0 < Argx¡ < 2n/kj for all ; = 1,..., m + r}.
That is, if kj = 1 there is no restriction on Xj (in C or R), while, if kj = 00, then
Xj e and, if Xj € IR and kj = 2, then x¡ is positive. Let
Δν = (X e V : 0 < ArgXj < 2π/kj}.
Then Ay is a cone of dimension equal to dim V — k. The set Δ y will be called the
fundamental cell. It will enable us to compute all the equivariant homotopy extensions
and to classify their classes in Chapter 3.
Lemma 2.2 (Fundamental cell lemma). The images of A under Γ /Η cover properly
G (i.e., ina 1-1 fashion).
Proof. The proof will be by induction on m + r. If there is only one coordinate, then
Γ/Η\ acts freely on V| \{0}. If this group is S1, then the image of e¡ under it will
generate G, while if this group is Z*,, > 1, then one has to cut G into k\ equal
pieces in order to generate G.
If the result is true for η - 1, let G = Gn~ 1 χ {\xn\ = 1}, Δ = Δ„_ι χ {0 <
Arg*„ < 2n/kn) and write Γ/Η = (Τ/Hn-)(Hn-\/Η), recalling that these groups
are abelian. By the induction hypothesis, the images of Δη_ι under Τ/Ηη-\ cover
properly Gn-\. Furthermore, from the case η = 1, the set [xn : |jcrt[ = 1} is covered
properly by the images of {jc„ : 0 < Argxn <2π/kn] under Hn-\/H, a group which
fixes all points of Cn-\. Hence, if (X„_i, jc„) is in G, there are y„_i in T/Hn-\ and
Yn in Hn-i/H such that Χ„_ι = γη-\Χ°η_ν with in C„_ 1, γ~\χη = Yn
with 0 < Argx^ < 2ix¡kn and γηΧη-\ = Χη-\·
Then (Χ„_ι,λ„) = (γη-ιΧ°_ν Yn-ìY'^Xn) = yn-iVn(X%-i,xj}), i-e-, G is
covered by the images of Δ under Γ/Η.
If(Xn-i,xn) = γ(Χ\χι) = γ2(Χ2,χ2), with (Xj,xj) in Δ and y¡ in Γ/Η,
then (Χ',λ;1) = yf'^ÍX2,χ2). Thus, Χ1 = γΧ2, χ1 = γχ2. By the induction
hypothesis, Χ1 = X2 and γ belongs to Hn-1, but then χ1 = χ2 and γ belongs to H.
•
This fundamental cell lemma will be the key tool in computing the homotopy
groups of Chapter 3.

1.2 The fundamental cell lemma 7
Example 2.1. Let S1 act on e} via βη'ψ, with nj > 0. Then, Hj — {φ = Ink/tij, k —
0,...,«/ — 1} = Z„.. Let ñj — (η \ : • · • : rij) be the largest common divisor (l.c.d.)
of n\,..., nj, then Hj = [φ = 2πk/ñj, k = 0,..., ñj — 1} = Ζf¡ . Thus, k\ = oo,
kj = ñj-i/ñj.
Note that, since Γ/Η = (T/Hi)x(Hi/H2)x---x(.Hm+r-i/H) if dim Γ/Η = k,
then there are exactly k coordinates (which have to be complex) with kj = oo. In
fact, since Hj is the isotropy subgroup for the action of Hj-\ on χy, each factor, by
Lemma 1.1, is at most one-dimensional.
Lemma 2.3. Under the above circumstances, one may reorder the coordinates in such
a way that kj = oo for j = 1,..., k and kj < oo for j > k.
Proof. Assuming k > 0, there is at least one coordinate with dim Γ/Hj = 1: if
not, Hj > T" for all j's and hence Η > Tn with \Γ/Η\ < oo. Denote by z\ this
coordinate, then Γ/Η = (Γ/Ηι)(Ηι/Η), with dim H\/H = k - 1. If Hy/H is a
finite group, i.e., k = 1, then one has a decomposition into finite groups with kj < oo
for j > 1. On the other hand, if k > 1, then, by repeating the above argument, one
has a coordinate Z2 with H\ /Hi of dimension 1. •
The following result will be used very often in the book.
Lemma 2.4. Let T" act on V = Cm via expi (Nj, Φ), j = 1,..., m. Let A be the
m χ η matrix with NJ as its j-th row. Then:
(a) dim Γ/Η = k if and only if A has rank k.
(b) Assuming kj — oo for j = 1,..., k and that the k χ k matrix Β with B¡j = nlj,
1 < i, j < k, is invertible, then one may write ΛΦ = , with Φ = Φ + ΛΦ,
where Φτ — (ΦΓ, ΦΓ) and Φτ — {ψ\,..., <pk).
(c) With the same hypothesis, there is an action ofTk on Cm, generated by ΨΓ =
(Φι,..., Φ*) such that {NJ, Φ) = (Mj, Φ), with Mj = (m[,...,m[) such
thatmj — SijMjfor j = 1k, i.e., the action of Τk on the first k coordinates
reduces to elMi*j.
Proof (a) The relation {Ν^, Φ) ξ 0, [2ττ] gives parallel hyperplanes in M" with
normal Ν i. Thus, dim Η = η — k is equivalent to dim ker A = η — k.
(b) Write A = ^ ^ and let A = B~lC. Then, Λ Φ = 0 means Φ = -AÔand
(E - DA)Φ = 0. Since dim ker A = η - k, one has E = DA, ker A = (—ΑΦ, Φ)
and ΑΦ has the form given in the lemma.
(c) Let M be a k χ k diagonal matrix such that B~x M has integer entries. Define
Φ = Then, ΑΦ = Qß-'ΜΨ = (ο^Μα/ψ) 8ives the action of Tonce
one has noticed that the entries of DB~l M are integers. •

8 1 Preliminaries
Another simple but useful observation is the following
Lemma 2.5. Let Tn act on V as before. Then there is a morphism S] —>• Tn given
by ψ] = Mjtp, Mj integers, such that {7Vy, Μ) ψ 0, [2π], unless Ν J = 0 and
Vs' = VT". The vector M is (Mi,..., Mn)T.
Proof. As before, the congruences (NJ, Φ) ξ 0, [2π] give families of hyperplanes
with normal Ν i, if this vector is nonzero. From the denseness of Q in M it is clear
that one may find integers (Mi,..., Mn) such that the direction {(pj = Mj<p] is not
on any of the hyperplanes {Ν·*, Φ) = 0, for j = 1,..., m. Thus, Σηι Μι φ 0 and,
being an integer, this number cannot be another multiple of 2n, unless N^ = 0 and
the corresponding coordinate is in VT". •
Definition 2.1. Let Κ be a subgroup of Γ (not necessarily an isotropy subgroup) and
let Η = P| Vej D K, where {ej} span VK. We shall call Η the isotropy subgroup of
VK. Note that Κ < Fe and that VH = VK.
A final technical result is the following:
Lemma 2.6. Let Η be an isotropy subgroup with dim W(H) = k. Then there are
two isotropy subgroups H_ and Η, both with Wey I group of dimension k, such that
H_ < Η < Η. The group Η is maximal among such subgroups and H_ is the unique
minimal such subgroup. H_ will be called the torus part of Η.
Proof. Let Η be such a maximal element, for example given by H\ Π · · · Π Hk
as in Lemma 2.3. Then, Γ/Η = (Γ/H)(H/H) and H/H is a finite group. If
Η = Tn~k χ Z„[ χ · · · χ Zn,, then, from Lemma 2.1 applied to Η, one has that VT"
is the linear space of all points with W(HX) finite. If H_ is the isotropy subgroup of
Tti—k IT T*t\'—ic
V1 , then, since V" is contained in V , one has that H_ is a subgroup of Η and
contains Tn~k (from Definition 2.1) and is clearly unique. •
Remark 2.1. If A is the matrix generated by the action of Tn on V and AH its
restriction on V H (as in Lemma 2.4), then A H and A— have rank k. Furthermore, from
Lemma 2.4 (b), Α—Φ = (ββ//)Ψ with Ψ = Φ + Λ Φ and the torus part corresponds
to Φ ξ 0. It is easy to see that on V— one has exactly nj = Y^!¡=\ η]ιλ\ for i > k and
7 = 1,..., dim V—, where λ\, I = 1,..., k, i — k + 1,..., η are the elements of the
k χ (n — k) matrix Λ.
1.3 Equivariant maps
A look at the heading of this book tells us that perhaps it is time to get started with
some formal definitions.

1.3 Equivariant maps 9
Definition 3.1. Let £ be a Γ-space. A subset Ω of £ is said to be Τ-invariant if for
any χ in Ω, the orbit Γ(χ) is contained in Ω.
Definition 3.2. If Β and E are Γ-spaces, with actions denoted by γ and γ respectively,
then a map / : Β E is said to be Τ-equivariant if
f(yx) = γ fix)
for all λ: in B.
Definition 3.3. Let Γ act trivially on E. A map / : Β E is said to be Τ-invariant
if f(Yx) = f(x), for all xe B.
Example 3.1. Let act on Β = E as —/, then an odd map, f(—x) = — f(x), is
Z2-equivariant. On the other hand, an even map, f(—x) = /(*), with a trivial action
on E is Γ-invariant. In general, if Β — BZl φ B\, E = EZl φ E\, with an action of
Z2 as —I on B\ and E\, then an equivariant map f(xο, jci) = (/o, /i)(*o> will
have the property that fo(xo, -*i) = /oC*o, -*i) and fi(xo, -xi) = -/i(xo, x)· In
particular, f\ (jco, 0) = 0, that is, / maps ß^2 into E^2. We shall see below that this
is a general property of equivariant maps.
Example 3.2. Let C^OR^), respectively ϋ\π (E^), be the space of continuous, re-
spectively differentiable, 27Τ-periodic functions X(t) in RN, with the action
p(tp)X(t) — X(t + φ). Let f(X) be a continuous vector field on independent
of t. Then
dX
F(X) = — - f(X)
at
is S '-equivariant.
In terms of Fourier series, X(t) = Σ Xne'nl with X-n = Xn, one has the equiv-
alent formulation
inX„-fn(X ο,Χι,...), η = 0,1,2,...,
with fn(Xo, X\,··.) = ¿ fo* f(X(t))e~in'dt. In this case the action of S1 on
Xn is given by em<pXn, and it is an easy exercise of change of variables to see that
fn(X 0, e^Xx, envx2,... ) = ein«>fn(X0, XltX2t... ), i.e., that the map F is equiv-
ariant.
Note that the isotropy group of Xn is the set Η = {φ = 2kn/n, k = 0, ...,
η — 1} = Z„ and that VH = {Xm, m = 0 or a multiple of «}.
Example 3.3. Let Γο be a group acting on RN and let f(yoX) = yof (X) be a Γο-
equivariant vector field.
If Γ = S1 χ Γο one may consider the Γ-equivariant map
F(X) = — - f(X)

