Algebraic And Geometric Surgery Andrew Ranicki

Algebraic And Geometric Surgery Andrew Ranicki
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ALGEBRAIC AND GEOMETRIC SURGERY
by Andrew Ranicki
Oxford Mathematical Monograph (OUP), 2002
This electronic version (February 2007) incorporates the errata which were
included in the second printing (2003) as well as the errata found subsequently.
Note that the pagination of the electronic version is somewhat dierent from
the printed version. The list of errata is maintained on
http://www.maths.ed.ac.uk/~aar/books/surgerr.pdf

For Frank Auerbach

CONTENTS
Preface vi
1 The surgery classication of manifolds 1
2 Manifolds 14
2.1 Dierentiable manifolds 14
2.2 Surgery 16
2.3 Morse theory 18
2.4 Handles 22
3 Homotopy and homology 29
3.1 Homotopy 29
3.2 Homology 32
4 Poincare duality 48
4.1 Poincare duality 48
4.2 The homotopy and homology eects of surgery 53
4.3 Surfaces 61
4.4 Rings with involution 66
4.5 Universal Poincare duality 71
5 Bundles 85
5.1 Fibre bundles and brations 85
5.2 Vector bundles 89

CONTENTS iii
5.3 The tangent and normal bundles 105
5.4 Surgery and bundles 112
5.5 The Hopf invariant and theJ-homomorphism 118
6 Cobordism theory 124
6.1 Cobordism and transversality 124
6.2 Framed cobordism 129
6.3 Unoriented and oriented cobordism 133
6.4 Signature 135
7 Embeddings, immersions and singularities 143
7.1 The Whitney Immersion and Embedding Theorems 143
7.2 Algebraic and geometric intersections 149
7.3 The Whitney trick 156
7.4 The Smale-Hirsch classication of immersions 161
7.5 Singularities 167
8 Whitehead torsion 170
8.1 The Whitehead group 170
8.2 Theh- ands-Cobordism Theorems 175
8.3 Lens spaces 185
9 Poincare complexes and spherical brations 193
9.1 Geometric Poincare complexes 194
9.2 Spherical brations 198
9.3 The Spivak normal bration 205

iv CONTENTS
9.4 Browder-Novikov theory 209
10 Surgery on maps 218
10.1 Surgery on normal maps 220
10.2 The regular homotopy groups 226
10.3 Kernels 230
10.4 Surgery below the middle dimension 238
10.5 Finite generation 240
11 The even-dimensional surgery obstruction 246
11.1 Quadratic forms 246
11.2 The kernel form 255
11.3 Surgery on forms 280
11.4 The even-dimensionalL-groups 287
11.5 The even-dimensional surgery obstruction 294
12 The odd-dimensional surgery obstruction 301
12.1 Quadratic formations 301
12.2 The kernel formation 305
12.3 The odd-dimensionalL-groups 316
12.4 The odd-dimensional surgery obstruction 319
12.5 Surgery on formations 322
12.6 Linking forms 332
13 The structure set 338
13.1 The structure set 338

CONTENTS v
13.2 The simple structure set 342
13.3 Exotic spheres 344
13.4 Surgery obstruction theory 355
References 360
Index 366

PREFACE
Surgery theory is the standard method for the classication of high-dimen-
sional manifolds, where high means>5. The theory is not intrinsically dicult,
but the wide variety of algebraic and geometric techniques required makes heavy
demands on beginners. Where to start?
This book aims to be an entry point to surgery theory for a reader who already
has some background in topology. Familiarity with a book such as Bredon [10]
or Hatcher [31] is helpful but not essential. The prerequisites from algebraic and
geometric topology are presented, along with the purely algebraic ingredients.
Enough machinery is developed to prove the main result of surgery theory : the
surgery exact sequencecomputing the structure set of a dierentiable manifold
Mof dimension>5 in terms of the topologicalK-theory of vector bundles over
Mand the algebraicL-theory of quadratic forms over the fundamental group
ringZ[1(M)]. The surgery exact sequence is stated in Chapter 1, and nally
proved in Chapter 13. Along the way, there are basic treatments of Morse theory,
embeddings and immersions, handlebodies, homotopy, homology, cohomology,
Steenrod squares, Poincare duality, vector bundles, cobordism, transversality,
Whitehead torsion, theh- ands-Cobordism Theorems, algebraic and geometric
intersections of submanifolds, the Whitney trick, Poincare complexes, spherical
brations, quadratic forms and formations, exotic spheres, as well as the surgery
obstruction groupsL(Z[]).
This text introduces surgery, concentrating on the basic mechanics and work-
ing out some fundamental concrete examples. It is denitely not an encyclope-
dia of surgery theory and its applications. Many results and applications are not
covered, including such important items as Novikov's theorem on the topological
invariance of the rational Pontrjagin classes, surgery on piecewise linear and topo-
logical manifolds, the algebraic calculations of theL-groups for nite groups, the
geometric calculations of theL-groups for innite groups, the Novikov and Borel
conjectures, surgery on submanifolds, splitting theorems, controlled topology,
knots and links, group actions, stratied sets, the connections between surgery
and index theory, . . . . In other words, there is a vast research literature on
surgery theory, to which this book is only an introduction.
The books of Browder [14], Novikov [65] and Wall [92] are by pioneers of
surgery theory, and are recommended to any serious student of the subject.
However, note that [14] only deals with the simply-connected case, that only a
relatively small part of [65] deals with surgery, and that the monumental [92] is

CONTENTS vii
notoriously dicult for beginners, probably even with the commentary I had the
privilege to add to the second edition. The papers collected in Ferry, Ranicki and
Rosenberg [24], Cappell, Ranicki and Rosenberg [17] and Farrell and Luck [23]
give a avour of current research and include many surveys of topics in surgery
theory, including the history. In addition, the books of Kosinski [42], Madsen
and Milgram [45], Ranicki [70], [71], [74] and Weinberger [94] provide accounts of
various aspects of surgery theory.
On the afternoon of my rst day as a graduate student in Cambridge, in
October, 1970 my ocial supervisor Frank Adams suggested that I work on
surgery theory. This is still surprising to me, since he was a heavy duty homotopy
theorist. In the morning he had indeed proposed three topics in homotopy theory,
but I was distinctly unenthusiastic. Then at tea-time he said that I might look
at the recent work of Novikov [64] on surgery theory and hamiltonian physics,
draining the physics out to see what mathematics was left over. Novikov himself
had not been permitted by the Soviet authorities to attend the Nice ICM in
September, but Frank had attended the lecture delivered on Novikov's behalf
by Mishchenko. The mathematics and the circumstances of the lecture denitely
sparked my interest. However, as he was not himself a surgeon, Frank suggested
that I actually work with Andrew Casson. Andrew explained that he did not
have a Ph.D. himself and was therefore not formally qualied to be a supervisor
of a Ph.D. student, though he would be willing to answer questions. He went on
to say that in any case this was the wrong time to start work on high-dimensional
surgery theory! There had just been major breakthroughs in the eld, and what
was left to do was going to be hard. This brought out a stubborn streak in me,
and I have been working on high-dimensional surgery theory ever since.
It is worth remarking here that surgery theory started in 1963 with the classi-
cation by Kervaire and Milnor [38] of the exotic spheres, which are the dieren-
tiable manifolds which are homeomorphic but not dieomorphic to the standard
sphere. Students are still advised to read this classic paper, exactly as I was
advised to do by Andrew Casson in 1970.
This book grew out of a joint lecture course with Jim Milgram at Gottingen
in 1987. I am grateful to the Leverhulme Trust for the more recent (2001/2002)
Fellowship during which I completed the book. I am grateful to Markus Banagl,
Jeremy Brookman (who deserves special thanks for designing many of the dia-
grams), Diarmuid Crowley, Jonathan Kelner, Dirk Schuetz, Des Sheiham, Joerg
Sixt, Chris Stark, Ida Thompson and Shmuel Weinberger for various suggestions.
Any comments on the book subsequent to publication will be posted on the
website
http://www.maths.ed.ac.uk/eaar/books
2nd June, 2002

1
THE SURGERY CLASSIFICATION OF MANIFOLDS
Chapter 1 is an introduction to the surgery method of classifying manifolds.
Manifolds are understood to be dierentiable, compact and closed, unless oth-
erwise specied.
A classication of manifolds up to dieomorphism requires the construction
of a complete set of algebraic invariants such that :
(i) the invariants of a manifold are computable,
(ii) two manifolds are dieomorphic if and only if they have the same invariants,
(iii) there is given a list of non-dieomorphic manifolds realizing every possible
set of invariants.
One could also seek a homotopy classication of manifolds, asking for a complete
set of invariants for distinguishing the homotopy types of manifolds. Dieomor-
phic manifolds are homotopy equivalent.
The most important invariant of a manifoldM
m
is its dimension, the number
m>0 such thatMis locally dieomorphic to the Euclidean spaceR
m
. If
m6=nthenR
m
is not dieomorphic toR
n
, so that anm-dimensional manifold
M
m
cannot be dieomorphic to ann-dimensional manifoldN
n
. The homology
and cohomology of an orientablem-dimensional manifoldMare related by the
Poincare duality isomorphisms
H

(M)

=Hm(M):
Anym-dimensional manifoldMhasZ2-coecient Poincare duality
H

(M;Z2)

=Hm(M;Z2);
with
Hm(M;Z2) =Z2; Hn(M;Z2) = 0 forn > m :
The dimension of a manifoldMis thus characterised homologically as the largest
integerm>0 withHm(M;Z2)6= 0. Homology is homotopy invariant, so that
the dimension is also a homotopy invariant : ifm6=nanm-dimensional manifold
M
m
cannot be homotopy equivalent to ann-dimensional manifoldN
n
.

2 THE SURGERY CLASSIFICATION OF MANIFOLDS
There is a complete dieomorphism classication ofm-dimensional mani-
folds only in the dimensionsm= 0;1;2, where it coincides with the homotopy
classication. Form>3 there existm-dimensional manifolds which are homo-
topy equivalent but not dieomorphic, so that the dieomorphism and homotopy
classications must necessarily dier. Form= 3 complete classications are the-
oretically possible, but have not been achieved in practice { the Poincare con-
jecture that every 3-dimensional manifold homotopy equivalent toS
3
is actually
dieomorphic toS
3
remains unsolved!
Form>4 group-theoretic decision problems prevent a complete classica-
tion ofm-dimensional manifolds, by the following argument. Every manifoldM
can be triangulated by a nite simplicial complex, so that the fundamental group
1(M) is nitely presented. Homotopy equivalent manifolds have isomorphic fun-
damental groups. Every nitely presented group arises as the fundamental group
1(M) of anm-dimensional manifoldM. It is not possible to have a complete
set of invariants for distinguishing the isomorphism class of a group from a nite
presentation. Group-theoretic considerations thus make the following questions
unanswerable in general :
(a)IsMhomotopy equivalent toM
0
?
(b)IsMdieomorphic toM
0
?
since already the question
(c)Is1(M)isomorphic to1(M
0
)?
is unanswerable in general.
The surgery method of classifying manifolds seeks to answer a dierent ques-
tion :
Given a homotopy equivalence ofm-dimensional manifoldsf:M!M
0
isf
homotopic to a dieomorphism?
Every homotopy equivalence of 2-dimensional manifolds (= surfaces) is ho-
motopic to a dieomorphism, by the 19th century classication of surfaces which
is recalled in Chapter 3.
A homotopy equivalence of 3-dimensional manifolds is not in general homo-
topic to a dieomorphism. The rst examples of such homotopy equivalences
appeared in the classication of the 3-dimensional lens spaces in the 1930's : the
Reidemeister torsion of a lens space is a dieomorphism invariant which is not
homotopy invariant. AlgebraicK-theory invariants such as Reidemeister and

THE SURGERY CLASSIFICATION OF MANIFOLDS 3
Whitehead torsion are signicant in the classication of manifolds with nite
fundamental group, and in deciding if `h-cobordant' manifolds are dieomorphic
(via thes-Cobordism Theorem, stated in 1.11 below), but they are too special to
decide if an arbitrary homotopy equivalence of manifolds is homotopic to a dif-
feomorphism. Chapter 8 deals with the main applications of Whitehead torsion
to the topology of manifolds.
In 1956, Milnor [49] constructed an exotic sphere, a dierentiable manifold

7
with a homotopy equivalence (in fact a homeomorphism)
7
!S
7
which is
not homotopic to a dieomorphism. The subsequent classication by Kervaire
and Milnor [38] form>5 of pairs
(m-dimensional manifold
m
, homotopy equivalence
m
!S
m
)
was the rst triumph of surgery theory. It remains the best introduction to
surgery, particularly as it deals with simply-connected manifoldsM(i.e. those
with1(M) =f1g) and so avoids the fundamental group. The surgery classi-
cation of homotopy spheres is outlined in Section 13.3.
Denition 1.1An (m+ 1)-dimensional cobordism(W;M; M
0
) is an (m+
1)-dimensional manifoldW
m+1
with boundary the disjoint union of closedm-
dimensional manifoldsM,M
0
@W=M[M
0
:.
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......................................................................................................................................................................................................................................................
......................................................................................................................................................................................................................................................
M W
M
0
2
The cobordism classes of manifolds are groups, with addition by disjoint
union. The computation of the cobordism groups was a major achievement of
topology in the 1950's { Chapter 6 is an introduction to cobordism theory. The
cobordism classication of manifolds is very crude : for example, the 0- and 2-
dimensional cobordism groups have order two, and the 1- and 3-dimensional
cobordism groups are trivial. Surgery theory applies the methods of cobordism
theory to the rather more delicate classication of the homotopy types of mani-
folds.
What is surgery?