10 1 Preliminaries
on the space of 2π -periodic functions in RN. If Η is the isotropy subgroup of a Fourier
component Xn, then the space VH of "twisted orbits" has an interesting description
given in the last section of this chapter.
We are going now to describe some of the simplest consequences of the equivari-
ance.
Property 3.1 (Orbits of zeros). If f(yx) = yf(x) and f(xo) = 0, then f(yxo) = 0,
for all γ in Γ.
Property 3.2 (Stratification of the space). If f : Β E is Γ-equivariant, then if
Η < Γ, f maps BH into EH. The map fH = f\gH is Ν (H)-equivariant.
Proof. For χ in BH and γ in H, one has f(yx) = f(x) = yf(x). Hence, f(x)
is fixed by H, i.e., it belongs to EH. Now, since N(H) is the largest group which
keeps BH invariant, this implies that yx is in BH for γ in N(H) and χ in BH, and
the remaining part of the statement follows. •
Note that, in particular, if Γ is abelian, then fH is Γ-equivariant. This simple
property implies that one may try to study / by looking for zeros with a given sym-
metry (for example, radial solutions). It is then convenient to reduce the study to the
smallest possible BH, i.e., the largest H, in particular to maximal isotropy subgroups,
where one knows that W(H) acts freely on Btì and which are completely classified.
If, furthermore, one decomposes BH into irreducible representations of W(H) (see
Section 5), one may determine, not only the linear terms, but also higher order terms
in the Taylor series expansion, if the number of representations is small. These ideas
have been used extensively, in particular in the physics literature, in order to give
normal form expansions. The information obtained this way is very precise but, from
the requirements of genericity and low dimension, it does not allow for a complete
study of stability, symmetry breaking or period doubling, when one has to consider
perturbations with a symmetry different from the one for the given solutions. Hence,
in these cases, it is convenient not to fix a priori the symmetry of the solution and to
treat the complete equivariant problem. Then one will have a more general vision, but
probably less precise. This is the point of view adopted in this book.
Property 3.3 (Linearization). If f(yx) = yf(x) and f is C1 at χ o, with Γ\0 = Η,
then
Df(yxo)y = yDfix 0),
for all γ in Γ. In particular, Df(xo) is Η-equivariant.
Proof. Since f(yx0 + yx) - fiyxo) = 9(f(xo + x)~ f(xo)) = yDf(x0)x + ••·,
one has that / is linearizable at yxq and the above formula holds. •
This implies, if Β — E = RN, that Df(yxo) is conjugate to Df(xo) with the
same determinant.

1.3 Equivariant maps 11
On the other hand, if the dimension of the orbit of is positive, i.e., if dim Γ/Η =
k with Η = ΓΛ0, then one may choose a differentiate path y(t), with y(0) =
I, y'(0) Φ 0, such that f(y(t)xo) = f(xo). Differentiating with respect to t and
evaluating at t = 0, one has
Hence, γ'(0)χο is in the kernel of Df(xo), for each direction y'(0) such that
}/'(0)λ:ο Φ 0. Since the orbit is a differentiable manifold, this will be true for any
direction tangent to the orbit. Hence one has at least a i-dimensional kernel. For
example, if Γ is abelian and Tn acts, as in Example 1.4, by e\pi{NJ, Φ), then one
may take y(t) = (0,..., i, 0,... ) i.e., <pj = 0 except φι = t. In this case, y'(0);co is
i(n\x\,...,nfxm)T.
A property which will be used frequently in this book is the following:
Property 3.4 (Diagonal structure). If Β = BH ® Bj_, E = EH ® Ej_ with Β χ and
E± being Ν (H)-topological complements and f = fH 0 /j_, then at any χ η in BH
one has
where χ = xh®x _l and D u, D± standfor differentials with respect to these variables.
Proof One has that
From the fact that f±(xH) = 0, one has Du f±(xfí) — 0. Since the decomposition
of Β and E is /V(//)-invariant (hence //-invariant), the action of H on these spaces
is diagonal. The //-equivariance of Df{x¡j) implies that D±f±y = yDj_/j_, and
D±fH = DxfHγ for any γ in Η. Let A denote D±fH, then, since Ay — A,
one has that ker Λ is a closed //-invariant subspace of B±. Assume there is x± with
Ax± φ 0. Let V be the subspace of B± generated by x± and ker A. Defining ζ by the
relation yx± — x± + z(y), one has that ζ is in ker A and for any χ = ax± φ y in V
(i.e., y belongs to ker Λ) one gets yx = ax± + az,(y) + y, proving that V is also an
//-invariant subspace, with ker A as a one-codimensional subspace. This implies (see
any book on Functional Analysis) that there is a continuous projection Ρ from V onto
ker A. As a matter of fact, we shall prove below (in Lemma 4.4.) that one may take Ρ
to be equivariant. Then, if ij_ = (/ — P)x±, one has Ax± = Ax± (since Px± belongs
to ker A) and yχχ = (/ — P)yx±, from the equivariance of P, and yx± = k(y)x±
since (/ — P)V is one-dimensional. Applying A to this relation, one obtains k(y) = 1
and Jcx is fixed by H, i.e., x± belongs to BH Π B± — {0}, a contradiction. Hence
A = 0. •
Df(x0)y'(0)x0 = 0.
For the last property of this section, we shall assume that £ is a Γ-Hilbert space
and the action of Γ is via orthogonal operators, i.e., yTy = / (in finite dimensional

12 1 Preliminaries
spaces one may always redefine the scalar product in such a way that the representation
turns out to be orthogonal: see below, Lemma 5.1).
Property 3.5 (Gradients). If J : E —* M /s α C1, Γ-invariantfunctional, then f (χ) =
VJ(jc) is equivariant.
Proof. Since J(yx) = J(x), one has, from Property 3.3, that DJ(yx)y = DJ(x),
since the action y on Κ is trivial. But, DJ(x) = S/J(x)T, hence VJQc) = γ DJT (yx),
giving the result. •
Remark 3.1. If Γ has positive dimension and one takes a path y(t) with y (0) = /,
then, differentiating the identity J(y(t)x) = J(x), one obtains
V/OO · γ(0)χ = 0,
that is y(0)x is orthogonal to the field V7(x) = f{x). If one looks for critical points
of J, i.e., such that VJ(x) = 0, this orthogonality may be regarded as a reduction
in the number of "free" equations. From the analytical point of view, one may use
some analogue of the Implicit Function Theorem and reduce the number of variables.
Or, one may use, as in conditioned variational problems, a "Lagrange multiplier", i.e.,
one may add a new variable μ and look for zeros of the equation
f{x)+ßy{0)x =0.
In fact, if f(x) = 0, then μ = 0 gives a solution of the above equation. Con-
versely, if (μ, χ) is a solution, then by taking the scalar product with y(0)x, one has
μ\γ(0)χ\ζ = 0, hence/(χ) = 0 and ¿¿y (Ο)* = 0, in particular μ = 0 if γ(0)* φ 0.
This argument can be repeated for each subgroup y(t) and one obtains y¡(0) for
j = I,..., dim Γ. Considering the equation
f(x) + J2vYj( 0)*=0,
one obtains a problem with several parameters. A solution of this problem will give
that
(a) f(x) = 0 and (b) y, (0)* = 0.
One will conclude that μj —Oiiyj (0)x are linearly independent. This will depend on
the isotropy subgroup of x. This point of view will be taken when studying orthogonal
maps (see § 7).
1.4 Averaging
At this stage the reader may be puzzled why we insist on working with compact Lie
groups. As a matter of fact, up to now, the compactness of the Lie group Γ was not

1.4 Averaging 13
used in our considerations and seems to bear only a decorative aspect in the whole
business. Almost the same can be said about linear actions. Now, the consistency of
these two features namely, compactness of Γ and linearity of the actions, becomes
evident when you realize that, under these two conditions, a powerful instrument is
at hand. Precisely, the existence of an integration on Γ, the Haar integral, such that
Jr dy — 1, which is Γ-invariant on the class of continuous real-valued functions g on
Γ, under both left and right actions, i.e.,
f gir''1 Y)dy = f^g(r)dy = J g(Yy')dy.
The first important consequence of this fact is that, provided £ is a Banach Γ-space,
one may define a new norm, say
\\x\w = j \yx\ dy,
satisfying, |||y';t||| = |||x|||, i.e., the action of Γ is an isometry.
This allows us to assume in the rest of the book that the action is an isometry. In
particular, the ball
Br — {x : H* H < /?} is Γ-invariant.
Using Pettis integrals and standard averaging, one has the following remarkable
result.
Lemma 4.1 (Gleason's Lemma). If Β and E are Γ -spaces and f(x) is a continuous
map from Β into Ε, then
f(x) = J f (yx) dy is Γ-invariant
and
f(x) = J y-1 /(yx) dy is V-equivariant.
Furthermore, if f is compact, then so are f and f.
Proof From the change of variables yy', one has
fiy'x) = f f{yy'x)dy = j f{y"x)dy" = f(x).
Also, f(y'x) = fry-lf(yy'x)dy = y'fr(99Tlf(YY'x)dy = γ'fix),
under the same change of variables. See [Br. p. 36].
The continuity of / and / follows from the compactness of Γ. In fact, the orbit
Γχο is compact and hence / is uniformly continuous on it. Moreover, if χ is close

14 1 Preliminaries
to jcfj (therefore, the orbit Γ* is close to Γχο, taking into account that the action is an
isometry), one gets
f(x) - fixo) = J y~iyx) - fiyxo))dy.
Also,
II fix) - fix o> It < max Wfiyx) - fiyx0) II-
As far as compactness is concerned, recall that / is said to be compact if it is
continuous and if fiK) is compact, for any bounded set Κ in B.
Therefore, the sets A = (JrxA- f(Vx) and Â ξ |JpXÄ: Ϋ~ι fiyx) are precom-
1 Λ
pact. In fact, if you have a sequence [y~ fiynXn)} in A then, by the compactness of
Γ, you get a subsequence {ynj} converging to some γ and {f iynjXnj)}, converging to
some y. Thus,
Yñ/fiYnjXnj) ~ Y~ly = iyñ/ ~ V~l)f (YnjXnj) + Y~lifÍYnjXnj) ~ y)
yields the convergence, since || y^J1 — y-1 II tends to 0, as operators, and, since Γ A' is
bounded, A is compact and Â is bounded.
Now, cover A and A with balls of radius and extract a finite subcover
based at fiyjXj), j = ],..., k, and y¡~1 fiyixi), I = 1,. • •, r, respectively. Let {φ}}
be a partition of unity associated to the covering, i.e., ιpj : E —>• [0, 1], with support
in a ball centered at yj = fiyjxj), respectively yj~l fiyjXj), of radius 1 /2N and such
thatE<P/(y) = 1·
Define,
InÌX) = J^^2vjifiYx))f(YjXj)dy,
/λ/Μ = J ^φάγ~ιfÍYx))y¡~1 f(yix¡)dy.
Then, /ν (x) belongs to the space generated by {fiyjXj)}, while /ν(χ) belongs to
the finite dimensional space generated by {y¡~1 fiyixi)}· Hence, /nÍK) and /ν(Κ)
are precompact. Furthermore,
fix) - fN(x) = J Y,<Pjifiyx))ifiyx) - fiyjXj))dy,
fix) - fNix) = ΐ Σφι(γ~ιf(YxMY~lHyx) - Y[~lfiyixi))dy.
Now, since q)j (y) is non-zero only if ||y — yj || < 1 /2N and Σ <Pj 00 = 1 » one gets
ix) - fN(x)\ < l/2N and ||/(x) - /„«Il < I/2N.