4 THE SURGERY CLASSIFICATION OF MANIFOLDS
Denition 1.2Asurgeryon anm-dimensional manifoldM
m
is the procedure
of constructing a newm-dimensional manifold
M
0m
= cl:(MnS
n
D
mn
)[
S
n
S
mn1D
n+1
S
mn1
by cutting outS
n
D
mn
Mand replacing it byD
n+1
S
mn1
. The surgery
removesS
n
D
mn
Mandkillsthe homotopy classS
n
!Minn(M).2
Terminology: given a subsetYXof a spaceXwrite cl:(Y) for theclosureof
YinX, the intersection of all the closed subsetsZXwithYZ.
At rst sight, it might seem surprising that surgery can be used to answer such
a delicate question as whether a homotopy equivalence of manifolds is homotopic
to a dieomorphism, since an individual surgery has such a drastic eect on the
homotopy type of a manifold :
Example 1.3(i) View them-sphereS
m
as
S
m
=@(D
n+1
D
mn
) =S
n
D
mn
[D
n+1
S
mn1
:
The surgery onS
m
removingS
n
D
mn
S
m
converts them-sphereS
m
into
the product of spheres
D
n+1
S
mn1
[D
n+1
S
mn1
=S
n+1
S
mn1
:
(ii) Form= 1,n= 0 the surgery of (i) converts the circleS
1
into the disjoint
unionS
0
S
1
=S
1
[S
1
of two circles..
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S
1
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1
[ S
1
(iii) Form= 2,n= 0 the surgery of (i) converts the 2-sphereS
2
into the torus
S
1
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S
2
S
1
S
1

THE SURGERY CLASSIFICATION OF MANIFOLDS 5
(iv) Form=nthe surgery of (i) converts them-sphereS
m
into the empty set
;. 2
There is an intimate connection between surgery and cobordism. A surgery
on a manifoldMdetermines a cobordism (W;M; M
0
) :
Denition 1.4Thetraceof the surgery removingS
n
D
mn
M
m
is the
cobordism (W;M; M
0
) obtained by attachingD
n+1
D
mn
toMIat
S
n
D
mn
f1g M f1g: 2....
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.....................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................
M = M f 0 g
M I
W = M I [ D
n +1
D
m n
D
n +1
D
m n
M
0
Here is a more symmetric picture of the trace (W;M; M
0
) :.
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........................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................................................................................................................................................................................................................................................................................................................
D
n +1
D
m n
S
n
D
m n
D
n +1
S
m n 1
M W
M
0
M
0
I
M
0
f 0 g M
0
f 1 g
Them-dimensional manifold with boundary
(M0; @M0) = (cl.(M
m
nS
n
D
mn
); S
n
S
mn1
)
is obtained fromMby cutting out the interior ofS
n
D
mn
M, with
M=M0[@M0
S
n
D
mn
;
M
0
=M0[@M0
D
n+1
S
mn1
;
W= (M0I)[(D
n+1
D
mn
);
(M0I)\(D
n+1
D
mn
) =S
n
S
mn1
I :
Note thatMis obtained fromM
0
by the opposite surgery removingD
n+1

S
mn1
M
0
.

6 THE SURGERY CLASSIFICATION OF MANIFOLDS
In fact, twom-dimensional manifoldsM
m
,M
0m
are cobordant if and only if
M
0
can be obtained fromMby a nite sequence of surgeries.
Denition 1.5Abordismof mapsf:M
m
!X,f
0
:M
0m
!Xfromm-
dimensional manifolds to a spaceXis a cobordism (W;M; M
0
) together with a
map
(F;f; f
0
) : (W;M; M
0
)!X(I;f0g;f1g):.
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f
F
f
0
M W
M
0
X f 0 g
X I
X f 1 g
2
Example 1.6A homotopyh:f'f
0
:M
m
!Xcan be regarded as a bordism
(F;f; f
0
) : (W;M; M
0
) =M(I;f0g;f1g)!X(I;f0g;f1g);
with
F:MI!XI; (x; t)7!(h(x; t); t): 2
Dieomorphic manifolds are cobordant. Homotopy equivalent closed man-
ifolds are cobordant, but in general only by a nonorientable cobordism. It is
possible to decide if two manifolds are cobordant, but it is not possible to de-
cide if cobordant manifolds are homotopy equivalent, or if homotopy equivalent
manifolds are dieomorphic. Given that cobordism is considerably weaker than

THE SURGERY CLASSIFICATION OF MANIFOLDS 7
dieomorphism and that cobordism drastically alters homotopy types, it may
appear surprising that cobordism is a suciently powerful tool to distinguish
manifolds within a homotopy type. However, surgery theory provides a system-
atic procedure for deciding if a map ofm-dimensional manifoldsf:M!M
0
satisfying certain bundle-theoretic conditions is bordant to a homotopy equiva-
lence, and if the bordism can be chosen to be a homotopy (as in 1.6), at least
in dimensionsm>5. This works because surgery makes it comparatively easy
to construct cobordisms with prescribed homotopy types. The applications of
cobordism theory to the surgery classication of high-dimensional manifolds de-
pend on the following fundamental result :
Whitney Embedding Theorem 1.7 ([97], [99], 1944)
Iff:N
n
!M
m
is a map of manifolds such that
either2n+ 16m
orm= 2n>6and1(M) =f1g
thenfis homotopic to an embeddingN
n
,!M
m
. 2
The proof of 1.7 will be outlined in Chapter 7.
Denition 1.8(i) Anh-cobordismis a cobordism (W
m+1
;M
m
; M
0m
) such
that the inclusionsM ,!W,M
0
,!Ware homotopy equivalences.
(ii) Anh-cobordism (W;M; M
0
) istrivialif there exists a dieomorphism
(W;M; M
0
)

=M(I;f0g;f1g)
which is the identity onM, in which case the composite homotopy equivalence
M'W'M
0
is homotopic to a dieomorphism. 2
Theh-Cobordism Theorem was the crucial rst step in the homotopy classi-
cation of high-dimensional manifolds :
h-Cobordism Theorem 1.9 (Smale [83], 1962)
A simply-connected(m+1)-dimensionalh-cobordism(W
m+1
;M; M
0
)withm>5
is trivial. 2
Thus form>5 simply-connectedm-dimensional manifoldsM; M
0
are dif-
feomorphic if and only if they areh-cobordant.
Theh-Cobordism Theorem was subsequently generalised to non-simply-conn-
ected manifolds, using Whitehead torsion (which is described in Chapter 8). The
Whitehead groupW h() of a groupis an abelian group which measures the

8 THE SURGERY CLASSIFICATION OF MANIFOLDS
extent to which Gaussian elimination fails for invertible matrices with entries
in the group ringZ[]. The Whitehead torsion of a homotopy equivalencef:
M
m
!M
0m
of manifolds (or more generally of niteCWcomplexes) is an
element(f)2W h(1(M)). A homotopy equivalencefissimpleif(f) = 0.
Denition 1.10Ans-cobordismis a cobordism (W
m+1
;M
m
; M
0m
) such that
the inclusionsM ,!W,M
0
,!Ware simple homotopy equivalences. 2
A dieomorphismf:M
m
!M
0m
ofm-dimensional manifolds determines
an (m+ 1)-dimensionals-cobordism (W;M; M
0
) with
W= (MI[M
0
)=f(x;1)f(x)jx2Mg
the mapping cylinder, such that there is dened a dieomorphism
(W;M; M
0
)

=M(I;f0g;f1g):
Thes-Cobordism Theorem is the non-simply-connected version of theh-
Cobordism Theorem :
s-Cobordism Theorem 1.11 (Barden-Mazur-Stallings, 1964)
An(m+1)-dimensionalh-cobordism(W
m+1
;M; M
0
)withm>5is trivial if and
only if it is ans-cobordism. 2
It follows that form>5h-cobordantm-dimensional manifoldsM; M
0
are
dieomorphic if and only if they ares-cobordant. The proofs of theh- ands-
Cobordism Theorems will be outlined in Chapter 8.
The Whitehead group of the trivial group is trivial,W h(f1g) = 0, and theh-
Cobordism Theorem is just the simply-connected special case of thes-Cobordism
Theorem. The conditionm>5 in theh- ands-Cobordism Theorems is due to
the use of the Whitney Embedding Theorem (1.7) in their proof. It is known
that theh- ands-Cobordism Theorems for (m+ 1)-dimensional cobordisms are
true form= 0;1, and are false form= 4 (Donaldson [21]),m= 3 (Cappell and
Shaneson [18]). It is not known if they are true form= 2, on account of the
classical 3-dimensional Poincare conjecture.
Milnor [58] used the lens spaces to constructh-cobordisms (W
m+1
;M; M
0
)
of non-simply-connected manifolds which are not dieomorphic.
One way to prove that manifolds are dieomorphic is to rst decide if they are
cobordant, and then to decide if some cobordism can be modied by successive
surgeries on the interior to be ans-cobordism.

THE SURGERY CLASSIFICATION OF MANIFOLDS 9
The tangent bundle of anm-dimensional manifoldM
m
is classied by the
homotopy class of a map
M:M!BO(m):
(See Chapter 5 for some basic information on bundles, including the classify-
ing spaceBO(m)). Iff:M!M
0
is a homotopy equivalence ofm-dimen-
sional manifolds which is homotopic to a dieomorphism there exists a homo-
topyf

M
0'M:M!BO(m). The tubular neighbourhood of an embedding
M
m
,!S
m+k
(klarge) is ak-plane bundleM:M!BO(k) which is a stable
inverse ofM. The stable normal bundle ofMis classied by a map
M:M!BO= lim
!
k
BO(k):
By the result of Mazur [47] form>5 a homotopy equivalence ofm-dimensional
manifoldsf:M!M
0
is covered by a stable bundle mapb:M!M
0if and
only iff1 :MR
k
!M
0
R
k
is homotopic to a dieomorphism for some
k>0. It is possible to extendfto a homotopy equivalence ofh-cobordisms
(F;f;1) : (W;M; M
0
)!M
0
(I;f0g;f1g)
if and only iff1 :MR!M
0
Ris homotopic to a dieomorphism.
The surgery theory developed by Browder, Novikov, Sullivan and Wall in the
1960's provides a systematic solution to the problem of deciding if a homotopy
equivalencef:M!M
0
ofm-dimensional manifolds is homotopic to a dieo-
morphism, with obstructions taking values in the topologicalK-theory of vector
bundles and the algebraicL-theory of quadratic forms. The obstruction theory
was obtained as the relative version of the systematic solution to the problem of
deciding if a spaceXwithm-dimensional Poincare dualityH

(X)

=Hm(X)
is homotopy equivalent to anm-dimensional manifold. The theory thus deals
both with the existence and the uniqueness of manifold structures in homotopy
types.
Denition 1.12Anm-dimensional geometric Poincare complexis a -
niteCWcomplexXwith a fundamental homology class [X]2Hm(X) (using
twisted coecients in the nonorientable case) such that the cap products are
isomorphisms
[X]\ :H

(X; )!Hm(X; )
for everyZ[1(X)]-module . 2
Example 1.13Anm-dimensional manifold is anm-dimensional geometric Poin-
care complex. 2

10 THE SURGERY CLASSIFICATION OF MANIFOLDS
The property of being a geometric Poincare complex is homotopy invariant,
unlike the property of being a manifold. Thus any niteCWcomplex homotopy
equivalent to a manifold is a geometric Poincare complex. In order for a space
to have a ghting chance of being homotopy equivalent to anm-dimensional
manifold it must at least be homotopy equivalent to anm-dimensional geomet-
ric Poincare complex. Geometric Poincare complexes which are not homotopy
equivalent to a manifolds may be obtained by glueing togetherm-dimensional
manifolds with boundary (M; @M), (M
0
; @M
0
) using a homotopy equivalence
@M'@M
0
which is not homotopic to a dieomorphism.
Denition 1.14LetXbe anm-dimensional geometric Poincare complex.
(i) Amanifold structure(M; f) onXis anm-dimensional manifoldMto-
gether with a homotopy equivalencef:M!X.
(ii) Themanifold structure setS(X) ofXis the set of equivalence classes of
manifold structures (M; f), subject to the equivalence relation :
(M; f)(M
0
; f
0
) if there exists a bordism
(F;f; f
0
) : (W;M; M
0
)!X(I;f0g;f1g)
withFa homotopy equivalence,
so that (W;M; M
0
) is anh-cobordism.
2
Surgery theory asks : isS(X) non-empty? And if so, then how large is it?
In any case, it is clear from the denition thatS(X) is a homotopy invariant
ofX, i.e. that a homotopy equivalenceX!Yinduces a bijectionS(X)!
S(Y). Surgery theory reducesS(X) to more familiar homotopy invariant objects
associated toX. A homotopy equivalencef:M
m
!N
m
ofm-dimensional
manifolds determines an element (M; f)2S(N), such thatfish-cobordant to
1 :N!Nif and only if
(M; f) = (N;1)2S(N):
In particular, iffis homotopic to a dieomorphism thenfish-cobordant to
1 :N!N, and (M; f) = (N;1)2S(N).
The determination ofS(X) is closely related to the bundle properties of
manifolds and geometric Poincare complexes.
A niteCWcomplexXis anm-dimensional geometric Poincare complex if
and only if a regular neighbourhood (Y; @Y)S
m+k
of an embeddingX ,!
S
m+k
is such that
mapping bre (@Y!Y)'S
k1
:
A regular neighbourhood is theP Lanalogue of a tubular neighbourhood. The
(k1)-spherical bration

THE SURGERY CLASSIFICATION OF MANIFOLDS 11
S
k1
!@Y!Y'X
is theSpivak normal brationof a geometric Poincare complexX, with a
classifying map
X:X!BG= lim
!
k
BG(k):
(See Section 9.2 for an exposition of brations). The Spivak normal bration is
the homotopy theoretic analogue of the stable normal bundleM=Mof a
manifoldM.
The classifying spacesBO,BGfor stable bundles and spherical brations
are related by a bration sequence
G=O!BO!BG!B(G=O);
withG=Othe classifying space for stable bundles with a bre homotopy trivial-
isation. The homotopy class of the composite map
t(X) :X
X
//
BG //B(G=O)
is the primary obstruction toXbeing homotopy equivalent to anm-dimensional
manifold. There exists a null-homotopyt(X)' fgif and only if the Spivak
normal brationXadmits a vector bundle reduction ~X:X!BO. Surgery
theory oers a two-stage programme for deciding if a geometric Poincare complex
Xis homotopy equivalent to a manifold, involving the concept of a normal map :
Denition 1.15Adegree 1 normal mapfrom anm-dimensional manifold
M
m
to anm-dimensional geometric Poincare complexX
(f; b) :M
m
!X
is a mapf:M!Xsuch that
f[M] = [X]2Hm(X);
together with a stable bundle mapb:M!overf, from the stable normal
bundleM:M!BOto a stable bundle:X!BO. 2
The two stages of the obstruction theory for deciding if anm-dimensional ge-
ometric Poincare complexXis homotopy equivalent to anm-dimensional man-
ifold are :
(i) DoesXadmit a degree 1 normal map (f; b) :M
m
!X? This is the case
precisely when the mapt(X) :X!B(G=O) is null-homotopic.
(iii) If the answer to (i) is yes, is there a degree 1 normal map (f; b) :M
m
!X
which is bordant to a homotopy equivalence (f
0
; b
0
) :M
0m
!X?

12 THE SURGERY CLASSIFICATION OF MANIFOLDS
The extent to which a degree 1 normal map (f; b) :M
m
!Xof connected
M; Xfails to be a homotopy equivalence is measured by the relative homotopy
groupsn+1(f) (n>0) of pairs of elements
(mapg:S
n
!M, null-homotopyh:fg' :S
n
!X) .
By J.H.C. Whitehead's Theorem,fis a homotopy equivalence if and only if
(f) = 0. Letm= 2nor 2n+ 1. It turns out that it is always possible to
`kill'i(f) fori6n, meaning that there is a bordant degree 1 normal map
(f
0
; b
0
) :M
0
!Xwithi(f
0
) = 0 fori6n. There exists a normal bordism of
(f; b) to a homotopy equivalence if and only if it is also possible killn+1(f
0
). In
general there is an obstruction to killingn+1(f
0
), which form>5 is essentially
algebraic in nature :
Wall Surgery Obstruction Theorem 1.16 ([92], 1970)
For any groupthere are dened algebraicL-groupsLm(Z[])depending only
onm( mod 4), as groups of stable isomorphism classes of(1)
n
-quadratic forms
overZ[]form= 2n, and as groups of stable automorphisms of such forms for
m= 2n+ 1. Anm-dimensional degree 1 normal map(f; b) :M
m
!Xhas a
surgery obstruction
(f; b)2Lm(Z[1(X)]);
such that(f; b) = 0if(and form>5only if) (f; b)is bordant to a homotopy
equivalence. 2
Ifm= 2n>6 and (f; b) :M
2n
!Xis a degree 1 normal map such that
i(f) = 0 fori6nthe surgery obstruction is largely determined by the (1)
n
-
symmetric pairing
:KK!Z[1(X)]
dened on the kernelZ[1(X)]-moduleK=n+1(f) by the intersection of im-
mersionsS
n
#M
2n
which are null-homotopic inX. In order to killKit is
necessary that there be a sucient number of elementsx2Kwith(x; x) = 0
which are represented by embeddingsx:S
n
D
n
,!M
2n
. The even-dimensional
surgery obstruction will be obtained in Chapter 11, and involves a (1)
n
-quadratic
renementof the (1)
n
-symmetric form (K; ). The odd-dimensional surgery
obstruction form= 2n+ 1>5 will be obtained in Chapter 12.
Example 1.17The simply-connected surgery obstruction groups are given by :
m(mod 4)0123Lm(Z)Z0Z20

THE SURGERY CLASSIFICATION OF MANIFOLDS 13
The surgery obstruction of a 4k-dimensional normal map (f; b) :M
4k
!Xwith
1(X) =f1gis
(f; b) =
1
8
signature (K2k(M); )2L4k(Z) =Z
withthe nonsingular symmetric form on the middle-dimensional homology
kernelZ-module
K2k(M) = ker(f:H2k(M)!H2k(X)):
The surgery obstruction of a (4k+ 2)-dimensional normal map (f; b) :M
4k+2
!
Xwith1(X) =f1gis
(f; b) = Arf invariant (K2k+1(M;Z2); ; )2L4k+2(Z) =Z2
with; the nonsingular quadratic form on the middle-dimensionalZ2-coecient
homology kernelZ2-module
K2k+1(M;Z2) = ker(f:H2k+1(M;Z2)!H2k+1(X;Z2)):
2
Surgery Exact Sequence 1.18(Browder, Novikov, Sullivan, Wall 1962{1970)
Letm>5.
(i)The manifold structure setS(X)of anm-dimensional geometric Poincare
complexXis non-empty if and only if there exists a normal map(f; b) :M
m
!
Xwith surgery obstruction
(f; b) = 02Lm(Z[1(X)]):
(ii)The structure setS(M)of anm-dimensional manifoldM
m
ts into the
surgery exact sequenceof pointed sets
: : :!Lm+1(Z[1(M)])!S(M)![M; G=O]!Lm(Z[1(M)]): 2
The surgery exact sequence will be obtained in Chapter 13. The restriction
m>5 is due to the use of the Whitney Embedding Theorem (1.7) in the proof,
exactly as in theh- ands-Cobordism Theorems.
The geometric surgery construction works just as well in the low dimensions
m64. However, the possible geometric surgeries and their eect (e.g. on the
fundamental group) are much harder to relate to algebra than in the higher
dimensions. The type of algebraic surgery considered in the book thus only deals
with the `high-dimensional' part of 3- and 4-dimensional topology.

2
MANIFOLDS
This chapter gives the basic constructions of geometric surgery.
Sections 2.1 and 2.2 give the ocial denitions of manifolds and surgery.
Section 2.3 is a rapid introduction to Morse theory, including the key fact that
every manifoldMadmits a Morse functionf:M!R. Section 2.4 introduces
handles, which are the building blocks of manifolds and cobordisms. The main
result of this chapter is the Handle Decomposition Theorem 2.22 : a Morse
functionf:M!Ron a manifoldMdetermines a handle decomposition
M=
m
[
i=0
(h
i
[h
i
[: : :[h
i
)
with onei-handleh
i
=D
i
D
mi
for each critical point offof indexi. More
precisely, ifa < b2Rare regular values offthenf
1
(a); f
1
(b)Mare codi-
mension 1 submanifolds such thatf
1
(b) is obtained fromf
1
(a) by a sequence
of surgeries, one for each critical value in [a; b]R, andf
1
([a; b]) is the union
of the traces of the surgeries..
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f
1
( a ) f
1
( b )
M
R

a
b
Morse functions and handle decompositions are highly non-unique. Indeed,
surgery theory is essentially the study of the possible handle structures on man-
ifolds and cobordisms.
2.1 Dierentiable manifolds
It is assumed that the reader has already had a rst course on dierentiable
manifolds, such as Bredon [10] or Hirsch [33].

DIFFERENTIABLE MANIFOLDS 15
Denition 2.1Anm-dimensionaldierentiable manifoldM
m
is a paracom-
pact Hausdor topological space with a maximal atlas of charts
U=f(UM; :R
m
!U)g
of open neighbourhoodsUMwith a homeomorphism:R
m
!U, such that
for every (U; ), (U
0
;
0
)2Uthetransition function

01
j:
1
(U\U
0
)!U\U
0
!
01
(U\U
0
)
is a dieomorphism of open subsets ofR
m
. 2
There is a corresponding notion of a dierentiable manifold with boundary
(M; @M).
Denition 2.2(i) Adierentiable mapof manifoldsf:N
n
!M
m
is a map
such that for any charts (UM; :R
m
!U)2UM, (VN; :R
n
!V)2
UNwithf(V)Uthe function
1
(fj) :R
n
!R
m
is dierentiable.
(ii) Adieomorphismf:N!Mis a dierentiable map which is a homeo-
morphism with a dierentiable inversef
1
:M!N.
(iii) Anembeddingof manifoldsf:N
n
,!M
m
is a dierentiable map which is
injective, i.e. the inclusion of a submanifold.
(iv) Anisotopybetween embeddings of manifoldsf0; f1:N
n
,!M
m
is a
homotopy
f:NI!M; (x; t)7!ft(x)
which is an embeddingft:N ,!Mat each levelt2I.
(v) Animmersionof manifoldsf:N
n
#M
m
is a dierentiable map which
is locally an embedding, i.e. such that for everyx2Nthere exists a chart
(V; )2UNsuch thatx2Vand (fj) :R
n
!Mis an embedding.
(vi) Aregular homotopyof immersionsf0; f1:N
n
#M
m
is a homotopy
f:NI!M; (x; t)7!ft(x)
which is an immersionft:N#Mat each levelt2I. 2
In particular, an embedding is an immersion, and isotopic embeddings are
regular homotopic.
In Chapter 5 we shall describe the neighbourhoods of submanifolds in terms
of vector bundles. The following is an important special case :
Denition 2.3An embedding of a submanifoldN
n
,!M
m
isframedif it
extends to an embeddingND
mn
,!M. 2

16 MANIFOLDS
2.2 Surgery
The input of a surgery on a manifold is a framed embedding of a sphere :
Denition 2.4(i) Ann-embeddingin anm-dimensional manifoldM
m
is an
embedding
g:S
n
,!M :
(ii) Aframedn-embeddinginMis an embedding
g:S
n
D
mn
,!M ;
withcoren-embedding
g= gj:S
n
=S
n
f0g,!M : 2
The eect of a surgery is another manifold :
Denition 2.5Ann-surgeryon anm-dimensional manifoldM
m
is the surgery
removing a framedn-embedding
g:S
n
D
mn
,!M, and replacing it with
D
n+1
S
mn1
, witheectthem-dimensional manifold
M
0m
= cl.(M
m
n
g(S
n
D
mn
))[
S
n
S
mn1D
n+1
S
mn1
: 2
The applications of surgery to the classication of manifolds require a plenti-
ful supply of framedn-embeddingsS
n
D
mn
,!M
m
. The Whitney Embedding
Theorem (7.2) shows that for 2n < mevery mapS
n
!M
m
can be approxi-
mated by ann-embedding. However, in general ann-embeddingS
n
,!Mcannot
be extended to a framedn-embeddingS
n
D
mn
,!M{ see 5.66 below for a
specic example.
The notion of a cobordism (W;M; M
0
) was dened in 1.1. The trace of an
n-surgery removingS
n
D
mn
,!M
m
was dened in 1.4 to be the cobordism
(W;M; M
0
) with
W
m+1
=M
m
I[
S
n
D
mn
f1gD
n+1
D
mn
:
Denition 2.6Thedual(mn1)-surgeryonM
0
removes thedual framed
(mn1)-embeddingD
n+1
S
mn1
,!M
0
with eectMand trace
(W;M
0
; M). 2
IfMis closed (@M=;) then so isM
0
. IfMhas non-empty boundary@M
the embeddingS
n
D
mn
,!Mis required to avoid@M, so that
@M
0
=@M :