1.4 Averaging 15
But then, for any bounded sequence {*„}, one has a subsequence {xn(N)} such that
Ιν(χπ(Ν)), respectively /ν(χπ{Ν)), is convergent. Using a Cantor diagonal process,
one obtains, due to the uniform approximation of fix) by //v (*), respectively of fix)
by fy(x), a convergent subsequence for f (XN(N)), respectively f (XN(N))· •
Remark 4.1. If fiyx) = fix), then f(x) = f(x), while, if fiyx) = γ fix), then
f(x) = f(x).
Example 4.1. If Γ = S1 acts on C^ (K) via time translation as in Example 1.5, and
f it, x) is continuous and 2π-periodic in t, then / induces a mapping from C^OR)
into itself, via fit, χ it)). Then
fixit)) = (1/2*) f2* fit, xit + <p))d<p = i 1/2π) f * f(t,x(<p)) dip,
Jo Jo
fixit)) = (1/2 π) ¡2π fit-φ, xit))d<p.
Jo
Example 4.2. If Γ = Zm is generated by yo, then
- m — 1
/ giY)dY = a/m)Ys(Yj)·
J Γ V
Remark 4.2. In the proof of the compactness of fix) and fix), we have seen that a
map / is compact if and only if it can be uniformly approximated on bounded sets by
finite dimensional maps. The reader may recover this important result by forgetting
the action of Γ. Now, for the case of a non-trivial action of Γ on Ε, a word of caution is
necessary: The map fyix) is invariant and belongs to a finite dimensional subspace.
However, /¿vM is not equivariant. One could have tried to use the set A also for this
case and define
ÂM = jrY,<PÁf(Yx))Y~Xf{YjXj)dY
which is Γ-equivariant and approximates, within \/2N on K, the map /(x), but which
is not necessarily finite dimensional, as the following example shows, since the orbit
of fiVjXj) may not span a finite dimensional space.
Example 4.3. On I2 — {(-to, χι, Χ2, • • • ), -*o e R, xj e C for j > 1 with Σ \xj\2 <
00}, consider the action of S1 given by
el9ix0,x\,x2, ...) = (*o, el<fix\, β2,φΧ2,...).
Consider the point x0 = (1, 1/2, 1/22,..., 1/2",... ) = (a0, a, a2, a3,... ).
Then, for any n, e,9lxo,..., el<t>»xo, for φ\,..., <pn different, are linearly indepen-
dent. In fact, taking the first η components, one obtains a Van der Monde matrix, with

16 1 Preliminaries
y'-th row equal to (1, aj, aj,... a"-1), where a7 = el<pi /2 and determinant equal to
Yli>j(ai ~~ aj)· Hence, the closure of the linear space generated by the orbit of xo
is h•
However, we will show in the next section that the set of points in E whose orbit
is contained in a finite dimensional Γ-invariant subspace is dense in E. Thus, in the
definition of f'N take yj such that Γ^ C Mj, a finite dimensional Γ-invariant subspace,
with IIyj - fiyjxj)|| < 1/2W, and define
/*(*) = ¡Σψ}(γχ)?-χν}Ίγ.
Thus, since γ~1 yj c Mj, the Γ-map has range in the finite dimensional Γ-invariant
subspace generated by the Mfs and (x) — f^(x)\ < l/2N~l.
We have thus proved the following result, which will be crucial for the extension
of the Γ-degree to the infinite dimensional setting.
Theorem 4.1. A continuous Γ-equivariant map f from Β into E is compact if and
only if for each bounded subset Κ of Β, there is a sequence of Γ-equivariant maps
fy, with range in a finite dimensional Γ-invariant subspace Μ Ν of E, such that, for
all χ in Κ, one has
(x)~ fN(x)\ < 1/2".
In our construction of the Γ-degree, we shall also need the following consequences
of averaging:
Lemma 4.2 (Invariant Uryson functions). If A and Β are closed Τ-invariant subsets
ofE, with ΑΠΒ = φ, then there is a continuous Τ-invariantfunction φ : E —> [0, 1],
with φ(χ) — 0 if χ e A and φ(χ) = 1 if χ e Β.
Proof Indeed, let φ be any Uryson function relative to A and Β (for instance
dist(jc, A)/(dist(;c, A) + dist(;c, Β))), then
φ(χ) = J φ(γχ)άγ
has the required properties. Note that, if one has renormed E in such a way that the
action is an isometry, then distQc, A) = dist(/jc, Λ) and ψ(χ) can be chosen to be the
above map. •
Lemma 4.3 (Invariant neighborhood). If A C E is a Γ-invariant closed set and U,
containing A, is an open, Γ-invariant set, then there is a Γ-invariant open subset V
such that A C V C V C U.
Proof In fact, let φ : E ^ [0, 1] be a Γ-invariant Uryson function with <p\A=0 and
φ\vc = 1. Then, V = <p_1([0, 1/2)) has the required properties. •

1.5 Irreducible representations 17
Lemma 4.4 (Equivariant projections). If EQ is a closed Γ-invariant subspace of E
and Ρ is a continuous projection from E onto EQ, then
Px = j Y'^Pyxdy
is α Γ-equivariant projection onto EQ. If EQ = ET, then
Px = J γχ άγ
is a Γ-invariantprojection onto ER. Moreover, E\ = (/ — P)E and (/ — P)E are
closed Τ-invariant complements of EQ and ER.
Proof. The first part is clear since fr άγ = 1 and EQ is Γ-invariant. As far as the
second part is concerned, notice that Ρ χ is in ER and Ρ χ = χ for je in ER. •
1.5 Irreducible representations
A good deal of this book is based on the decomposition of finite dimensional repre-
sentations into irreducible subrepresentations and the corresponding form of linear
equivariant maps.
Definition 5.1. Two representations of Β and E are equivalent if there is a continuous
linear invertible operator Τ from Β onto E such that γ Τ = Τ γ.
Lemma 5.1. Every finite dimensional representation is equivalent to an orthogonal
representation, i.e., with γ in 0(n).
Proof. In fact, the bilinear form
β(·*,>ο = fr(yx> yy)dY
is positive definite, symmetric and invariant. Hence, there is a positive definite matrix
A such that B(x, y) = (Ax, y). One may define a positive symmetric matrix Τ such
that Τ2 = A, by diagonalizing A. Hence B(x, y) = (Τ χ, Ty). Since Β(γχ, yy) =
B(x, y), one has that (ΤγΤ~ιχ, TyT~ly) = Β(γΤ~ιχ, yT~ly) = (x, y), which
implies that ΤγΤ~χ is in 0(n). •
Remark 5.1. The same result is true in any Hilbert space. The existence of the self-
adjoint bounded positive operator A follows from Riesz Lemma and that of Τ from
the spectral decomposition of A.
Definition 5.2. A representation £ of Γ is said to be irreducible if E has no proper
invariant subspace (not necessarily closed).

18 1 Preliminaries
This implies that ER = {0} unless Γ acts trivially on E and dim E = 1.
Definition 5.3. A subrepresentation EQ of Γ in E is a closed proper invariant subspace
EQ of E.
Lemma 5.2. If E isa finite dimensional representation of Γ, then there are irreducible
subrepresentations E\,..., Ek, such that Ε = E\ ® · · · ® E^.
Proof From Lemma 5.1 it is enough to consider the case where the representation
is orthogonal. Then, if E\ is Γ-invariant, the orthogonal complement E^ is also
Γ-invariant, since (γχ, y) = (χ, γτy) = (χ, γ~ìy)• Hence, if χ e Ej- and y is in
E ι (hence also y~ly e E), this scalar product is 0 and γχ is in Ε¡Κ Applying this
argument a finite number of times one obtains a complete reduction of E. •
The above arguments can be extended to the infinite dimensional setting in the
following form.
Lemma 5.3. (a) If EQ is an invariant subspace of the representation Ε, then EQ is a
subrepresentation. If furthermore E is a Hilbert space, then E = EQ φ E\, where E
is also a subrepresentation.
(b) If E is an orthogonal representation (hence E is Hilbert) and EQ is an invariant
subspace, then EQ is a subrepresentation.
Proof, (a) If {xn} in EQ converges to x, then {γχη}, which is in EQ, converges to γχ
and ËQ is invariant. The second part follows from Lemma 4.4, since there is always
a projection on ËQ.
(b) follows from the argument used in Lemma 5.2 and the fact that EQ is closed.
•
Lemma 5.4 (Schur's Lemma). If Β and E are irreducible representations of Γ and
there is a linear equivariant map A from Β into Ε, such that Α γ = γ A for all γ in
Γ, then either A = 0, or A is invertible.
Proof Note first that the statement is purely algebraic and no topology is used. Since
the domain of A is linear and Γ-invariant (so that the equivariance makes sense), one
has that the domain of A is all of B. Furthermore, since ker A is Γ-invariant, then
either it is Β (and A = 0) or it reduces to {0} and A is one-to-one. But then Range A
which is also Γ-invariant and non-trivial (since Αφ 0) must be E. Hence A is also
onto and invertible. •
Remark 5.2. If E is not irreducible, then either A = 0, or A is one-to-one and
onto Range A. This last subspace is (algebraically) irreducible since A-1 is clearly
equivariant.

1.5 Irreducible representations 19
Corollary 5.1. (a) If E is an irreducible representation of Γ and A is α Γ-equivariant
linear map from E into E, i.e., Ay = γ A with a real eigenvalue λ, then A — λ/.
(b) If E has no proper subrepresentations and A is a bounded Τ-equivariant linear
map with eigenvalue λ, then A = λΐ. Any bounded Τ-equivariant linear map Β is
either 0 or one-to-one.
(c) If furthermore E is a Hilbert space with no proper subrepresentations and
equivalent to an orthogonal representation of Γ (i.e., there is a continuous isomor-
phism Τ on E such that, if γ Ξ T~lyT, then γτγ = I), and A is a bounded
Γ-equivariant linear map from E into Ε, then
Τ~ίΑΤ = μΙ + νΒ
with Β2 = —I, Β + BT = 0. Moreover, Τ = I if the representation is orthogonal.
Proof, (a) In fact, A — XI is Γ-equivariant, with a non-trivial kernel, hence, from
Schur's Lemma, it must be 0.
(b) Since ker(A — λ/) is closed, the previous argument gives the result. Similarly,
if ker Β φ {0}, then Β = 0.
(c) One has T~l ATy — T~x ΑγΤ — yT~x AT, hence T~x AT is Γ-equivariant
with respect to the orthogonal representation. Let A = T~x AT, then A + ÄT and
AT A are self-adjoint and equivariant. Hence, 2μ — ±|| A + AT || is an eigenvalue for
Ä + ÄT. From (b), one has Ä + ÄT = 2μΙ or, else (Ä - μΐ) + (Ä - μΙ)τ = 0.
Furthermore, (A — μΙ)τ(Α — μΐ) = ν2/, since this operator is either positive, or
identically 0 if it has a kernel (again from (b)). If υ = 0, then (Â — μ,/)2 = 0 and
A — μΐ must have a non-trivial kernel, i.e., from (b), A — μΐ. On the other hand, if
ν φ 0, let Β = (i - μΙ)/ν. Then, Βτ + Β = 0 and ΒτΒ = I, i.e., Β2 = -I. •
Corollary 5.2. If E is a finite dimensional irreducible representation of an abelian
group Γ, then either E = IR and Γ acts trivially or as Ία, or E = C and Γ acts as in
(1.4).
Proof. Since Γ is abelian, one has that γγ\ = y¡y, where γ is the equivalent or-
thogonal representation given in the preceding corollary. Furthermore y, a matrix, is
Γ-equivariant, hence
γ =μΙ + vB,
where^, υ, ßdependony. Sinceyry = / one has μ2 + υ2 = 1. If yi = μ\Ι + ν\Β
and Y2 = μι I + V2B2, from γ\γι = γιγ\, one obtains, if i>i vi φ 0, that B\B2 =
B2B\ Ξ Β. But then, BT = Β and Β2 = I. From Schur's Lemma, the self-adjoint
matrix Β must be of the form λ/, with λ2 = 1. If λ = 1, then B1B2 = I implies (by
multiplying with B) that Bj — —B\ and then one may change V2 to — \>2. While, if
λ = — 1, then one obtains B2 — B\. That is, one has a unique Β such that any γ is
written as μΐ + vB.