SURGERY 17
Example 2.7A (1)-surgery on anm-dimensional manifoldMhas eect the
disjoint union
M
0
=M[S
m
:
The dualm-surgery onM
0
has eectM. 2
Example 2.8(i) The eect onS
m
of then-surgery removing the framedn-
embedding dened in 1.2
g:S
n
D
mn
,!S
n
D
mn
[D
n+1
S
mn1
=@(D
n+1
D
mn
) =S
m
is them-dimensional manifold
D
n+1
S
mn1
[D
n+1
S
mn1
=S
n+1
S
mn1
:
The trace of then-surgery (W
m+1
;S
m
; S
n+1
S
mn1
) can be viewed as
W= cl:(S
n+1
D
mn
nD
m+1
);
using any embeddingD
m+1
,!S
n+1
D
mn
in the interior.
(ii) The special casem= 1,n= 0 of (i) gives a framed 0-embedding inS
1
g:S
0
D
1
=D
1
[D
1
,!S
1
:
The 0-surgery onS
1
removingg(S
0
D
1
) has eect the disjoint union of two
circles
S
1
S
0
=S
1
[S
1
:
The trace is the cobordism (W
2
;S
1
; S
1
[S
1
) obtained fromS
2
by punching out
the interior of an embeddingD
2
[D
2
[D
2
,!S
2
.S
1
S
1
I [ D
1
D
1
S
1
[ S
1
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(iii) Modify the framed 0-embeddingg:S
0
D
1
,!S
1
in (ii) by twisting one of
the two embeddings ofD
1
by the dieomorphism
!:D
1
!D
1
;t7! t ;

18 MANIFOLDS
dening a dierent 0-embedding
g!:S
0
D
11[!
//
S
0
D
1
g
//
S
1
with the same core asg. The 0-surgery onS
1
removingg!(S
0
D
1
) has eect
a single circleS
1
. The trace is the cobordism (N
2
;S
1
; S
1
) obtained from the
Mobius bandM
2
by punching out the interior of an embeddingD
2
,[email protected]
1
S
1
×I∪D
1
×D
1
S
1
............................................................................................................................................................................................................................................................................................................................................................................ ................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..........................................................................................................................................................................................................
2
Denition 2.9Theconnected sumof connectedm-dimensional manifolds
M
m
,M
0m
is the connectedm-dimensional manifold
(M#M
0
)
m
= cl:(MnD
m
)[(S
m1
I)[cl:(M
0
nD
m
)
obtained by excising the interiors of embedded discsD
m
,!M,D
m
,!M
0
and
joining the boundary componentsS
m1
,!cl:(MnD
m
),S
m1
,!cl:(M
0
nD
m
)
byS
m1
I...............................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................
.......................................................................................................................
......................................................................................................................................................
.......................................................................................................................
......................................................................................................................................................
M#M
0
M M
0
S
m−1
×I
2
Example 2.10The connected sumM#M
0
is the eect of the 0-surgery on the
disjoint unionM[M
0
which removes the framed 0-embeddingS
0
D
m
,!
M[M
0
dened by the disjoint union of embeddingsD
m
,!M,D
m
,!M
0
.2
2.3 Morse theory
Morse theory studies dierentiable manifoldsMby considering the critical points
of dierentiable functionsf:M!Rfor which the second dierential is non-
trivial. This section is only a very rudimentary account { see Milnor [55] for the
classic exposition of Morse theory.

MORSE THEORY 19
It is assumed that the reader is already acquainted withCWcomplexes,
which are spaces obtained from;by successively attaching cells
X=
1
[
i=0
(D
i
[D
i
[: : :[D
i
)
of increasing dimensioni. Morse theory is used to prove that anm-dimensional
manifoldMcan be obtained from;by successively attaching handles
M=
m
[
i=0
(D
i
D
mi
[D
i
D
mi
[: : :[D
i
D
mi
)
of increasing indexi, givingMthe structure of aCWcomplex. However, there
is an essential dierence : the cell structure of aCWcomplex is part of the
denition, whereas a handle decomposition of a manifold has to be proved to
exist.
Here is the basic connection between Morse theory, handles and surgery. If
a < b2Rare regular values of a Morse functionf:M
m
!Rthen
(M[a; b];Na; Nb) =f
1
([a; b];fag;fbg)
is a cobordism of (m1)-dimensional manifolds. If everyt2[a; b] is a regular
value then eachNt=f
1
(t) is dieomorphic toNa, with a dieomorphism
(M[a; b];Na; Nb)

=Na(I;f0g;f1g):
If [a; b] consists of regular values except for one critical value of indexithen
(M[a; b];Na; Nb) is the trace of an (i1)-surgery onNa, with
M[a; b]

=NaI[D
i
D
mi
obtained fromNaIby attaching ani-handle. ThusMis obtained from;by
attaching ani-handle for each critical value offwith indexi, andMhas the
structure of a niteCWcomplex with onei-cell for eachi-handle.
The dierential of a dierentiable functionf:N!Matx2Nis dened
using any charts (UM; :R
m
!U)2UM, (VN; :R
n
!V)2UN
such thatx2V,f(x)2U. The function
()
1
f :R
n
!R
m
;x= (x1; x2; : : : ; xn)7!(f1(x); f2(x); : : : ; fm(x))
is dierentiable, and the dierential offatxis the linear map given by the
Jacobianmnmatrix
df(x) =

@fi
@xj

:R
n
!R
m
;
h= (h1; h2; : : : ; hn)7!df(x)(h) =
n
X
j=1
@f1
@xj
hj;
n
X
j=1
@f2
@xj
hj; : : : ;
n
X
j=1
@fm
@xj
hj

:

20 MANIFOLDS
Denition 2.11Letf:N
n
!M
m
be a dierentiable map.
(i) Aregular pointoffis a pointx2Nwhere the dierentialdf(x) :R
n
!R
m
is a linear map of maximal rank, i.e.
rank(df(x)) = min(m; n):
(ii) Acritical pointoffis a pointx2Nwhich is not regular.
(iii) Aregular valueoffis a pointy2Msuch that everyx2f
1
(y)Nis
regular (including the empty casef
1
(y) =;).
(iv) Acritical valueoffis a pointy2Mwhich is not regular. 2
Implicit Function Theorem 2.12The inverse image of a regular valuey2
Mof a dierentiable mapf:N
n
!M
m
is a submanifold
P=f
1
(y)N
with
dim(P) =nmin(m; n) =

nmifm6n
0 ifm > n.
ProofSee Chapter II.1 of Bredon [10]. 2
The Taylor expansion of a dierentiable functionf:M
m
!Ratx2Mis
given in local coordinates by
f(x1+h1; x2+h2; : : : ; xm+hm)
=f(x1; x2; : : : ; xm) +
1P
k=1
1
k!
X
16i1;i2;:::;ik6m
@
k
f
@xi1@xi2: : : @xik
hi1hi2: : : hik
2R;
((h1; h2; : : : ; hm)2R
m
):
The linear term in the Taylor series
L(h1; h2; : : : ; hm) =
m
X
i=1
@f
@xi
hi
is determined by the dierential offatx(= the gradient vectorrf(x)2R
m
)
df(x) =

@f
@x1
@f
@x2
: : :
@f
@xm

:R
m
!R; (h1; h2; : : : ; hm)7!
m
X
i=1
@f
@xi
hi;
which is either 0 or has the maximal rank 1. Thusx2Mis a regular point off
ifdf(x)6= 0, andx2Mis a critical point offifdf(x) = 0. The quadratic term
in the Taylor series
Q(h1; h2; : : : ; hm) =
m
X
i=1
m
X
j=1

@
2
f
@xi@xj

hihj=2

MORSE THEORY 21
is determined by the Hessianmmmatrix of second partial derivatives
H(x) =

@
2
f
@xi@xj

:
Denition 2.13Letf:M
m
!Rbe a dierentiable function on anm-dimensional
manifold.
(i) A critical pointx2Moffisnondegenerateif the Hessian matrixH(x) is
invertible.
(ii) TheindexInd(x) of a nondegenerate critical pointx2Mis the num-
ber of negative eigenvalues inH(x), so that with respect to appropriate local
coordinates the quadratic term in the Taylor series offnearxis given by
Q(h1; h2; : : : ; hm) =
Ind(x)
X
i=1
(hi)
2
+
m
X
i=Ind(x)+1
(hi)
2
2R:
(iii) The functionfisMorseif it has only nondegenerate critical points.2
In dealing with Morse functionsf:M!Rit will be assumed that the
critical pointsx1; x2; : : :2Mhave distinct imagesf(x1); f(x2); : : :2R. The
index of a critical valuef(xj)2Ris then dened by
Ind(f(xj)) = Ind(xj)>0:
Theorem 2.14(Morse)
Everym-dimensional manifoldM
m
admits a Morse functionf:M!R.
ProofThere exists an embeddingM
m
,!S
m+k
forklarge, by the Whitney
Embedding Theorem (1.7). The function dened for anya2S
m+k
nMby
fa:M!R;x7! kxak
is Morse for allaexcept for a set of measure 0. See Milnor [55] or Chapter 6 of
Hirsch [33] for more detailed accounts! 2
In fact, the set of Morse functions is dense in the function space of all dier-
entiable functionsf:M!R.
Example 2.15The height function on them-sphere
S
m
=f(x0; x1; : : : ; xm)2R
m+1
j
m
X
k=0
x
2
k= 1g
is a Morse function

22 MANIFOLDS
f:S
m
!R; (x0; x1; : : : ; xm)7!xm
with a critical point (0; : : : ;0;1) of index 0 and a critical point (0; : : : ;0;1) of
indexm. 2
Example 2.16Them-dimensional real projective spaceRP
m
is the quotient of
S
m
by the antipodal map
RP
m
=S
m
=f(x1; x2; : : : ; xm+1)(x1;x2; : : : ;xm+1)g:
Equivalently,RP
m
is the space with one point for each 1-dimensional subspace
of the (m+ 1)-dimensional real vector spaceR
m+1
[x0; x1; : : : ; xm] =f(x0; x1; : : : ; xm)j2Rg R
m+1
(xk2R;not all 0):
For any real numbers0< 1< : : : < mthere is dened a Morse function
f:R P
m
!R; [x0; x1; : : : ; xm]7!
mP
k=0
k(xk)
2
mP
k=0
(xk)
2
with (m+ 1) critical points [0; : : : ;0;1;0; : : : ;0] of index 0;1; : : : ; m. 2
Example 2.17Them-dimensional complex projective spaceC P
m
is the 2m-
dimensional manifold with one point for each 1-dimensional subspace of the
(m+ 1)-dimensional complex vector spaceC
m+1
[z0; z1; : : : ; zm] =f(z0; z1; : : : ; zm)j2Cg,!C
m+1
(zk2C;not all 0):
For any real numbers0< 1< : : : < mthere is dened a Morse function
f:C P
m
!R; [z0; z1; : : : ; zm]7!
mP
k=0
kjzkj
2
mP
k=0
jzkj
2
with (m+ 1) critical points [0; : : : ;0;1;0; : : : ;0] of index 0;2; : : : ;2m. 2
2.4 Handles
Denition 2.18(i) Given an (m+ 1)-dimensional manifold with boundary
(W; @W) and an embedding
S
i1
D
mi+1
,!@W(06i6m+ 1)
dene the (m+1)-dimensional manifold with boundary (W
0
; @W
0
) obtained from
Wbyattaching ani-handleto be

HANDLES 23
W
0
=W[
S
i1
D
mi+1D
i
D
mi+1
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W
D
i
D
m i +1
W
0
(ii) Anelementary(m+ 1)-dimensional cobordism of indexiis the cobor-
dism (W;M; M
0
) obtained fromMIby attaching ani-handle at
S
i1
D
mi+1
,!M f1g;
with
W=MI[D
i
D
mi+1
:
(iii) Thedualof an elementary (m+ 1)-dimensional cobordism (W;M; M
0
) of
indexiis the elementary (m+ 1)-dimensional cobordism (W;M
0
; M) of index
(mi+ 1) obtained by reversing the ends, and regarding thei-handle attached
toMIas an (mi+ 1)-handle attached toM
0
I. 2
Lemma 2.19For any06i6m+ 1the Morse function
f:D
m+1
!R; (x1; x2; : : : ; xm+1)7!
i
X
j=1
x
2
j+
m+1
X
j=i+1
x
2
j
has a unique interior critical point02D
m+1
, which is of indexi. The(m+ 1)-
dimensional manifolds with boundary dened for0< <1by
W=f
1
(1;]; W=f
1
(1; ]
are such thatWis obtained fromWby attaching ani-handle
W=W[D
i
D
mi+1
: 2
Here is an illustration in the casem=i= 1 :

24 MANIFOLDSy
2
x
2
>
y > 0
y
2
x
2
>
y < 0
D
1
D
1
y
2
x
2
6
x > 0
y
2
x
2
6
x < 0
D
1
f 0 g
f 0 g D
1
W

= f ( x; y ) 2 D
2
j y
2
x
2
6 g
D
1
D
1
= f ( x; y ) 2 D
2
j 6 y
2
x
2
6 g
W

= f ( x; y ) 2 D
2
j y
2
x
2
6 g = W

[ D
1
D
1
.............................................................................................................................................................................................................................................................................................................
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Proposition 2.20Letf:W
m+1
!Ibe a Morse function on an(m+ 1)-
dimensional manifold cobordism(W;M; M
0
)with
f
1
(0) =M ; f
1
(1) =M
0
;
and such that all the critical points offare in the interior ofW.
(i)Iffhas no critical points then(W;M; M
0
)is a trivialh-cobordism, with a
dieomorphism
(W;M; M
0
)

=M(I;f0g;f1g)
which is the identity onM.
(ii)Iffhas a single critical point of indexithenWis obtained fromMI
by attaching ani-handle using an embeddingS
i1
D
mi+1
,!M f1g, and
(W;M; M
0
)is an elementary cobordism of indexiwith a dieomorphism
(W;M; M
0
)

=(MI[D
i
D
mi+1
;M f0g; M
0
):
Proof(i) See Milnor [55].
(ii) In a neighbourhood of the unique critical pointp2W
f(p+ (x1; x2; : : : ; xm+1)) =f(p)
i
X
j=1
(xj)
2
+
m+1
X
j=i+1
(xj)
2

HANDLES 25
with respect to a coordinate chartR
m+1
,!Wsuch that 02R
m+1
corresponds
top2W, withf(p)2Rthe critical value. For any >0 there are dened
dieomorphisms
f
1
(1; c]

=MI ; f
1
[c+;1)

=M
0
I
by (i), and by 2.19 there is dened a dieomorphism
f
1
[c; c+]

=MI[D
i
D
mi+1
:
2
Attaching a handle (2.18) to a manifold with boundary is a surgery on the
boundary :
Proposition 2.21If an(m+1)-dimensional manifold with boundary(W
0
; @W
0
)
is obtained from(W; @W)by attaching ani-handle
W
0
=W[
S
i1
D
mi+1D
i
D
mi+1
then@W
0
is obtained from@Wby an(i1)-surgery
@W
0
= cl.(@Wn(S
i1
D
mi+1
))[
S
i1
S
miD
i
S
mi
:
ProofBy construction. 2
Somewhat by analogy with the result that every nite-dimensional vector
space has a nite basis :
Handle Decomposition Theorem 2.22 (Thom [87], Milnor [53])
(i)Every cobordism(W
m+1
;M
m
; M
0m
)has a handle decomposition as the union
of a nite sequence
(W;M; M
0
) = (W1;M0; M1)[(W2;M1; M2)[: : :[(Wk;Mk1; Mk)
of adjoining elementary cobordisms(Wj;Mj1; Mj)with indexij, such that
06i16i26: : :6ik6m+ 1; M0=M ; M k=M
0
:M
0
W
1
M
1
W
2
M
2
W
k
M
k
: : :
: : :
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......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ...........................................................................................................................
......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ...........................................................................................................................
(ii)Closedm-dimensional manifoldsM; M
0
are cobordant if and only ifM
0
can
be obtained fromMby a sequence of surgeries.

26 MANIFOLDS
Proof(i) By the relative version of (2.14) any cobordism admits a Morse func-
tion
f: (W;M; M
0
)!I
withM=f
1
(0),M
0
=f
1
(1), and such that all the critical values are in the
interior ofI. SinceWis compact there is only a nite number of critical points :
label thempj2W(16j6k). Write the critical values ascj=f(pj)2R, and
letijbe the index ofpj. It is possible to choosefsuch that
0< c1< c2: : : < ck<1;06i16i26: : :6ik6m+ 1:
Letrj2I(06j6k) be regular values such that
0 =r0< c1< r1< c2< : : : < rk1< ck< rk= 1:
By 2.20 (i) each
(Wj;Mj1; Mj) =f
1
([rj1; rj];frj1g;frjg) (16j6k)
is an elementary cobordism of indexij.
(ii) The trace of a surgery is an elementary cobordism (2.18). Thus surgery-
equivalent manifolds are cobordant. Conversely, note that every elementary cobor-
dism is the trace of a surgery, and that by (i) every cobordism (W;M; M
0
) is a
union of elementary cobordisms. 2
If (W;M; M
0
) has a Morse functionf:W!Iwith critical points of index
06i06i16: : :6ik6m+ 1 thenWhas a handle decomposition
W=MI[h
i0
[h
i1
[: : :[h
ik
withh
i
=D
i
D
mi+1
a handle of indexi.
Corollary 2.23Every closedm-dimensional manifoldM
m
can be obtained from
;by attaching handles. A Morse functionf:M!Rwith critical points of index
06i06i16: : :6ik6mdetermines a handle decomposition
M=h
i0
[h
i1
[: : :[h
ik
;
so thatMis a nitem-dimensionalCWcomplex with onei-cell for each critical
point of indexi.
ProofApply 2.22 to the cobordism (M;;;;). 2
Strictly speaking, the above result shows thatMcontains a subspace (not a
submanifold)
L=D
i0
[D
i1
[: : :[D
ik
which is a niteCWcomplex, and such that the inclusionL!Mis a homotopy
equivalence.

HANDLES 27
Example 2.24(i) Them-sphereS
m
has a handle decomposition consisting of
a 0-handle and anm-handle
S
m
=h
0
[h
m
;
given by the upper and lower hemispheres.
(ii) The cobordism (D
m+1
;;; S
m
) has a handle decomposition with one 0-handle
D
m+1
=h
0
:
The dual cobordism (D
m+1
;S
m
;;) has a handle decomposition with one (m+1)-
handle
D
m+1
=S
m
I[h
m+1
:
(iii) The torusM=S
1
S
1
has a Morse functionf:M!Rwith 4 critical
values. Here is a picture of the corresponding handle decomposition with the
corresponding 4 handles :.
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h
0
h
1
h
1
h
2
M
R

r
0
c
1
r
1
c
2
r
2
c
3
r
3
c
4
r
4
M = S
1
S
1
= h
0
[ h
1
[ h
1
[ h
2
Ind( c
1
) = 0 ; Ind( c
2
) = Ind( c
3
) = 1 ; Ind ( c
4
) = 2
2
Example 2.25The Morse functionf:R P
m
!Rof Example 2.16 has one crit-
ical point of indexifori= 0;1; : : : ; m, so thatR P
m
has a handle decomposition
of the type
R P
m
=h
0
[h
1
[: : :[h
m
: 2

28 MANIFOLDS
Example 2.26The Morse functionf:C P
m
!Rof Example 2.17 has one crit-
ical point of indexifori= 0;2; : : : ;2m, so thatC P
m
has a handle decomposition
of the type
C P
m
=h
0
[h
2
[: : :[h
2m
: 2

3
HOMOTOPY AND HOMOLOGY
In order to understand how surgery theory deals with the homotopy types
of manifolds it is necessary to understand how algebraic topology deals with the
homotopy types of more general spaces such asCWcomplexes. This chapter
provides some of the necessary background, assuming that the reader already
has some familiarity with the homotopy theory ofCWcomplexes. See Bredon
[10], Hatcher [31], Whitehead [96],: : :for considerably more detailed accounts
of algebraic topology.
Section 3.1 reviews the homotopy groups(X) and the stable homotopy
groups
S
(X), and the Freudenthal Suspension Theorem. Section 3.2 deals with
the homology and cohomology groupsH(X),H

(X), the Steenrod squares, as
well as the Universal Coecient Theorem, the Theorems of J.H.C. Whitehead
and Hurewicz, and the method of killing homotopy classes ofCWcomplexes
by attaching cells, the cellular chain complex of aCWcomplex and the handle
chain complex of a manifold.
3.1 Homotopy
Denition 3.1(i) Apointed spaceXis a space together with a base point
x02X. Apointed mapf:X!Yis a map of pointed spacesf:X!Y
such that
f(x0) =y02Y :
Apointed homotopybetween pointed mapsf; g:X!Y
h:f'g:X!Y
is a maph:XI!Ysuch that
h(x0; t) =y02Y(t2I):
(ii) Thehomotopy set[X; Y] of pointed spacesX; Yis the set of pointed
homotopy classes of pointed mapsf:X!Y.
(iii) Thehomotopy groupsof a pointed spaceXare
n(X) = [S
n
; X] (n>0)
with0(X) the set of path components, the fundamental group1(X) nonabelian
(in general), and the higher homotopy groupsn(X) (n>2) abelian. 2

30 HOMOTOPY AND HOMOLOGY
Example 3.2Here are some homotopy groups of spheres :
m(S
1
) =

Zifm= 1
0 ifm>2 ,
m(S
n
) =

0 ifm < n
Zifm=n,
n+1(S
n
) =

Zifn= 2
Z2ifn>3 ,
n+2(S
n
) =

0 ifn= 1
Z2ifn>2 .
See Section 6.1 for an account of the degree invariant detectingn(S
n
). See
Section 5.5 for an account of the Hopf invariantHused to detectn+1(S
n
). See
Section 11.4 for an account of the Arf invariant detectingn+2(S
n
). 2
Remark 3.3As already noted in Chapter 1 the applications of the homotopy
groups to surgery on non-simply-connected manifolds will make use of the action
of the fundamental group1(X) on the higher homotopy groupsn(X)
1(X)n(X)!n(X) (n>2):
This action can be dened by considering elements ofn(X) as homotopy classes
of pairs (; ) consisting of an unpointed map:S
n
!Xand a path:I!X
from(0) =xto(1) =(1S
n), with 1S
n2S
n
a base point, and letting1(X)
act on. Alternatively, use the universal cover
e
XofX(which may be assumed
connected), identifyn(X) =n(
e
X) with the set of unbased homotopy classes
of mapsS
n
!
e
X, and let1(X) act on
e
Xas the group of covering translations.
2
Denition 3.4Therelative homotopy groupsn(f) (n>1) of a pointed
mapf:X!Yconsist of the pointed homotopy classes of pairs
( pointed map:S
n1
!X, pointed null-homotopy:f' :S
n1
!Y) ,
designed to t into a long exact sequence
: : ://
n(X)
f
//
n(Y)
//
n(f)
//
n1(X)
//: : ://
1(Y):
2
For a pair of pointed spaces (Y; XY) the relative homotopy groups(f)
of the inclusionf:X!Yare denoted(Y; X).

HOMOTOPY 31
For any pointed mapf:X!Yan element (; )2n(f) can be represented
by a commutative diagram
S
n1

//

X
f

D
n

//
Y
with:S
n1
!X,:D
n
!Ypointed maps.
Denition 3.5Given a spaceXand a map:S
n1
!Xlet
Y=X[D
n
be the space obtained fromXbyattaching ann-cellat. 2
ACWcomplex is a space obtained from;by successively attaching cells of
non-decreasing dimension
X= (
[
D
0
)[(
[
D
1
)[(
[
D
2
)[: : : :
The images of the mapsD
n
!Xare called then-cells ofX.
Theorem of J.H.C.Whitehead 3.6 The following conditions on a mapf:
X!Yof connectedCWcomplexes are equivalent :
(i)fis a homotopy equivalence,
(ii)finduces isomorphismsf:(X)!(Y),
(iii)(f) = 0.
ProofSee Theorem VII.11.2 of Bredon [10]. 2
Denition 3.7Letn>1.
(i) A spaceXisn-connectedif it is connected and
i(X) = 0 (i6n):
(ii) A mapf:X!Yof connected spaces isn-connectediff:i(X)!i(Y)
is an isomorphism fori < nandf:n(X)!n(Y) is onto, or equivalently if
i(f) = 0 (i6n):
(iii) A pair of connected spaces (Y; XY) isn-connectedif the inclusion
f:X!Yisn-connected, or equivalently if
i(Y; X) = 0 (i6n): 2

32 HOMOTOPY AND HOMOLOGY
Example 3.8(i)S
n
is (n1)-connected.
(ii) (D
n
; S
n1
) is (n1)-connected. 2
Denition 3.9(i) Thesuspensionof a pointed spaceXis the pointed space
X=S
1
X=(S
1
fg [ f1g X);
with 2X, 12S
1
base points.
(ii) Thesuspension mapin the homotopy groups is dened by
E:m(X)!m+1(X) ; (f:S
m
!X)7!(f: (S
m
) =S
m+1
!X):
2
Freudenthal Suspension Theorem 3.10 IfXis an(n1)-connected space
for somen>2then the suspension mapE:m(X)!m+1(X)is an isomor-
phismm <2n1and a surjection form= 2n1.
ProofSee Whitehead [96, VII.7.13]. 2
Denition 3.11Thestable homotopy groups of spheresare

S
n= lim
!
k
n+k(S
k
) (n>0)
with
2n+2(S
n+2
) =2n+3(S
n+3
) =: : :=
S
n
by 3.10. 2
Remark 3.12(i) The stable homotopy groups
S
nare nite forn >0 (Serre).
(ii) The low-dimensional stable homotopy groups of spheres are given by :
n0123456789
S
nZZ2Z2Z2400Z2Z240(Z2)
2
(Z2)
3
2
3.2 Homology
This section summarises the aspects of (co)homology which are particularly im-
portant in keeping track of the eects of surgeries : the Universal Coecient
Theorem, relative groups, cup and cap product pairings