20 1 Preliminaries
Now, if for all γ 's the corresponding ν is 0, then γ — ±1 (since μ2 = 1) and any
one-dimensional subspace is invariant. Then Ε = M and Γ acts trivially if μ = 1 for
all γ, or Γ acts as Z2 if, for some y, μ is — 1.
On the other hand, if there is a non-zero v, then from B2 = —/, one has (det B)2 =
(-l)dim£ and hence E is even-dimensional. Furthermore, if e φ 0, then the subspace
generated by e and Be is Γ-invariant and of dimension 2, since Be is orthogonal to
e : (e, Be) = (BTe, e) = —{Be, e). Thus, from the irreducibility of E, one has that E
is equal to this subspace. Take e of length 1 and define a complex structure by defining
Be — i. Then, γ = μ + vi, with μ2 + ν2 = 1, is a unit complex number. Remark
1.1 and the fact that any compact abelian group can be represented as a product, ends
the proof. •
Remark 5.3. Another way of seeing the above argument is the following: γ, as an
orthogonal real matrix, has two-dimensional invariant eigenspaces, where γ acts as a
rotation. Since y¡ commutes with γ, these invariant subspaces are also invariant for
Y\. Hence, the action of Γ on this subspace can be written as Rv (*), where Rv is a
rotation by an angle φ. Writing ζ = x + iy, this vector can be identified with el(pz.
Clearly, we could have taken ζ = χ — iy. Then this action would have been
e~l<pz· These two representations are equivalent as real representations, since the map
rÇ) = corresponding to conjugation, is equivariant. Of course, they are not
equivalent as complex representations.
The next set of results in this section will concern the fact that any irreducible rep-
resentation (in the sense of our definition) of a compact Lie group is finite dimensional.
We shall begin with the Hilbert space case.
Theorem 5.1. If E is an orthogonal irreducible representation ο/Γ, with no proper
subrepresentations, then E is finite dimensional. Furthermore, one has the equality
j ((γχι, yiKyx2, yi) + (yxi,y2)(vx2, y))dy = 2{x\,x2)(y\,yi)/àim E,
forallx\,X2,y\,y2.
Proof. The left hand side of the above equality is a continuous linear functional on Ε,
as a function of x\ alone. Hence, from Riesz Lemma, it has the form (x\, z) for some
ζ which depends upon )>i, X2, y2· For fixed yi, y2, the vector ζ depends linearly and
continuously on Therefore one may write ζ = A-xi, where the operator A depends
on >»i and ^2· From the invariance of the Haar integral, one has that
(γχ\,Αγχ2) = (xi,Ax2),
hence γτ Αγ = A and A is equivariant. Furthermore, by interchanging x\ with x2,
one has that A = AT. Thus, from Corollary 5.1 (b), one has that A = λ/, where, of
course, λ depends on ji and y2 but the left hand side is λ(*ι, x2).

1.5 Irreducible representations 21
By using the same argument with y ι and y>2, one has that the left hand side is
μ(γΐ, >'2), hence it is of the form c(x\, X2){y\, yi), where c is independent of x\, X2,
yi, yi. Taking = χι, y\ = yi, the left hand side is /Γ 2(yx\, yi)2dy and c is
positive.
Take now, ej, ei,e ν an arbitrary collection of orthonormal vectors in E. Then,
from Parseval's inequality, one has
Ν
Y^yx,ejf< \γχ\2<\χ\2.
ι
Taking χι — x^ = χ and y\ = y2 = e¡, and integrating the above equality, one obtains
N f
2Σ eifdy = Arelix»2 < 2||*||2.
Hence, c < 2/N. From this it follows that E is finite dimensional. Furthermore, if
dim Ε — Ν, one gets an equality, and one obtains c = 2/N. •
Corollary 5.3. If E is a Y-Banach space with no proper subrepresentations, then E
is finite dimensional.
Proof. For a general Banach space E, take X a non-zero element of E*, i.e., a
continuous linear functional on E. Consider
(x,y)x = f^X(Yx)X(Yy)dY.
Then, (x, y)x is bilinear, continuous in χ and y and (χ, χ)χ > 0. Hence, E is given
the structure of a pre-Hilbert space: define the equivalence relation χ ^ y if and only
if {x — y, χ — y)x = 0, i.e., iff X(y(x — y)) = 0 for all γ in Γ. Taking the set
of equivalence classes and completing with respect to the || ||χ-ηοπη, one obtains
a Hilbert space Η χ and a natural mapping ψχ from E into Ηχ. Define an action
y of Γ on Ηχ by factorization and extension by continuity of the action of Γ on
E. Since, (yx, yy)x = (x, y)x, one has that Ηχ is an orthogonal representation
of Γ. Furthermore, ψχγ = γφχ, by construction, and ψχ is a linear mapping, with
\φ χ Ο) \2X — fr X(yx)2dy < ||X||2||x||2, i.e., ψχ is continuous (||X|| is the norm of
X in £*).
Now, since E has no proper subrepresentations, one has, from Schur's Lemma, that
ψχ is one-to-one (since Χ φ 0, at least for some χ one has ψχ(χ) Φ 0). Now, if Ηχ
contains a proper subrepresentation M, we may assume that M is finite dimensional
(the precise argument will be given in the next corollary). Let Ρ be an equivariant
orthogonal projection from Ηχ onto M (see Lemma 4.4.). Then, Ρψχ is a continuous
linear map from E into M. From Corollary 5.1 (b), Ρψχ is either one-to-one, or

22 1 Preliminaries
identically 0. In the first case, this implies that E is finite dimensional. In the second
case, <ρχ(Ε) c M1, which contradicts the fact that ψχ(Ε) is dense in Ηχ. •
Note that, if E = (IR) and X(x(t)) = jc(0), then, under the time shift, one has
II*III = SF Ιθ2π x2(<P)d<P and Ηχ is L2[0, 2π],
Corollary 5.4. (a) Any infinite dimensional Banach Y-space E contains finite dimen-
sional irreducible representations.
(b) The set of points whose orbits are contained in a finite dimensional invariant
subspace is dense in E.
Proof, (a) If E has all its subrepresentations of infinite dimension, take a sequence
Μ ι D A/2 D · • • of subrepresentations and let M^ = p| Mn. Then, M^ is a closed
linear invariant subspace of E. By ordering such sequences by inclusion, one should
have, by Zorn's Lemma, a maximal element. For this element, the corresponding Mœ
is an infinite dimensional subrepresentation. If £ is a Hilbert space (with orthogonal
action), the above conclusion contradicts the maximality, since either Mm has a proper
subrepresentation M' and then {Mn Π M'} is strictly "larger" than {Mn}, or, Mm is
finite dimensional. This implies that the argument in Corollary 5.3 is complete and
one may repeat it for a general Banach space.
(b) Take a finite dimensional subrepresentation M\ of E and N\ an invariant closed
complement (which exists, by Lemma 4.4). Since N\ is an infinite dimensional rep-
resentation, it contains a finite dimensional representation M2 (of course, if E is finite
dimensional, there is nothing to prove). Let N2 be an invariant closed complement of
M2 in Ν ι. Continuing this process, one obtains a sequence Mn of finite dimensional
invariant subspaces and complements Nn such that Mn+\ φ Nn+\ = Nn. Moreover,
there are equivariant projections Pn from E onto φ" Mj such that / — Pn projects
onto Nn. Let Ν = f]Nn. Then, it is easy to see that Ν is a closed, linear and
invariant subspace of E. Ordering sequences of such {7V„} by inclusion, construct
the corresponding Ν for a maximal sequence. Then, if /V / {0}, Ν contains a finite
dimensional subrepresentation M and its corresponding complement Ν (take Μ — Ν
in case Ν is finite dimensional). But then [Nn Π Ν) is strictly "larger" than {Nn},
contradicting the maximality. Hence, Ν = {0} and, for any χ in E, one has that
(/ — P„)x goes to 0, i.e., Pnx, which belongs to 0" Mj, approximates x. Note that,
for a Hilbert space, one may take the space Eo of all points whose orbits lie in a finite
dimensional invariant subspace. Clearly, Eo is an invariant linear subspace and £0
is a closed invariant subrepresentation. If ËQ is a proper subrepresentation, then EQ
contains a finite dimensional subrepresentation N, which is a contradiction, since Ñ
should be in Eo. Hence, Eo is E. Here the maximal Ν is £Q-, the intersection of all
the orthogonal complements of finite dimensional invariant subspaces. •
Remark 5.4. In a finite dimensional irreducible representation, the set of finite linear
combinations of points on a given orbit is dense: if not, the closure of the linear space

1.5 Irreducible representations 23
generated by such combinations would be a proper subrepresentation.
Our last set of results of this section concerns the form of a linear equivariant map
between two finite dimensional representations V and W.
Let V = V] ®· · -®Vq and W = W^©· · φ W¡ be a decomposition of V and W into
irreducible subspaces. Let P¡ : V V, and Qj : W Wj be equivariant projections,
i.e., γ Pi — P¡ γ and γ Qj = Qjy. Assume that there is a linear map A : V —>• W, such
that Ay = γ A. Let AI; = QjAP¡ : V, Wj. Then, A^y — γ Ai} and, from Schur's
Lemma, either Αη — 0 or A¡j is an isomorphism, in which case dim V, = dim Wj
and Vi and Wj are equivalent representations. Hence, if one considers all possible
A's, it follows that one has to look only at the subrepresentations of V which are
equivalent to those of W. Furthermore, since an equivalent representation amounts to
a choice of bases (in V and W) and since ker A as well as Range A are also representa-
tions, with complements which are representations, the problem can be reduced to the
study of A from V into itself, with γ A = Ay and A,y = 0 if V, and Wj are not
equivalent.
As in Corollary 5.1, one may assume that γ is in 0(V) (again a choice of basis).
Then Ajj = μί} I + VijB¡j, with £?· = -/ and BtJ + Bf. = 0.
Theorem 5.2. Let V be a finite dimensional irreducible orthogonal representation.
Then exactly one of the following situations occurs.
(a) Any equivariant linear map A is of the form A = μΐ, i.e., V is an absolutely
irreducible representation.
(b) There is only one equivariant map B, such that Β2 = —I, BT + Β = 0. Then,
any equivariant linear map A has the form A = μΐ + vB. In this case, V has
a complex structure for which A = (μ + iv )I.
(c) There are precisely three equivariant maps B\, B2, Bj withthe above properties.
Then, B¡Bj = — B¡ B¿ and Βτ, = Β] Β^· In this case, V has a quaternionic
structure and any equivariant linear map A can be written as A = μ I +
viZ?i + V2B2 + V3Bj = ql,where q = μ + νμ'ι + V2¿2 + V3Í3 « in H.
Proof. If Γ is abelian, this result was proved in Corollary 5.2, where only (a) and (b)
occur. Since the abelian case is the main topic of our book, we shall not give the proof
of Theorem 5.2 here. However, an elementary proof is not easy to find. Thus, we give
a proof in Appendix A. •
In the same vein, one has the following result (with an easy proof in the abelian
case) which will be proved in Appendix A.
Theorem 5.3. Let V be decomposed as

24 1 Preliminaries
where V® are the absolutely irreducible representations of real dimension m, repeated
η i times, V^ are complex irreducible representations of complex dimension mj re-
peated nj times, while V® are quaternionic representations of dimension (over H) m¡
and repeated n¡ times. Then, there are bases of V such that any equivariant matrix
has a block diagonal form
A =
(A*
A®
Au
2
Af
Af
where A® are real n¡ χ n¡ matrices repeated m¡ times, A^ are complex nj χ nj
matrices, repeated mj times and Aj® are n¡ χ n¡ quaternionic matrices repeated m¡
times.
On the new basis, the equivariance of A and the action have the following form: γ
is block diagonal on each subspace corresponding to the repetition of the same matrix,
i.e., ifBnxn is repeated m times, on W corresponding to the same representation, then
γ = (Yijl)i<ij<m, with Yij in Κ = R, C or H, and / the identity on IK", where the
product, for the quaternionic case, is on the right.
Remark 5.5. If Γ is abelian, the irreducible representations of Γ are either one-
dimensional and Γ acts trivially or as Z2, or two-dimensional and Γ acts as Z„, η > 3
or S1. Of course, in this case there are no quaternionic components.
Note also that the equivariance of A and the action of Γ on the new basis will
be important when considering Γ-equivariant deformations of A: any deformation of
A®, A J" or A}®, in the corresponding field, will give rise, by repeating the deformation
on the m replicae, to a Γ-deformation of A. This will be the situation when computing
the Γ-index of 0, when A is invertible, or when studying the Γ-bifurcation with several
parameters, as in [/].

1.6 Extensions of Γ-maps 25
1.6 Extensions of Γ-maps
Many of our constructions are based upon extensions of equivariant maps, in particular
when possible, by non-zero maps. As a matter of fact, the equivariant degree will
consist of obstructions to such non-zero equivariant extensions. Thus, the key to
our computations of homotopy groups will be a step by step extension of Γ-maps,
subtracting "topologically" multiples of generators along the way, in order to get a
formula for the class of each map.
Our first result is a simple extension of Dugundji's theorem.
Theorem 6.1 (Dugundji-Gleason extensions). Let A\ C A 2 be Τ-invariant closed
subsets of Β. If f : A\ —> E is α Γ-equivariant continuous map, then there is a
Γ-equivariant continuous extension f : A2 —»• E. Furthermore, f is compact if so
is f.
Proof. From Dugundji's theorem, / has a continuous extension / from A2 into E
which is compact if / is compact. From Lemma 4.1, the map
f(x) = J 9~lf(yx)dY
is Γ-equivariant (and compact if / is compact). Furthermore, if χ is in A\, then
Πγχ) = fiyx) = Y fix) and fix) = fix). •
In case Β and E are infinite dimensional, we shall look at maps with the following
compactness property.
Definition 6.1. If Β = U χ W and E = V χ W, where U, V are finite dimensional
representations of Γ and W is an infinite dimensional representation, an equivari-
ant map /, from a closed Γ-invariant subset A of Β into E is called a Τ-compact
perturbation of the identity if / has the form
fiu, w) = igiu, w), w - hiu, w)),
where g is Γ-equivariant from Β into V and h in W is compact and Γ-equivariant.
Definition 6.2. If /o and f\ are Γ-maps from a closed invariant subset A of Β into
£\{0} (Γ-compact perturbations of the identity if Β and E are infinite dimensional),
then /O is said to be T-homotopic to f\, if there is fit, JC), Γ-equivariant, from I χ A
into £Λ{0} (and a Γ-compact perturbation of the identity), where I — [0, 1], with
fi0,x) = Mx) and fil, x) = Mx).
One then has the following crucial result:
Theorem 6.2 (Equivariant Borsuk homotopy extension theorem). Let A\ c A2 be
Γ-invariant closed subsets of B. Assume that fo and f\,from A1 into £\{0}, are

26 1 Preliminaries
Γ-equivariant maps which are T-homotopic. Then fo extends Γ-equivariantly to A2
without zeros if and only if f\ does. If this is the case, then the extensions are Γ-
homotopic. Similarly, if fo, f\ and the Γ-homotopy are Γ-compact perturbations of
the identity, then the extensions and the homotopy must be taken Τ-compact pertur-
bations of the identity.
Proof. Let fQ : A2 £\{0} be the Γ-extension of fo and f(t, χ) : I χ A\ £\{0}
be the Γ-homotopy from fo to f\. Let, by Dugundji-Gleason Theorem 6.1, g(t, x)
be any Γ-equivariant extension to / χ of the map defined as /(?, x) on / χ A] and
fo{x) on {0} χ A2.
It is easy to see that, in the infinite dimensional case, one preserves the compactness
of the perturbations.
Let A be the subset of A 2 consisting of all χ for which there is a t with g(t, x) — 0.
Then, by construction, Α Π A\ = φ. Furthermore, from the compactness of [0, 1],
if {jc,j} is in A, converging to χ in A2, then g(tn,xn) = 0, {/„} has a subsequence con-
verging to some t and g(t, x) = 0. Thus, A is closed. Furthermore, the equivariance
of g, with respect to x, implies that A is invariant.
From Lemma 4.2, there is an invariant Uryson function ^ : A2 -»• [0, 1] such that
<p(A) = 0and<p(A]) = 1.
Define f(t, χ) = g(<p(x)t, x). Then the Γ-equivariance of / follows from that
of g (and of the invariance of φ), as well as the compactness property. Furthermore,
/(0, x) — g(0, χ) = fo(x). Finally, if f(t, x) = 0 for some t, then χ belongs to
Α, φ(χ) = 0, but g(0, x) = fo(x) φ 0. The map f(t,x) gives a Γ-homotopy on
A2, from fo(x) to fi(x) = g(<p(x), x), which provides an extension of f\, since, on
Α\,ψ{χ) = 1. •
Another useful fact is the following observation:
Lemma 6.1. Let S" be the unit sphere in the Γ -space V = and f : 5"
W\{0} (another finite dimensional representation) a Τ-map. Then any Τ-equivariant
extension f of f to the unit ball has a zero if and only if f is not Γ-deformable to a
non-zero constant map.
Proof. Note first that a non-zero constant equivariant map may exist only if Wr φ {0}.
In other words, if Wr is reduced to 0, any equivariant extension / must have /(0) = 0
(see also Property 3.2).
Now, if / is such an extension, define the Γ-homotopy / : / x S" W\{0}, by
f(t, x) = /(( 1 — t)x), deforming radially and equivariantly /(0, x) = f (x) to /(0).
On the other hand, if f(t,x) Γ-deforms fix), for t = 1, to the constant /(0, x),
define f(x) = /(||x||, x/||x||) which will provide the appropriate Γ-extension of /.
•
One of the key tools which will be used in our computations of equivariant homo-
topy groups of spheres is the existence of complementing maps, which will play the

1.6 Extensions of Γ-maps 27
role of a suspension (defined in Section 8). In order to be more specific, let us assume
that U and W are finite dimensional orthogonal representations of an abelian compact
Lie group Γ, with action given as in Example 1.4. Suppose that an equivariant map
is given from U into WH, for some subgroup Η of Γ. The problem is then the
following: is it possible to give a "complementing" Γ-equivariant map from (UH)1
into (W0)1 which is zero only at zero? Recall that, since Γ is abelian and the action
is orthogonal, all the above subspaces are representations of Γ. The answer to the
question is in general negative, as the following example shows.
Example 6.1. On C2, consider the following action of ^p2q, where ρ and q are rela-
tively prime: On {z\, z2) in U, Γ acts via (e2jrik/P2, ¿>2*<*/(Ρ<?)) for k = 0,..., p2q-1.
On ξ2) in W, Γ acts as (e2nik/P, ^'¿/(Λ)). The isotropy subgroups for the ac-
tion of Γ on U are as follows:
H = Zg, for k a multiple of p2 and UH = {(Zl, 0)},
Κ = Zp, for k a multiple of pq and UK = {(0, Z2)},
L = {e}, for k = 0 and U[e] = U.
One has WH = WK = {(|i, 0)}, but there is no non-zero equivariant map between
(•UH)1 and (W*7)1, since {UH)L C\UK = UK and (W")1 Π WK = {0}. On the
other hand, if aq + βρ = 1, the map
f(Zl,Z2) = (zf + zq2,z"z2)
(where a negative power is interpreted as a conjugate: z_1 = z), is an equivariant map
from U into W with only one zero at the origin.
One of our main hypotheses in Chapter 3 will be the following:
For any pair of isotropy subgroups Η and Κ for U, one has
(H) dim UH Π UK — dim WH Π WK.
Note that in Example 6.1, hypothesis (H) fails, although there dim UH = dim WH,
for all isotropy subgroups of Γ on U.
Lemma 6.2. Hypothesis (H) holds if and only if both (a) and (b) hold:
(a) dim UH = dim WH, for all isotropy subgroups H on U.
(b) There are integers l\,... ,ls such that the map F : (x ..., —>• (x[l,..., x[s )
is Γ-equivariant. Here χ¡ is a (real or complex) coordinate of Ό on which Γ
acts as in Example 1.4, and a negative power means a conjugate. Furthermore,
for all γ in Γ one has det γ det γ > 0.