HOMOLOGY 33
[:H
m
(X)ZH
n
(X)!H
m+n
(X);
\:Hm(X)ZH
n
(X)!Hmn(X);
as well as the Steenrod squares
Sq
i
:H
r
(X;Z2)!H
r+i
(X;Z2) (i>0):
See Chapters 3,4 of Hatcher [31] and/or Chapter VI of Bredon [10] for more
detailed accounts.
Here are the basic denitions and properties of chain complexes, chain maps,
chain contractions etc.
LetAbe an associative ring with 1. AnA-moduleKis understood to have
a leftA-action
AK!K; (a; x)7!ax
unless a rightA-action is specied. IfAis a commutative ring there is no dier-
ence between left and rightA-modules.
A morphism of direct sums ofA-modules
f:K=K1K2: : :Kn!L=L1L2: : :Lm
is given by anmnmatrixf= (fij) with entriesfij2HomA(Kj; Li), such
that
f:K!L; (x1; x2; : : : ; xn)7!(
n
X
j=1
f1j(xj);
n
X
j=1
f2j(xj); : : : ;
n
X
j=1
fmj(xj)):
Denition 3.13(i) AnA-modulechain complexis a sequence ofA-module
morphisms
C:: : :
//
Ci+1
dC
//
Ci
dC
//Ci1
//: : :
such that (dC)
2
= 0. The chain complex isniteiffi2ZjCi6= 0gis nite.
(ii) Thehomologyof anA-module chain complexCis the collection ofA-
modules
Hi(C) =
ker(dC:Ci!Ci1)
im(dC:Ci+1!Ci)
(i2Z):
(iii) Achain mapf:C!Dis a sequence ofA-module morphismsf:Ci!Di
such that
dDf=fdC:Ci!Di1:
(iv) Achain homotopyg:f'f
0
between chain mapsf; f
0
:C!Dis a
sequence ofA-module morphismsg:Ci!Di+1such that
ff
0
=dDg+gdC:Ci!Di:

34 HOMOTOPY AND HOMOLOGY
(v) Achain equivalenceis a chain mapf:C!Dwith a chain homotopy
inverse, i.e. a chain mapf
0
:D!Cwith chain homotopies
g:f
0
f'1 :C!C ; g
0
:ff
0
'1 :C
0
!C
0
:
(vi) Achain contractionof anA-module chain complexCis a chain homotopy
: 0'1 :C!C :
(vii) Thealgebraic mapping coneof anA-module chain mapf:C!Dis
the chain complexC(f) with
d
C(f)=

dD(1)
i1
f
0 dC

:C(f)i=DiCi1!C(f)i1=Di1Ci2:
2
Proposition 3.14(i)A chain mapf:C!Dinduces morphisms in homology
f:H(C)!H(D)which depend only on the chain homotopy class off.
(ii)For any chain mapf:C!Dthe short exact sequence of chain complexes
0
//
D
//
C(f)
//C1
//
0
induces a long exact sequence of homologyA-modules
: : :
//
Hi(C)
f
//
Hi(D)
//
Hi(f)
//
Hi1(C)
//
: : :
with
Hi(f) =Hi(C(f)):
(iii)A chain mapf:C!Dis a chain equivalence if and only ifC(f)is chain
contractible.
(iv)A nite chain complexCof projectiveA-modules is chain contractible if and
only ifH(C) = 0.
(v)A chain mapf:C!Dof nite chain complexes of projectiveA-modules
is a chain equivalence if and only if the morphismsf:H(C)!H(D)are
isomorphisms.
Proof(i) By construction.
(ii) Every short exact sequence ofA-module chain complexes
0
//
D
//
D
0 //
D
00 //
0
induces a long exact sequence of homologyA-modules
: : ://
Hi(D)
//
Hi(D
0
)
//
Hi(D
00
)
//
Hi1(D)
//: : : :
(iii) Given a chain contraction : 0'1 :C(f)!C(f) letg; h; kbe the
morphisms dened by

HOMOLOGY 35
=

k ?
(1)
i
g h

:C(f)i=DiCi1!C(f)i+1=Di+1Ci:
Theng:D!Cis a chain homotopy inverse forf:C!D, with chain
homotopies
h:gf'1 :C!C ; k:fg'1 :D!D :
Conversely, iff:C!Dis a chain equivalence with chain homotopy inverse
g:D!Cand chain homotopiesh:gf'1,k:fg'1 then theA-module
morphisms
=

1 (1)
i+1
(fhkf)
0 1

k 0
(1)
i
g h

:
C(f)i=DiCi1!C(f)i+1=Di+1Ci
dene a chain contraction
: 0'1 :C(f)!C(f):
(iv) IfCis any contractible chain complex thenH(C) = 0.
Conversely, suppose thatCis a nite projectiveA-module chain complex
withH(C) = 0. Assume inductively that there existA-module morphisms :
Ci!Ci+1fori < ksuch that
dC + dC= 1 :Ci!Ci:
TheA-module morphism 1dC:Ck!Ckis such that
dC(1dC) = (1dCdC)dC= 0 :Ck!Ck1
so that
im(1dC:Ck!Ck)ker(dC:Ck!Ck1) = im(dC:Ck+1!Ck):
SinceCkis projective there exists anA-morphism :Ck!Ck+1such that
dC = 1dC:Ck!Ck;
giving the inductive step in the construction of a chain contraction : 0'1 :
C!C.
(v) A chain equivalencef:C!Dinduces isomorphismsf:H(C)!H(D).
For the converse apply (iv) to the algebraic mapping coneC(f). 2
Denition 3.15(i) Thedualof anA-moduleKis the rightA-module
K

= HomA(K; A)
withAacting on the right by

36 HOMOTOPY AND HOMOLOGY
K

A!K

; (f; a)7!(x7!f(x)a):
(ii) Thedualof anA-module morphismf:K!Lis the rightA-module
morphism
f

:L

!K

;g7!(x7!g(f(x))):
(iii) Thecohomologyof anA-module chain complexCis the collection of right
A-modules
H
i
(C) =
ker(d

C
:C
i
!C
i+1
)
im(d

C
:C
i1
!C
i
)
(i2Z)
whereC
i
= (Ci)

. 2
The homology and cohomology groups of a spaceXare dened using the
singular chain complexS(X), with
S(X)n= free abelian group generated by maps:
n
!X ;
d:S(X)n!S(X)n1;7!
nP
i=0
(1)
i
@i
using the standardn-simplices

n
=f(x0; x1; : : : ; xn)2R
n+1
j06xi61;
n
X
i=0
xi= 1g
and the inclusion maps
@i:
n1
!
n
; (x0; x1; : : : ; xn1)7!(x0; x1; : : : ; xi1;0; xi; : : : ; xn1):
The (singular)homologyandcohomologygroups ofXare dened by
Hn(X) =Hn(S(X))
= ker(d:Sn(X)!Sn1(X))=im(d:Sn+1(X)!Sn(X));
H
n
(X) =H
n
(S(X))
= ker(d

:S
n
(X)!S
n+1
(X))=im(d

:S
n1
(X)!S
n
(X))
withS
n
(X) = HomZ(Sn(X);Z). For any abelian groupGtheG-coecient sin-
gular homology groupsH(X;G) are dened using theG-coecient singular
chain complex
S(X;G) =GZS(X)
and theG-coecient singular cohomology is dened using
S
n
(X;G) = HomZ(Sn(X); G):
ForG=Zthese are justH(X;Z) =H(X),H

(X;Z) =H

(X).

HOMOLOGY 37
Example 3.16TheZ- andZ2-coecient homology and cohomology groups are
related by exact sequences
: : :!Hn(X)
2
//
Hn(X)!Hn(X;Z2)!Hn1(X)!: : : ;
: : :!H
n
(X)
2
//
H
n
(X)!H
n
(X;Z2)!H
n+1
(X)!: : : :
2
For a commutative ringR
S
n
(X;R) = HomR(Sn(X;R); R);
so thatS
n
(X;R) is theR-module dual ofSn(X;R) as in 3.15. TheR-coecient
homology and cohomology groupsH(X;R),H

(X;R) areR-modules which are
related by evaluation morphisms
H
n
(X;R)!HomR(Hn(X;R); R) ;f7!(x7!f(x)):
Given anR-moduleAletT AAbe the torsion submodule
T A=fx2Ajsx= 02Afor somes6= 02Rg:
Universal Coecient Theorem 3.17
(i) (F-coecient)For any eldFand anyn>0the evaluation morphism
e:H
n
(X;F)!HomF(Hn(X;F); F) ;f7!(x7!f(x))
is an isomorphism.
(ii) (Z-coecient)For anyn>0the evaluation morphism
e:H
n
(X)!HomZ(Hn(X);Z) ;f7!(x7!f(x))
is onto, and the morphism
ker(e) =T H
n
(X)!HomZ(T Hn1(X);Q=Z) ;f7!(x7!
f(y)
s
)
(f2S(X)
n
; x2S(X)n1; y2S(X)n; s6= 02Z; sx=dy)
is an isomorphism, so that there is dened a short exact sequence
0
//
HomZ(T Hn1(X);Q=Z)
//
H
n
(X)
e
//
HomZ(Hn(X);Z)
//
0:

38 HOMOTOPY AND HOMOLOGY
ProofThese results follow from the structure theorem for f.g.R-modules with
Ra principal ideal domain. Every f.g.R-moduleShas a presentation of the type
0
//
R
k
d
//
R
` //
S
//
0
with
d(0; : : : ;0;1;0; : : : ;0) = (0; : : : ;0; si;0; : : : ;0)2R
`
for somesi6= 02R(16i6k), andSis a direct sum of cyclicR-modules
S=R
`k

k
M
i=1
R=si:
More generally, every nite chain complexCof f.g. freeR-modules is isomorphic
to a direct sum of chain complexes of the type
E[n] :: : :!0!En+1!En=R!0!: : :(n2Z)
withEn+1=Ror 0. For anyn2Zthe evaluation map
e:H
n
(C)!HomR(Hn(C); R) ;f7!(x7!f(x))
is onto, with a naturalR-module isomorphism
ker(e) =T H
n
(C)!HomR(T Hn1(C); K=R) ;f7!(x7!
f(y)
s
)
(f2C
n
; x2Cn1; y2Cn; s6= 02R ; sx=dy)
whereKis the quotient eld ofR.
(i) IfR=Fis a eld thenT H
n
(C) = 0.
(ii) The ring of integersZis a principal ideal domain, with quotient eldQthe
rationals. 2
The homological properties of intersections of subspaces of a spaceXare
derived from the homological properties of the diagonal map
:X!XX;x7!(x; x);
using diagonal chain approximations :
Diagonal Chain Approximation Theorem 3.18 The singular chain com-
plexS(X)of any topological spaceXis equipped with a natural chain map
0:S(X)!S(X)ZS(X)
and natural higher chain homotopies
i:S(X)r!(S(X)ZS(X))r+i(i>1)

HOMOLOGY 39
such that
i+ (1)
i+1
Ti=d
S(X)ZS(X)i+1+ (1)
i
i+1d
S(X):
S(X)r!(S(X)ZS(X))r+i(i>0)
withTthe transposition automorphism
T:S(X)pZS(X)q!S(X)qZS(X)p;xy7!(1)
pq
yx
such thatT
2
= 1. Naturality means that for every mapf:X!Ythere is
dened a commutative square
S(X)r
i
//
f

(S(X)ZS(X))r+i
ff

S(Y)r
i
//
(S(Y)ZS(Y))r+i
ProofSee Chapter VI.16 of Bredon [10]. 2
Remark 3.19Diagonal chain approximations were rst constructed by Alexan-
der, Whitney and Steenrod in the 1930's using explicit formulae in simpli-
cial homology. The singular complex diagonal chain approximationsfigare
constructed by acyclic model theory. For any spacesX; Ythere is a natural
Eilenberg-Zilber chain equivalence
E0:S(XY)'S(X)ZS(Y)
with natural higher chain homotopies
Ei:S(XY)r!(S(X)ZS(Y))r+i(i>1)
such that
EiT+ (1)
i+1
T Ei=d
S(X)ZS(Y)Ei+1+ (1)
i
Ei+1d
S(XY):
S(XY)r!(S(X)ZS(Y))r+i(i>0)
with
T:XY!YX; (x; y)7!(y; x);
T:S(X)pZS(Y)q!S(Y)qZS(X)p;ab7!(1)
pq
ba
the transposition maps. The diagonal chain approximation is obtained by taking
X=Yand setting
i=Ei :S(X)r

//
S(XX)r
Ei
//
(S(X)ZS(X))r+i:

40 HOMOTOPY AND HOMOLOGY
LetWbe the standard freeZ[Z2]-module resolution ofZ
W:: : :
//
Z[Z2]
1T
//Z[Z2]
1 +T
//Z[Z2]
1T
//Z[Z2]:
The collectionfEigdenes a natural chain map
E:S(XX)!Hom
Z[Z2](W; S(X)ZS(X))
andfigdenes a natural chain map
:S(X)!Hom
Z[Z2](W; S(X)ZS(X)):
2
Denition 3.20(i) Thecup productpairing is
[:H
m
(X)ZH
n
(X)!H
m+n
(X) ;ab7!