28 1 Preliminaries
Proof. Let Hj be the isotropy subgroup of χj and HQ = fi Hj. Then UH° = U
and any isotropy subgroup Η = TX = Ç\Hj, where the intersection is on the j 's for
which the coordinate x¡ of χ is non-zero (see § 2), is such that Ho < Η.
Hence, if (H) holds, one obtains (a), since WH C WH°. Note that any equivariant
map from U into W will have its image in WH°. For notational purposes, define, for
Κ > Η, (UK)±H as UHn(UK)i-. Then, hypothesis (H) implies that dim(UK)±H =
dim (WK)±H.
Now, if Γ/Hj = Z2 and γ acts as -I on (Uv)X"J , then on (Wv)Í!'i, γ must
also act as —/, since if not one would violate the equality of the dimensions. Since
the action on a complex coordinate is a multiplication by a unit complex number, i.e.,
corresponding to a rotation with determinant equal to 1, then det γ and det γ (restricted
to WH°) have the same sign.
We may now begin to build up the map F. We shall identify UT and Wr and take
lj = 1 for these components.
Let H be maximal among the Hj's. Then, from Lemma 1.1, Γ/ H = Zn, n > 2
or S1 and acts freely on (i/r)-L/,\{0} and without fixed points on (Wr)Xw\{0}, as it
follows from Lemma 1.2, since no point in the second set, fixed by Η, may be fixed
by Γ /Η without being in Wr.
Thus, if γ generates Z„, one has yxj = e27Ttmj/nXj with 1 < rrij < n, rrij and
η relatively prime and yfy — e2jnn^n^j, with 1 < rij < n. Now, there is a unique
Pj, 1 < pj < n, such that pjmj = 1, [«]. Let lj be the residue class, modulo n, of
Pjtij. Then, (yxj)1' = γχ1· • Note that, if« = 2, then n¡ = m7 = 1 and lj = 1. That
is, on the real representations of Γ, where it acts as Z2, the map F is the identity.
On the other hand, ιΐΓ/Η = S1, acting as βιφ (or e'1*) on (Ur)±H and as ein>f
on (ννΓ)Χ/ί, then lj = rij (or — rij) will give the equivariant map (with negative lj
meaning conjugates).
Let now Κ and L be isotropy subgroups for (UH)L. Let H\ be the isotropy
subgroup for UK Π UL, i.e., Η ι is the intersection of the isotropy subgroups for all
the coordinates in that subspace. Then, UK Π UL C UH]. Since Κ and L are also
intersections of the corresponding subgroups, it is clear that Κ and L are subgroups
of Hi and then UH{ c UK Π UL, that is UHx = UK η UL, while WHi c WKnWL.
But, from (H), one has dim UH> = dim WHi and dim UK Π UL = dim WK Π WL,
then WHi = WKnWL. Since dim (UH)J-nUK <1UL = dim UK Π UL -dim UH Π
UK Π UL, one obtains that the hypothesis (H) is valid on (UH)L and (WH)L. Then,
one may repeat the above argument by choosing a maximal isotropy subgroup among
the remaining Hj's, proving the implication in a finite number of steps.
Conversely, if the map F exists, it is clear that dim UH < dim WH (and it is easy
to give examples with a strict inequality). While, if (a) and (b) hold, it is easy to see,
by direct inspection, that (H) is true. •
In order to construct the generators of the equivariant homotopy groups, in Chap-
ter 3, we shall need some invariant monomials. We shall again assume that the abelian
group Γ acts on U, with coordinates {x\,..., jc^}, with Hj = TXj. Let HQ be a

1.7 Orthogonal maps 29
subgroup of Γ and define, as in § 2, Hj = Hq Π HI Π · · · Π Hj. Let kj = \Hj-i/Hj\.
Lemma 6.3. There are integers a\,... ,as such that x"J ... x"s is Η-invariant.
j J
{If a is negative, xa means i'"'). If ks < oo, then one may take as = ks, while if
ks = oo, then as = 0. Furthermore, ifkj = 1 for j < s, then one may take a¡ = 0.
Proof. The proof will be by induction on j. \i j = s and ks = oo, any constant is
Γ-invariant, hence as — 0 will do. While, if ks is finite, then Hs-\/Hs acts freely on
xs (as in § 2) and any γ in Hs-\ can be written as γ = ßfS, for some S in Hs and a fixed
ßs such that ßsxs = el7lilk*xs. Hence, (yxs)ks = ß"ksxks5 = xkss is Hs-\-invariant.
Assume now that P(xJ+\,... ,xs) = ... x"s is H} -invariant, for some j > 1.
Then, iiHj-i/Hj = 51, this group acts freely on and as e'"'^ on x/, for/ = j,...,s,
with nj = 1. Since · · ·, ein*vxs) = e^ni°" P(xj+U ..., xs), one may
choose oij = —Σnlal and xj' • · • xs5 will be Hj-\-invariant. On the other hand, if
kj is finite, then any γ in Hj~\ is written as y = ßJS, with ßj generating Hj-\/H¡
and acting as β2πί^ι on x¡, 0 < a < kj and S in Hj. Then,
P(yxj+i,..., yxs) = ßjai+i (8xj+... ßjas(8xsr.
Now, as before, ßj = ßfkkη^, where ßk generates Hj-\/{Hk Π Hj-i), ßkXk =
e2x'/"kXkt where is the order of this group if finite (or ß^Xk = el<eXk if the group
is isomorphic to S1 and ßEkk means e2jTl£k^nk for some since ßjJ is in Hj, the
Hj -invariance of Ρ implies that the corresponding φ is a rational multiple of 2π); one
has 0 < < and ηϊ is in Hk Π Hj-\.
Thus, ßjak (Sx)ak =e2niaakEklnk{8xk)ak. Hence,
P(Yxj+l,..., Yxs) = e2niBaP{8xj+x,..., &xs) = e2niea P(xj+U .. .,xs),
with ε = ELy+i akSk/nk.
k- ~
Now, if γ = β^, i.e., a = kj, then this γ belongs to Hj-1 Π H¡ = Hj and
the corresponding skj must be an integer. Let eo be the non-integer part of ε and
define etj = —kjS o (it is an integer and aj = 0 if kj = 1). Then, if P(xj,... ,xs) =
Xj ... xs , one has
P(YXj,..., Yxs) = (ßjXj)aJe27li£aP(Xj+i, ...,xs)
= e27Tlaa^k>e27TlEaP(Xj, ...,xs) = P(xj,..., xs) •
1.7 Orthogonal maps
In the last chapters of the book, we shall be interested in a particular class of maps,
which we shall call orthogonal maps. The setting is the following: let Γ be a compact

30 1 Preliminaries
abelian group acting on the finite dimensional orthogonal representation V. Thus,
if Γ = Τ" χ Zm, χ · · · χ Zmj, with the torus Γ" generated by (φι,..., <pn), (pj in
[0, 2π], we shall define by
the infinitesimal generator corresponding to φ^.
Hence, if the action of <pj on the coordinate x¡ is as e'nJVj, then
Ajx = {in] χι,..., in™xm)T,
where inx stands for (—η Im χ, η Re Λ:)7.
Lemma 7.1. Let Η = Γχο. Then:
(a) There are exactly k linearly independent AJXQ if and only i/dim Γ/Η — k.
(b) In this case, if H_ is the torus part of Η and Η corresponds to the first k (non-
zero) coordinates of XQ, then for any χ in V— one has AjX = Y^¡= ¡ λ'A/x and
A\x,..., AkX are linearly independent whenever χι,... ,Xk are non-zero.
Proof (a) Since Η = (~]Hj, for the non-zero coordinates of XQ, one has from
Lemma 2.4 (a), that dim Γ/Η = k if and only if AH has rank k, where AH is the
matrix formed by n1·.
(b) follows from Remark 2.1 and the definition of λ', as given in Lemmas 2.4 (b)
and 2.6. Note that one may reparametrize T" by choosing Ψ7 = q>j + λ^ φι,
for j = 1,..., k and taking Ψ^+ι,..., Ψ„ acting trivially on V—. In this case, if Aj
is the diagonal matrix corresponding to the action of Η, that is, to the derivative with
respect to Ψ7, for j — k + I,..., η (since H_ corresponds to Ψ; = 0, j = 1,..., k),
then Aj is 0 on V— and, on any irreducible representation of Η in (V—)x, one of the
Aj, j = k + 1,..., n, will be invertible. •
Definition 7.1. A Γ-equivariant map /, from V into itself, is said to be Γ-orthogonal
if f(x) • AjX — 0, for all j = I,... ,n and all χ in the domain of definition of /. Here
the dot stands for the real scalar product. In terms of complex scalar product one has
Re(/(x) · Ajx) = 0.
Example 7.1 (Gradient maps). If f(x) = S7J(x), where J(yx) = J(x) is an invari-
ant function, we have seen in Remark 3.1, that f (γχ) = y/(JC) and that f(x)-AjX = 0,
i.e., that the gradient of an invariant function is an orthogonal map.
Linearizations of orthogonal maps have quite interesting properties. In fact:
Lemma 7.2. Assume that the Γ-orthogonal map f is Cl at XQ, with a k-dimensional
orbit. Let Η = Γ*0 and denote by D the matrix Df{XQ). Then:

Random documents with unrelated
content Scribd suggests to you:

The Project Gutenberg eBook of
Inducements to the Colored People of the
United States to Emigrate to British Guiana

This ebook is for the use of anyone anywhere in the United
States and most other parts of the world at no cost and with
almost no restrictions whatsoever. You may copy it, give it away
or re-use it under the terms of the Project Gutenberg License
included with this ebook or online at www.gutenberg.org. If you
are not located in the United States, you will have to check the
laws of the country where you are located before using this
eBook.
Title: Inducements to the Colored People of the United States to
Emigrate to British Guiana
Author: Richard Hildreth
Edward Carbery
Release date: January 21, 2019 [eBook #58749]
Language: English
Credits: Produced by hekula03, Martin Pettit and the Online
Distributed Proofreading Team at http://www.pgdp.net
(This
file was produced from images generously made
available
by The Internet Archive)
*** START OF THE PROJECT GUTENBERG EBOOK INDUCEMENTS
TO THE COLORED PEOPLE OF THE UNITED STATES TO EMIGRATE
TO BRITISH GUIANA ***

INDUCEMENTS
TO THE
COLORED PEOPLE OF THE UNITED STATES
TO EMIGRATE TO BRITISH GUIANA,
Compiled from Statements and Documents furnished by Mr.
Edward Carbery,
Agent of the "Immigration Society of British Guiana," and
a Proprietor in that Colony.
BY A FRIEND TO THE COLORED PEOPLE.
BOSTON:
PRINTED FOR DISTRIBUTION.
KIDDER AND WRIGHT, CONGRESS STREET.
1840.

INDUCEMENTS.

I. SITUATION, EXTENT,
GEOGRAPHICAL FEATURES,
CLIMATE, SOIL AND PRODUCTIONS
OF BRITISH GUIANA.
Guiana is a vast tract of territory situated on the north-east coast of
South America, between the mouths of those celebrated rivers, the
Oronoco and the Amazons.
British Guiana includes a portion of this coast, extending some two
hundred miles from east to west, bounded on the east by the river
Corentyn which separates it from Dutch Guiana, or Surinam, and on
the west by the Morocco creek, or the tract of country adjacent to it,
belonging to the republic of Venezuela. British Guiana extends inland
from the coast some two hundred miles, in a southerly direction, to
a chain of high mountains, by which it is bounded on the south, and
which separates it from Brazil. It thus includes an area of upwards of
forty thousand square miles, being about equal in extent to the
State of New York.
The whole country slopes gradually down from the mountains to the
sea. The back country is hilly and much diversified in surface; the
land along the sea-coast is flat, level, and extremely fertile. The
colony is watered by three large rivers, the Essequebo, the
Demarara, and the Berbice. These rivers descend from the
mountains, and run parallel to each other at nearly equal distances.
They are navigable for many miles, and together with numerous
smaller rivers and creeks, they not only afford great facilities for
internal navigation, but also for irrigating the land, a thing of great
importance in that climate.

British Guiana never suffers from those violent storms and
hurricanes with which other tropical regions are visited. Along the
whole coast, vessels can ride at anchor in perfect safety, at all
seasons of the year. The whole shore is a bed of deep soft mud, and
can be approached by vessels without danger.
The latitude of the coast, along which the settlements are situated,
is about seven degrees, north. The longitude of Georgetown, the
capital, is about fifty-seven degrees west from Greenwich. Its
direction from the city of New York is considerably east of south. The
distance is about two thousand miles, or twenty days' sail, very
nearly the same distance as New Orleans.
Situated under the tropic, Guiana enjoys a perpetual summer. The
thermometer generally ranges from 78° to 84°. The trade winds,
which blow constantly from the coast, render the climate
comfortable and salubrious. The year is divided into four seasons,
two rainy and two dry. The short rainy season usually commences
about December, and lasts four weeks: the long rainy season begins
in June, and lasts till the middle of August. But as regards these
seasons there is a good deal of variation. In the rainy season, the
rain falls violently during the forenoon, but the afternoons are clear
and pleasant. During the dry season occasional showers occur.
The only portion of this fertile country which has yet been settled
and cultivated, is a narrow strip extending along the coast, and a
little distance up the mouths of the principal rivers, together with
some islands at the entrance of the Essequebo. The plantations are
generally about half a mile wide, fronting on the sea, and extending
back two, three, four or five miles. This series of adjoining
plantations forms the only cultivated part of the country, which thus
resembles a long string of villages half a mile apart.
The soil of the plantations, which is very deep and rich, is divided by
canals into separate fields. The same fields are cultivated in constant
succession, and no manure is ever used. The canals not only serve
to drain and irrigate the land, but also to convey the canes, when

cut, to the sugar-house. Sugar and coffee are principally cultivated.
There are a few cotton plantations, and some devoted to the
cultivation of the plantain, which, with a rich variety of other
vegetables, such as the sweet potato, the banana, yams, the casava,
&c., furnish a large part of the food of the inhabitants. There are
also large cattle farms. Cattle are abundant, and beef is cheap.
The uncultivated tracts abound with a vast variety of useful plants
and trees. Many of the trees furnish excellent timber. There are in
the colony several steam mills employed in the manufacture of
lumber.

II. FORM OF GOVERNMENT,
ADMINISTRATION OF JUSTICE,
CIVIL DIVISIONS, POPULATION,
SOCIAL EQUALITY.
British Guiana is a colony, conquered some forty years since from
the Dutch, belonging to Great Britain. It is what is called a crown
colony, and all its laws are made, or revised in England.
The governor, whose authority is very extensive, is appointed by the
British queen. He is assisted in his administration by a council of nine
persons, called the Court of Policy, four of whom are high executive
officers appointed by the Crown. The other five are chosen by the
inhabitants. No law made by the Court of Policy can remain in force
unless it be approved in England by the queen in council.
Justice is administered by a Supreme Court consisting of three
Judges, who are always lawyers of high standing, sent out from
Great Britain. In the criminal trials which come before this court, the
judges are assisted by three assessors, who answer to our jurymen,
being persons chosen by lot from among the inhabitants,—who have
an equal vote with the judges. No prisoner can be found guilty,
except by at least four votes out of the six.
The colony is divided into three counties, Demarara, Berbice and
Essequebo. Each of these counties is again divided into parishes,
and the parishes are subdivided into judicial districts, each under the
superintendence of a Stipendiary Magistrate, appointed and paid by
the Crown. These stipendiary magistrates are persons of education
and character, sent out from Great Britain, and who, having no
interest or connections in the colony, and being frequently removed

from one district to another, may be expected to be impartial, and
not likely to be warped in their judgment by personal considerations.
These magistrates are under the sole control of the Governor, by
whom they can be suspended from office. They have exclusive
jurisdiction, as will presently appear, of all controversies, as to
contracts and labor, arising between employers and laborers. The
whole population of the three counties may be estimated at one
hundred thousand, of whom six or eight thousand are white, and all
the remainder, colored. The English language is now spoken by all,
and is the only language used in the colony.
Those distinctions which prevail to so great a degree in the United
States, between the free colored and the white population, and
which render the position of the colored man in the United States so
mortifying and uncomfortable, are wholly unknown in British Guiana.
In this respect all are equal: colonial offices and dignities are held
without distinction by white and colored. Colored men are
indiscriminately drawn to sit as assessors on the bench of the
Supreme Court. The colored classes in British Guiana are wealthy,
influential, and highly respectable. Many of them are magistrates,
proprietors, merchants with large establishments, and managers of
estates receiving liberal salaries. The collector of customs at one of
the principal ports, is a person of color, and many others hold public
stations. It is evident from these facts that color is no obstacle to
advancement or distinction. It is difficult and almost impossible for a
citizen of the United States, educated in the midst of distinctions and
prejudices, to realize the state of things so entirely different which
prevails in British Guiana.

III. SPECIAL LAWS FOR THE
PROTECTION OF LABORERS AND
EMIGRANTS.
The greater part of the laboring population of British Guiana were
formerly slaves. They have been lately set free by the justice and
bounty of the British government, which is very jealous of their
rights, and which has enacted many special laws for their protection.
A leading measure of this kind is, the appointment of the Stipendiary
Magistrates above described. These stipendiary magistrates have
exclusive jurisdiction over all controversies between employers and
laborers touching wages and contracts. It is provided by the fourth
chapter of the Orders in Council of Sept. 7th, 1838, which are the
supreme law in British Guiana, that any laborer, on complaint
preferred, and proof made before any stipendiary magistrate, that
his employer has not paid his wages, or delivered him the articles
agreed upon between them as a part of his wages, or that the
articles delivered were not of the quality or quantity agreed upon, or
that through the negligence of the master the contract has not been
properly performed, or that the laborer has been ill used,—upon
complaint preferred for any of these reasons, and proof made, the
stipendiary magistrate may, by summary process, order the payment
of the wages, the delivering of the stipulated articles, or
compensation to be made for any negligence or ill usage on the part
of the employer; and if the order be not complied with, the
magistrate has power to issue his warrant for the seizure and sale of
the goods of the employer, or so much as may be necessary; or if no
goods are to be found, the magistrate may commit the employer to
prison for any time not exceeding one month, unless compensation

be sooner made; and the magistrate may dissolve the contract if he
see fit.
To prevent contracts being made with emigrants, disadvantageous to
them or unfair in any respect, previous to their arrival in the colony,
it is provided in the same Orders in Council, chapter third, that no
contract of service made out of the colony shall be of any force or
effect in it; that no contract of labor shall remain in force for more
than four weeks, unless it be reduced to writing; and that no written
contract of service shall be binding, unless signed by the name or
mark of the persons contracting in the presence of a stipendiary
magistrate; nor unless the magistrate shall certify that it was made
voluntarily, and with a full understanding of its meaning and effect;
nor can any written contract of service remain in force for more than
one year.
It is evident from these statements with what careful safeguards
against fraud and oppression the benevolence of British law has
surrounded the laborer and the emigrant.
There is an Emigration Agent in British Guiana, who is a stipendiary
magistrate, and whose duty it is to furnish emigrants, arriving in the
colony, with every information, and to prevent any imposition from
being practiced upon them. It will appear, from an examination of
the above provisions, that all those colored persons from the United
States who may emigrate to Guiana, will go out perfectly free and
unshackled. On their arrival in the colony, they will be perfectly their
own masters, at full liberty to choose any kind of employment which
the colony offers; and should they be dissatisfied, or disappointed,
no obstacle will exist to their return.

IV. TAXES, MILITARY DUTY,
RELIGIOUS INSTRUCTION,
EDUCATION.
The revenue of British Guiana is chiefly derived from a tax on the
produce raised in the colony, and duties levied on the imposts.
Parish taxes are unknown, and the laborer is exempt from every
species of taxation, unless his income amount to five hundred
dollars. The militia laws were abrogated, and the colonial militia
disbanded soon after the emancipation took place, so that the poor
man is not compelled to contribute any portion of his time to the
public service.
There are Episcopalian, Presbyterian and Catholic church
establishments supported at an expense to the colony of upwards of
$113,000 per annum, as will appear by reference to the Royal
Gazette of May 7th, 1839, published in Georgetown, containing an
official estimate of the taxes to be raised for that year. There are
beside numerous Methodist and other dissenting religious teachers,
supported in part by charitable societies in England, and in part by
voluntary contributions in the colony. The laboring population of
Georgetown and its vicinity have erected several handsome chapels
at their own expense.
There are numerous Sunday, infant and day schools, for the
gratuitous diffusion of knowledge and moral education among the
people. On most of the principal estates a school-house is erected,
and a teacher provided, where the children of the laborers are
entitled to receive instruction free of expense. Great attention is paid
throughout the colony to the education of the rising generation.

V. DEMAND FOR LABOR, KINDS OF
LABOR, WAGES.
British Guiana possesses a superabundance of the most fertile land.
The planters are wealthy, and well provided with the most complete
machinery for the manufacture of sugar. The only deficiency is a lack
of labor. The harvest is abundant, but the laborers are few. For
example,—on a coffee plantation, called Dankbaarheid, in the county
of Berbice, belonging to Mr. Carbery, it was estimated by the owner
and other competent persons in September last, that the crop of
coffee on the trees exceeded one hundred thousand pounds weight.
Of this crop, through deficiency of labor, only forty thousand pounds
weight were gathered. Sixty thousand pounds of coffee on that
single plantation, worth, in the British market, sixty thousand sterling
shillings, or about fifteen thousand dollars, perished for lack of
hands to gather it. It is the same to a greater or less extent, on
every other plantation. Indeed this deficiency of labor is more
peculiarly felt on the sugar estates, upon many of which it is not
uncommon for ripe canes, which if manufactured would have
produced the value of several thousand pounds sterling, to perish in
the field for want of hands to gather it.
There is indeed a great opening for industry of every kind. All sorts
of mechanics are sure of steady employment at wages from one to
two dollars per day, according to their skill. Seamstresses and
domestics are much needed and will find full employment. Any
emigrant who can command a small capital, can open a shop, or set
up various kinds of business to good advantage. Georgetown, the
capital, situated at the mouth of the river Demarara, is a place of
about twelve thousand inhabitants, and furnishes abundant
employment in all those branches of business usually carried on in a
commercial town. New Amsterdam, at the mouth of the Berbice, has

about four thousand inhabitants, and there are besides several
villages, containing each some hundreds of inhabitants.
The greatest demand however for labor is, on the plantations.
Agricultural laborers are always sure of abundant employment and
high wages. The labor of agriculture is of various kinds, and may be
performed by any man accustomed to work, with little or no
previous instruction. It consists principally in cutting up weeds with
the hoe, cutting down sugar-cane, and throwing it into boats on the
canals, to be transported to the sugar-house; tending the sugar
boiling; packing away the sugar; boating it to market; picking and
curing coffee, which is very light work; tending cattle; cutting
timber; and a great variety of other labor, almost all of the simplest
kind.
Every laborer on a plantation has a comfortable house, with a plot of
ground annexed, capable of raising a much greater quantity of
provisions and poultry than the laborer can consume. For this he
pays nothing. He is also provided with medical attendance, medicine,
and a support at the expense of the estate, gratis, whenever he is
sick. Fuel is abundant, and close at hand. It is needed only for
cooking, and the laborer has but to help himself. Clothing, which in
that climate is very light, may be amply provided, at one-third the
expense incurred for that article in the United States. So many of the
wants of the laborer are thus supplied, free of expense to him, or at
a very trifling rate, that if he choose to do so, he can lay by a great
part of his wages.
The labor on a plantation is divided into tasks which a laborer of any
activity can easily perform in four hours. The lowest rate of wages
ever paid, is thirty-three and a third cents a task, and very
frequently, much more is given. For cutting cane, attending in the
boiling house, boating sugar, and several other kinds of labor, higher
wages are always paid. The people employed in making sugar, in
addition to their wages, are supplied with food at the expense of the
estate. This is in addition to the laborer's house, provision ground,