a[b:x7!
X
a(x
0
)b(x
00
)

with 0(x) =
P
x
0
x
00
.
(ii) Thecap productpairing is
\:Hm(X)ZH
n
(X)!Hmn(X) ;xy7!x\y=
X
y(x
0
)x
00
:
(iii) TheSteenrod squaresare the cohomology operations
Sq
i
:H
r
(X;Z2)!H
r+i
(X;Z2) ;x7!(y7! hxx;ri(y)i) (y2Hr+i(X;Z2));
identifyingH

(X;Z2) = HomZ2
(H(X;Z2);Z2) by the Universal Coecient
Theorem 3.17. 2
The cup and cap product pairings are also dened forR-coecient (co)homology
[:H
m
(X;R)RH
n
(X;R)!H
m+n
(X;R);
\:Hm(X;R)RH
n
(X;R)!Hmn(X;R)
for any commutative ringR.
Denition 3.21(i) Therelative homology groupsof a mapf:X!Yare
the relative homology groups of the induced chain mapf:S(X)!S(Y)
H(f) =H(f:S(X)!S(Y));
designed to t into a long exact sequence
: : :!Hn(X)
f
!Hn(Y)!Hn(f)!Hn1(X)!: : : :

HOMOLOGY 41
Iff:X!Yis the inclusion of a subspaceXYthe relative homology groups
are written
H(f) =H(Y; X):
(ii) Thereduced homology groups of a pointed space (X; x2X) are the
relative homology groups of the inclusioni:fxg,!X
_
H(X) =H(i) =H(X;fxg)
with
H(X) =
_
H(X)H(fxg);
so that
_
Hr(X) =Hr(X) forr6= 0, andH0(X) =
_
H0(X)Z. 2
For a triple of spaces (X; YX; ZY) the relative homology groups of the
associated pairs t into a long exact sequence
: : ://
Hr(Y; Z)
//
Hr(X; Z)
//
Hr(X; Y)
@
//
Hr1(Y; Z)
//: : : :
Therelative cohomology groupsH

(f) of a mapf:X!Yare dened
to t into a long exact sequence
: : ://
H
n1
(X)

////
H
n
(f)
//
H
n
(Y)
f

//
H
n
(X)
//: : : :
The relative cohomology groupsH

(Y; X) (XY) and thereduced coho-
mology groups
_
H

(X) are dened by analogy withH(Y; X) and
_
H(X).
Denition 3.22(i) Themapping cylinderof a mapf:X!Yis the iden-
tication space
M(f) = (XI[Y)=f(x;1)f(x)jx2Xg;
which containsYas a deformation retract.
(ii) Themapping coneof a mapf:X!Yis the pointed space
C(f) =M(f)=f(x;0)(x
0
;0)jx; x
0
2Xg
with base point [X f0g]2C(f). 2
Example 3.23The space obtained fromXby attaching ann-cell at:S
n1
!
X(3.5) is a mapping cone
X[D
n
=C(:S
n1
!X);
and ts into a cobration sequence
S
n1
//
X //X[D
n //S
n //: : : :
2

42 HOMOTOPY AND HOMOLOGY
Proposition 3.24The relative homology groups of a mapf:X!Yare the
reduced homology groups of the mapping cone
H(f) =
_
H(C(f:X!Y)):
Similarly for relative cohomology.
ProofThe cobration sequence of spaces
X
f
//
Y //C(f) //X //Y //: : :
induces a long exact sequence of homology groups
: : :
//
Hn(X)
f
//
Hn(Y)
//_
Hn(C(f))
//
Hn1(X)
//
: : : :
2
Homology is homotopy invariant : a homotopyh:f'g:X!Yinduces a
chain homotopyh:f'g:S(X)!S(Y), so thatf=g:H(X)!H(Y).
Iff:X!Yis a homotopy equivalence thenf:S(X)!S(Y) is a chain
equivalence, andf:H(X)!H(Y) is an isomorphism.
Denition 3.25TheHurewicz mapfrom the homotopy to the homology
groups is
n(X)!Hn(X) ; (f:S
n
!X)7!f[S
n
]
with [S
n
] = 12Hn(S
n
) =Z. 2
Forn>2 there is also aZ[1(X)]-module Hurewicz map
n(X) =n(
e
X)!Hn(
e
X) ; (f:S
n
!X)7!
e
f[S
n
]
with
e
Xthe universal cover ofX, and
e
f:S
n
!
e
Xthe lift offwhich sends the
base point ofS
n
to the base point of
e
X.
For a mapf:X!Ythere are also Hurewicz maps(f)!H(
e
f) from
the relative homotopy groups to the relative homology groups.
Hurewicz Theorem 3.26 (i)For a connected spaceXthe map1(X)!
H1(X)is onto, with kernel the commutator subgroup[1(X); 1(X)]/ 1(X)
generated by the commutators[g; h] =ghg
1
h
1
(g; h21(X)).
(ii)Ifn>2andXis an(n1)-connected space thenn(X)!Hn(X)is an
isomorphism of abelian groups.
(iii)Ifn>2andf:X!Yis an(n1)-connected map thenn(f)!Hn(
e
f)
is an isomorphism ofZ[1(X)]-modules, with
e
f:
e
X!
e
Ya1(X)-equivariant
lift offto the universal covers
e
X;
e
YofX; Y.

Other documents randomly have
different content

Philosophers, not capable of transmuting, by a touch, whole tons of
grossest substance into solid, shining gold, but of making it
chemically. Then there was the Magic Crystal Ball, in which the gazer
could behold whatever he wished to, that was then transpiring on
this earth, or any of the planets. ‘All this knowledge,’ said he, ‘I will
expound to you, on certain conditions to be hereafter mentioned.’
“I relate these things in the briefest possible manner, and make no
allusions to my feelings during the time I listened to the strange
being, Ettelavar, further than to remark, that during the—temptation,
shall it be called?—I seemed to be hovering in the aërial expanse,
and realized a fullness and activity of life never realized before, and
knew for the first time what it was to be a human being. My freed
spirit soared away into the superincumbent ether, and far, very far,
beneath us rolled the great revolving globe; while far away in the
black inane, twinkled myriads of fiery sparks—the starry eyes of
God, looking through the tremendous vault of Heaven. Picture to
yourself a soul, quitting earth, perhaps forever, and hovering over it
like a gold-crested cloud, at set of sun, when all the winds are
hushed to sleep on the still and loving bosom of its protecting God,
and thine!
“By the exercise of a power to me unknown, Ettelavar arrested our
motion, and the cloud on which we seemed to float stood still in
mid-air, and he said to me, ‘Look and learn!’
“Like busy insects in the summer sun, afar off in the distance I
beheld large masses of human beings toiling wearily up a steep
ascent, over the summit of which there floated heavily, thick, dense,
murky, gloom-laden clouds. Crimson and red on their edges were
they, as if crowned with thunder, and their bowels overcharged with
lightnings; and their sombre shadows fell upon the plains below,
heavy and pall-like, even as shrouds on the limbs of beauty, or the
harsh critic’s sentence upon the first fruits of budding and aspiring
genius. ‘It is nothing but a crowd,’ said I; and the being at my side
repeated, as if in astonishment, ‘Nothing but a crowd? Boy, the
destinies of nations centre in a crowd. Witness Paris. Look again!’

Obeying mechanically, I did so, and soon beheld a strange
commotion among the people; and I heard a wail go up—a cry of
deep anguish—a sound heavily freighted with human woe and
agony. I shuddered.
“On the extreme apex of the mountain stood a colossal monument,
not an obelisk, but a sort of temple, perfect in its proportions, and
magnificent to the view. This edifice was surmounted by a large and
highly polished golden pyramid in miniature. On all of the faces of
this pyramid was inscribed the Latin word Feliciíaë; I asked for an
explanation from my guide, but instead of giving it, he placed his air-
like hand upon my head, and drawing it gently over my brow and
eyes, said, ‘Look!’
“Was there magic in his touch? It really seemed so, for it increased
my visual capacity fifty-fold, and on again turning to the earth
beneath me, I found my interest almost painfully excited by a real
drama there and then enacting. It was clearly apparent that the
great majority of the people were partially, if not wholly blind; and I
observed that one group, near the centre of the plain below the
mountain, appeared to be under much greater excitement than most
of the others, and their turbulence appeared to result from the
desire of each individual to reach a certain golden ball and staff
which lay on a cushion of crimson velvet within the splendid open-
sided monument on the mountain. In the midst of this lesser crowd,
energetically striving to reach the ascending path, was one man who
seemed to be endowed with far more strength and resolution—not
of body, but of purpose—than those immediately around him.
Bravely he urged his way toward the mountain’s top, and, after
almost incredible efforts, succeeded. Exultingly he approached the
temple, by his side were hundreds more; he outran them, entered,
reached forth to seize the ball and sceptre—it seemed that the
courageous man must certainly succeed—his fingers touched the
prize, a smile of triumph illumined his countenance, and then
suddenly went out in the blight of death, for he fell to the earth from
a deadly blow, dealt by one treacherous hand from behind, while
others seized and hurled him down the steep abyss upon which the

temple abutted, and he was first dashed to pieces and then trampled
out of existence by the iron heels of advancing thousands—men who
saw but pitied not, rather rejoicing that one rival less was in
existence.
“ ‘Is it possible,’ cried I, internally, ‘that such hell-broth of
vindictiveness boils in human veins?’
“ ‘Alas, thou seest!’ replied Ettelavar, by my side. ‘Learn a lesson,’
said he, ‘from what you have seen. Fame is a folly, not worth the
having when obtained. ‘Felicitas’ is ever ahead, never reached,
therefore not to be looked for. Friendship is an empty name, or
convenient cloak which men put on to enable them to rob with
greater facility. No man is content to see another rise, except when
such rising will assist his own elevation; and the man behind will
stab the man in front, if he stands in his way. Human nature is
infantile, childish, weak, passionate and desperately depraved, and
as a rule, they are the greater villains who assume the most
sanctity; they the most selfish who prate loudest of charity, faith and
love. I begin my tutelage by warning, therefore arming you, against
the world and those who constitute it. If you wish to truly rise, you
must first learn to put the world and what it contains at its proper
value. Remember, I who speak am Ettelavar. Awake!’
“Like the sudden black cloud in eastern seas, there came a darkness
before me; my eyes opened, and fell upon the old clock face. Its
hands told me that it was exactly thirteen minutes since I had
marked the hour on the dial. Since that hour I have had much
similar experience, and it is this that affords ground for the unusual
powers in certain respects, not claimed by, but attributed to me.” ...
Such was the substance of the young man’s narrative, in answer to
questions propounded to him long before the date at which he is
introduced to the reader.

CHAPTER V.
LOVE. EULAMPÉA
[2]
—THE BEAUTIFUL.
The golden sun was setting, and day was sinking beneath his
crimson coverlets in the glowing west. The birds, on thousand green
boughs, were singing the final chorus of the summer opera; the
lambs were skipping homeward in the very excess of joy; while the
cattle on the hills lowed and bellowed forth their thanksgiving to the
viewless Lord of Glory. Man alone seemed unconscious of his duty
and the blessings he enjoyed. Toil-weary farmers were slowly
plodding their way supper and bed-ward, and all nature seemed to
be preparing to enjoy her bath of rest. Still sat the wanderer by the
highway side; still fell his tears upon the grateful soil; and as the
journeyers home and tavern-ward passed him by, many were the
remarks they made upon him, careless whether he heard them or
not. Some in cruel, heartless mockery and derision, some few in pity,
and all in something akin to surprise, for men of his appearance
were rarely seen in that neighborhood. At last there came along
three persons, two of whom were unmistakably Indians, and the
third, a girl of such singular complexion, grace, form, and
extraordinary facial beauty, it was extremely difficult to
ethnologically define what she was. This girl was about fourteen; the
boy who accompanied her and the grey-haired old Indian by her
side, was apparently about twelve years old. This last was the first
to notice the stranger.
“Oh, Evlambéa,” said he, “see! there’s a man crying, and I’m going
to help him!” The boy spoke in his own vernacular, for he was a full
blood of the Oneida branch of the Mohawks, fearless, honorable,

quick, impulsive, and generous as sunlight itself. To see distress and
fly to its relief was but a single thing for him, and used to be with his
people until improved and “civilized” with bad morals and worse
protection. The Indian was Ki-ah-wah-nah (The Lenient and Brave)
chief of the Stockbridge section of the Mohawks. The girl, Evlambéa,
nominally passed for his grandchild, but such was not the case, for
although she might well be taken for a fourth blood, she really had
not a trace of Indian about her, further than the costume, language,
and general education and habit. Her name was modern Greek, or
Romaic, but her features and complexion no more resembled that of
the pretty dwellers on Prinkipo or the shores of the Bosphorus, than
that of the Indians or Anglo Saxon. Many years previous to that day,
this girl, then a child of three or four months age, had been brought
to the chief and left in his care for a week, by a woman clad in the
garb of, and belonging to a wandering band of gipsies, who,
attracted by the universal reputation of the New World, had left
Bohemia and crossed the seas to reap a golden harvest. This band
had held its headquarters for nearly a year on Cornhill, Utica,
whence they had deployed about the country in a circle whose
radius averaged one hundred and twenty miles. The woman never
came back to claim the child, for the members of the band suddenly
decamped after having financiered a gullible old farmer out of
several thousands of dollars in gold, which they had persuaded him
it was necessary that he should put in a bag and bury in the ground
at a certain hour of a certain night, in order to the speedy discovery
of a large mine of diamonds that was certainly upon his farm, and
would as surely be brought to light when the gold was exhumed
after a certain time, which time was quite long enough for the band
to dig up the gold and disperse in all directions, to meet again three
thousand miles away. This bit of Cornhill swindling was considered
rather sharp practice, even for that locality, and ended by shrouding
the girl in an impenetrable mystery, and giving to the old chief a
child, who, as she expanded and grew up became quite as dear to
his heart as any one of his own offspring; and in fact, by reason of
her superior intelligence, she became far more so, for mind ever
makes itself felt and admired. Not one of the ethnological, physical,