fuel, medical attendance, gratuitous schooling for his children, and a
variety of other perquisites. The wages are paid weekly in cash.
I have now before me an original journal, for the month of October,
1839, of the plantation Thomas, adjoining Georgetown, owned by
Mr. Carbery. This journal is a printed form, with blanks filled up in
writing, containing an account for each day of the month, of the
number of laborers on the estate; the number actively employed,
and in what way; the number, sick, absent, or otherwise prevented
from working; the work done each day; with all the articles bought,
sold or shipped, and all the money paid on account of the plantation
during each day in the month,—in fact a complete history of all the
business of the estate for that time. Similar journals are kept on
every estate by the head manager, and are transmitted monthly to
the proprietor. This excellent custom was derived from the early
Dutch settlers.
On the plantation, Thomas, there are three hundred and twenty-five
acres of canes in cultivation. It appears by the journal above
referred to, that during the month of October, the number of
persons employed on the estate, varied from 163 to 176, of which
latter number 89 were men, 68 women, 14 boys, and 5 girls. Of
these, however, only 106, on an average, were daily at work on the
estate. To these laborers there was paid during the month of
October, in weekly payments, $1229 16, or an average of $11 60, to
each laborer, exclusive of house rent, provision grounds, fuel,
medical attendance, and many gratuities beside. It is to be
considered that this average amount of wages was earned by men,
women, boys and girls, including many old people and invalids, who
did but very little, and whose pay was therefore small. It therefore
must be obvious that the more active and industrious of the
laborers, earned from fifteen to twenty dollars, a month.
This single case, which is taken at random, will serve to show how
abundantly the laborer is rewarded. The laborers in this case did not
probably work on an average more than five hours per day. They
were employed in weeding and cutting cane, and making sugar, and

a portion of them as boatmen, watchmen, and mechanics. Though
they are all included under the class of agricultural laborers, only
about sixty out of the hundred and six were actually at work in the
fields. Many more are classed in the journal, "as jobbing and at work
about the buildings," that is, engaged in making sugar, and in a
great variety of other work necessary on such an estate.
To show with what rapidity the laborers grow rich and rise in the
world, I give the following extract from the Berbice Advertiser of
Nov. 1839. "Astonishing fact. Some negroes on the east coast, not a
dozen in all, have bought Northbroke (a plantation) for $10,000, of
which they paid down $8,000 last week, the remaining $2,000, is to
be paid this week." "What happiness," the editor justly observes,
"could our colony disseminate through the human species, did but
fresh importations of labor render the cultivation of the great staples
compatible with the formation of black villages and towns." It ought
to be mentioned that the people who clubbed together to buy this
estate had only been free since August, 1838. It may be well to
observe here that land in the colony is abundant and cheap; and
every laborer who is industrious, and will lay by his wages, has it in
his power to become a proprietor within a short period.
That there is no danger of overstocking British Guiana with
emigrants will appear by the following extract from an address of Mr.
John Scoble, delivered at Albany Tuesday evening, Aug. 1st, 1839.
He spoke of "British Guiana, a colony on the coast of South America,
and one which some think will ere long rival in its wealth and
population the State of New York. It is capable of sustaining a
population of forty millions, though the actual number of the
inhabitants is now only one hundred thousand."

VI. OFFERS MADE TO SUCH FREE
COLORED PERSONS OF THE UNITED
STATES, AS MAY CHOOSE TO
EMIGRATE TO BRITISH GUIANA.
Mr. Carbery arrived at Baltimore in September last. He came to the
United States partly for pleasure, and partly for the benefits of a
change of climate. He had been but a few days at Baltimore, when
his attention was attracted by the large number of free colored
persons in that city; the difficulty they seemed to have in gaining a
livelihood; and the discomforts of various kinds to which they are
subjected.
Knowing the great want of laborers in British Guiana, and the strong
disposition, existing there, to encourage immigration, it immediately
occurred to him, that by the transfer of a certain portion of the free
colored people of the United States to Guiana, not only might a
great benefit be done to that colony, but what all must regard as of
still greater importance, a boon of vast value might be conferred
upon the free colored people themselves.
Much impressed by these considerations, Mr. Carbery procured a
meeting of several of the free colored people of Baltimore, at which
he proposed to them to select two of their own number, in whom
they had confidence, whom he would send to British Guiana, free of
expense, in order to give them an opportunity to examine the
country, to judge for themselves, and to report to their brethren,
what the prospects for immigrants really are.
The free colored people of Baltimore, upon this suggestion and offer,
organized a Committee of Emigration, of which Mr. Green was

appointed chairman, and selected Messrs Peck and Price, two of
their number, as delegates to visit Guiana. These delegates sailed,
free of expense, in the barque Don Juan, from Boston, on the 21st
of December last. The result of their mission is not yet known, the
agents not having returned,—nor indeed has Mr. Carbery yet heard
of their arrival in the colony. The news however of their arrival and
reception is daily expected.
In the mean time certain letters which Mr. Carbery had previously
written to his friends in Guiana, giving an account of the numbers
and the condition of the free colored people in the United States,
had excited great attention and sympathy there. A public meeting
was held in Georgetown the capital; an "Immigration Society" was
established, and a very large sum of money was at once subscribed
to form a fund for paying the expenses of all such immigrants as
may choose to go to that colony. Of this sum, a considerable amount
has been already remitted to Mr. Carbery, who is appointed Agent of
the Society for the United States, to be applied towards the outfit of
emigrants,—the Society undertaking to pay the charter or passage
money on the arrival of the vessels, and to make all necessary
arrangements for the entertainment and comfort of the immigrants,
until such time, as they may select some regular employment. Mr.
Carbery is assured that should the colored people of the United
States or any part of them, be induced to accept the offer he now
makes, any amount necessary to carry his proposals into effect, will
be furnished as it may be needed.
As the agent of the above society Mr. Carbery offers to transport,
from the United States to British Guiana, free of any expense to
themselves, together with their baggage, all such sober and
industrious free colored people as shall see fit to embrace this
opportunity, so rare and extraordinary, of at once relieving
themselves from the great disabilities and disadvantages under
which they now labor, and of securing not only a comfortable
subsistence, and perhaps wealth, but what is of far greater
importance, both for themselves and their children,—a full

participation in all the rights, privileges and immunities of freemen,
and a standing and consideration in society, which at present is
wholly beyond their reach.
Mr. Carbery is also authorized by the society to guarantee to all
emigrants, who may accept his offers, maintenance at the colonial
expense, and comfortable and commodious lodgings, until they shall
succeed in obtaining such employment as they may prefer.
Transferred to a country which opens a vast field to labor, and to all
sorts of enterprise, relieved from a weight of prejudice which now
rests so heavily upon them, the free colored people of the United
States would have an opportunity which they do not now enjoy, of
proving, that when allowed to share the same moral and social
advantages, they are able successfully to compete with the white
man. It is indeed difficult to realize the effect often produced upon a
man's conduct and character, when he is removed from the
withering effect of the distinction of caste, and raised to an equality
of political and social privileges. Persons, who if they remain in the
United States, will be confined all their lives to menial and obscure
stations, by emigrating to British Guiana, which they may do in
twenty days, and without spending a cent, will alter the whole
course and prospect of their lives. With industry, application, and
sobriety, they will have a moral certainty of rising to a comfortable
competency if not to wealth, and of filling with pleasure to
themselves and benefit to the community, a respectable station in
society. Surely these considerations ought to have great weight with
all,—but more especially with the young, who are just coming
forward, and with those fathers and mothers who have families of
children growing up about them.
There is now opened to the free colored people of the United States,
a city of refuge in Guiana, of which it is to be hoped they will not fail
to avail themselves; and Mr. Carbery has reason to anticipate, should
the free colored people of the United States, and those persons
upon whose advice and opinions they most confidently rely, be led to
take the same view of the matter which he does, that his visit to the

United States may result in great good to a large body of his fellow
men, who at present are cut off from many of the chief benefits of
society, and by the unfortunate operation of circumstances over
which they have no control, are subjected to influences which crush
their energies, break their spirits, and prevent them from rising to
affluence or consideration. Relieved from these impediments,
transferred to a country where they will be secured in the enjoyment
of equal social and political rights, they will become new creatures,
and many of them will display talents and capacity of which they are
not now suspected.
Mr. Carbery, however, has no desire to induce any colored person, to
emigrate to Guiana, who is not well satisfied, and whose friends are
not also satisfied, that it will be for his benefit to go. Deeply
impressed as he is with the manifold advantages which the free
colored people of the United States may derive from closing with his
proposals, he submits them to the candid consideration of those
concerned, expressly desiring that before being adopted by any
body, they may be subjected to the closest scrutiny, and most
rigorous investigation.

VII. DIRECTIONS TO PERSONS
WISHING TO EMIGRATE.
Mr. Carbery is now in Boston, but intends to proceed immediately to
New York, Philadelphia, and Baltimore, for the purpose of
establishing Committees of Emigration in each of those cities,
whence persons desirous of emigrating may obtain all necessary
information. The address of those Committees will be published in
the principal newspapers, and due notice will be given of the
intended sailing of vessels with emigrants. Persons with families
desiring to emigrate will meet with particular encouragement, but no
person of good character will be refused a free passage.
Boston, Feb. 1st, 1840.

Welcome to our website – the perfect destination for book lovers and
knowledge seekers. We believe that every book holds a new world,
offering opportunities for learning, discovery, and personal growth.
That’s why we are dedicated to bringing you a diverse collection of
books, ranging from classic literature and specialized publications to
self-development guides and children's books.
More than just a book-buying platform, we strive to be a bridge
connecting you with timeless cultural and intellectual values. With an
elegant, user-friendly interface and a smart search system, you can
quickly find the books that best suit your interests. Additionally,
our special promotions and home delivery services help you save time
and fully enjoy the joy of reading.
Join us on a journey of knowledge exploration, passion nurturing, and
personal growth every day!
ebookbell.com

Equivariant Degree Theory Reprint 2012 Jorge Ize Alfonso Vignoli

About This Presentation

Slide Content

Tags

Categories

Download

Quick Actions

Statistics

Related Slideshows

Equivariant Degree Theory Reprint 2012 Jorge Ize Alfonso Vignoli

About This Presentation

Slide Content

Slide 1

Slide 2

Slide 3

Slide 5

Slide 7

Slide 8

Slide 9

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

Slide 53

Slide 54

Slide 55

Slide 59

Slide 60

Slide 61

Slide 62

Slide 63

Slide 64

Slide 65

Slide 66

Slide 67

Slide 68

Slide 69

Slide 70

Slide 72

Slide 73

Slide 74

Slide 75

Slide 76

Slide 77

Slide 78

Slide 79

Slide 80

Slide 81

Tags

Categories

Download

Quick Actions