moral, or mental characteristics which mark the Romany tribes was
to be noticed in this girl, and wise people concluded that she had
somewhere been stolen by the woman, who from fear or policy had
left her to her fate and the good old Indian’s care.
Esthetics is not my forte, hence I shall not attempt to describe the
young girl. The name she bore was marked on her clothing in Greek
letters, which were afterwards rendered into English by a professor
of a college whose assistance had been asked by the Indian.
Besides being known far and near as the most beautiful girl of her
age, she was also distinguished as by far the most intelligent. She
was undisputed queen on the Reservation, not by right, but by quiet
usurpation. She looked and acted the born Empress, and her
triplicate sceptre consisted of kindness, intelligence, and that
nameless dignity and presence inherent in truly noble souls.
Such was the bright-shining maiden, who, attracted by the boy’s cry
and actions, now crossed over to the side of young Beverly.
Observing his sorrowful appearance, she placed her soft hand
tenderly upon his head, and said in tones heart-felt and deeply
sympathetic, “Man of the heavy heart, why weep you here? Is your
mother just dead?”
The young man raised his head, saw the radiant girl before him,
and, after a moment’s hesitation, during which he shuddered as if at
some painful memory, murmuring, “No; it cannot be possible!—
cannot be—in this part of the world, too! no!” he replied to her,
saying, “Girl, I am lonely, and that is why I weep. I am but a boy,
yet the weight of years of grief rest on and bear me down. To-day is
the anniversary of my mother’s death, and, when it comes, I always
pass it in tears and prayer. Since she went home to heaven, I have
had no true friend, and my lot and life are miserable indeed. Men
call themselves my friends, and prove it by robbing me. Not long
ago, there came a man to me—he was very rich—and said, ‘People
tell me that you are very skillful with the sick. Come; I have a sister
whom the physicians say must die. I love her. You are poor; I am
rich. Save her; gold shall be yours.’ I went. She was beyond the

reach of medicine, and it was possible to prolong her life only in one
of two ways—either by the transfusion of blood from my veins to her
own, or by the transfusion of life itself. I was young and strong, and
we resolved to adopt the latter alternative, as being the only
possibly effective one; and for months, during three years, I sat
beside that poor sick girl, and freely let her wasted frame draw its
very life by magnetically sapping my own. Finally, I began to sink
with exhaustion and disease similar to her own, and, to save my life,
was forced to break the magnetic cord, and go to Europe. As soon
as it was severed she sunk into the grave, and then I returned, and
received a considerable sum of money in the nature of a loan. This
favor was granted me as a reward for my pains, time, and ruined
health. I was to return it from the proceeds of a business to be
immediately established. At that time I resolved to purchase a little
home for those who depended on my efforts for the bread they ate,
and so wrote to a man who called himself my friend, but who is the
direct cause of most of the evil I have for ten years experienced.
This fellow pretended to deal in lands. I put nine hundred dollars—
half I had in the world—in this man’s hands, to purchase a fine little
place of a few acres, which place he took me to see. I was pleased
with it, and saw a home for those who would be left behind me
when I was dead. A few days thereafter this ghoul came to me
again, and represented that gold bullion being down he could make
considerable profit for me in three days, would I make the
investment. I handed over the remainder of my money. The three
days lengthened into years. Instead of being a capitalist he was a
bankrupt—was not in the gold business, and had no more control of
the land he showed me than he had of Victoria’s crown. Meantime,
my furniture was seized; I lost my name with the friend who
advanced the sum; I became ill, and, in my agony, called this man a
swindler. To silence me, he gave me a check on a bank. I presented
it. ‘No funds!’ And yet he dared call himself an honest man. ‘You
have but to unsay the harsh things said about me,’ said this
semblance of a man to me one day, ‘and I am ready to pay you
everything I owe.’ My mind was unsettled; I listened to him, and the
result was that, by duplicity and fraud, more mean and despicable

than the first, if there be a depth of villainy more profound, he
obtained my signature to an acknowledgment that the money of
which he had openly swindled me, then in his hands, was ‘a friendly
loan.’ And then he laughed, ‘Ha! ha!’ and he laughed, ‘Ho! ho!’ at me
and my misery, and actually suffered a child in our family to perish
and wretchedly die for the want of food and medicine. But then he
told me that he had buried it properly, respectably, up there in the
cemetery, and it was the only truth I ever heard from his lips. But
then he sent the funeral bills for me to pay—all the while laughing at
my misery—while the lordly house he occupied was redeemed from
forced sale with my money, and himself and his feasted luxuriously
every day on what was the price of my heart’s blood! Still, they all
laughed, ‘Ha! ha!’ and grew fat on my blood. I still have the memory
of a dead child, up there in the cemetery. Poor starved child! It is no
satisfaction to me to know that this man will die a disgraced pauper,
dependent on charity for bread. Still less is it to realize, as I do, that
the brothel and the gibbet, the gambling hell and massive prisons,
are shadowed in the foreground of his line, and that it will utterly
perish from off the earth in ignominy and horror. I would not have it
so, but fate is fate; and I see, at least, one dangling form of his race
swinging in the air! My prophetic eye beholds——”
As the man uttered these terrible sentences, he shuddered as if
horror-stricken at the impending fate of this wronger of the living
and the dead, and it was clear to the girl that he would have freely
averted the doom, had such a thing been possible.
“Men and cliques,” said he, “have used me for their purposes—have,
like this ghoul, wormed themselves into my confidence, and then,
when their ends were served, have ever abandoned me to
wretchedness and misery.
“Rosicrucians, and all other delvers in the mines of mystery, all
dealers with the dead, all whose idiosyncracies are toward the ideal,
the mystic and the sublime, are debtors to nature, and the price
they pay for power is groans, tears, breaking hearts, and a misery
that none but such doomed ones can either appreciate or

understand. Compensation is an inexorable law of being, nor can
there, by any possibility, be any evasion of it. The possession of
genius is a certificate of perpetual suffering.
“You now know why I am sad, O girl of the good heart. I am weak
to-night; to-morrow will bring strength again. But, see! the golden
sun is setting in the west. Alas! I fear that my sun is setting also for
a long, long night of wretchedness.”
“You speak well, man of the sore spirit,” replied the girl. “You speak
well when you say the sun is setting; but you seem to forget that it
will rise again, and shine as brightly as he does to-day! He will shine
even though dark clouds hide him from us; and though you and I
may not behold his glories, some one else will see his face, and feel
his blessed heat. Old men tell us that the darkest hour is just before
the break of day. I bid you take heart. You may be happy yet!”
“The precise formula of the Mysterious Brotherhood!—the very
words uttered by the dead mother who bore me! How did this girl
obtain it? When? Where? From whom?”
Beverly started, gazed into the mighty depths of her eye, was about
to ask the questions suggested, but forbore.
“We may all be happy yet,” said she; “for the Great Spirit tells me
so!” And she crossed her hands upon her virgin breast—breast
glowing with immortal fervor and inspiration; and she threw, by a
toss of the head, her long, black sea of hair behind her, and stood
revealed the perfect incarnation of faith and hope, as if her upturned
eyes met God’s glance from Heaven. The old chief and the boy at his
side said nothing, but each instinctively folded his hands in the
attitude of confidence and prayer. The combined effect of all this
upon the young man was electric. The singular incident struck him
so forcibly that he rose to his feet, placed his hand upon the girl’s
head, uplifted his eyes and voice to heaven, and, from the depths of
his soul, responded “Amen, and Amen.”
It was at this critical instant that I, the editor of these papers,
chanced to come up to where this scene was being enacted. A few

words sufficed for an introduction, and on that spot begun a
friendship between us all that death himself is powerless to break.
Two hours thereafter, the chief, his son, the girl, the youth, were,
with myself, partaking of a friendly meal at the old man’s house.
After the repast was over, the conversation took a philosophic turn,
in which the chief, who was a really splendid specimen of the
cultivated Indian, took an active and interested part.
Presently the old people took their pipes, the younger ones went to
bed, and Beverly and ’Levambea, as she was almost universally
called, walked out, and sat them down beneath an old sycamore
that stretched its giant limbs like the genius of protection over the
cottage. There they talked gaily enough at first, but presently in a
tender and pathetic strain; and it was clear that there had sprung up
between them already something much warmer than friendship, yet
which was not love. When they rose to enter the house, the last
words uttered by the girl—uttered in the same singularly inspired
strain observed on their first meeting—were, “Yes! I will love you;
but not here, not now, perhaps not on this earth. Yet I will be your
prop, your stay, though deep seas between us roll. Listen! When I
am in danger you will know it, wherever you may be. When you are
in danger you will see me. Forget not what I say. Ask me no
questions. Your fate is a singular one, but not more so than my own.
Good night! Good-bye! We will see each other no more at present—
it is not permitted!” And without another word she abruptly left him,
darted into the house, passed up the stairs, and was gone like a
spirit.
Next day, at the solicitation of the chief and others who took an
interest in young Beverly, he consented to go with me to my home,
many leagues from that spot; and, accordingly, in due time we
arrived there, and for several months he was an inmate of my
house; and, while under the shadow of ill health and its consequent
sympathetic state, I became intimate with many of the loftier and
profound secrets of the celebrated Rosicrucian fraternity, with which
he was familiar, and which he gave me liberty to divulge to a certain

extent, conditioned that I forbore to reveal the locality of the lodges
of the Dome, or indicate the persons or names of its chief officers,
albeit, no such restriction was exacted in reference to the lesser
temples of the order—covering the first three degrees in this country
—to the acolytes of which the higher lodges are totally unknown.
Oh! how often have I sat beside him, on the green banks of a creek
that ran through my little farm, and raptly listened to the
profoundest wisdom, the most exalted conceptions and descriptions
of the soul, its origin, nature, powers, and its destinies—listened to
metaphysical speculations that fairly racked my brain to
comprehend, and all this from the lips of a man totally incapable of
grappling successfully with the money-griping world of barter and of
trade. Here was the most tremendous contradiction, in one man,
that I had ever known or heard of. One who revelled in mental
luxuries fit for an angel, yet had not forecast enough to foil a
common trickster;—who blindly, and for years, reposed his whole
trust in one whose sole aim was to rob him not only of his little
competence, but of his character as a man—who suffered one near
and dear to him to starve, literally starve to death, and then be
buried, at the very moment that himself and his were luxuriating on
the very money for which that man had bartered health, and almost
life itself! Was it not very singular? I have wondered, time and again,
how such things could be, and intensely so when he has been
revealing to me some of the loftier mysteries of the Order; when
talking of Apollonius of Tyanæ, the Platonists, the elder
Pythagoreans; of the Sylphs, Salamanders and Glendoveers; of
Cardan, and Yung-tse-Soh, and the Cabalistic Light; of Hermes
Trismegistus, and the Smaragdine Tables; of sorcery and magic,
white and black; of the Labyrinth, and Divine policy; of the God, and
the republic of gods; of the truths and absurdities of the gold-
seeking Hermetists and pseudo-Rosicrucians; of Justin Martyr,
Tertullian, Cyprian, Lactantius, and the Alexandrine Clement; of
Origen and Macrobius, Josephus and Philo; of Enoch and the pre-
Adamite races; of Dambuk and Cekus, Psellus, Jamblichus, Plotinus
and Porphyrius, Paracelsus, and over seven hundred other mystical
authors.

Said he to me one day, “Do you remember laughing at me when I
first began to talk about the Rosicrucians? and you asserted that, if
such a fraternity existed, it must be composed either of knaves or
fools, laughing heartily when informed that the order ramified
extensively on both sides of the grave, and, on the other shore of
time, was known in its lower degrees as the Royal Order of the Foli,
and, towering infinitely beyond and above that, was the great Order
of the Neridii; and that whoever, actuated by proper motives, joined
the fraternity on this side of the grave, was not only assured of
protection, and a vast amount of essential knowledge imparted to
him here, but also of sharing a lot on the farther side of life,
compared to which all other destinies were insignificant and crude. I
repeat this assertion now.”
FOOTNOTE:
[2] Romaic—Ευλαμπια—Eulampía—Evlambéah. “Bright-shining.”—
Lovely, mystically beautiful.

CHAPTER VI.
NAPOLEON III. AND THE ROSICRUCIANS—AN
EXTRAORDINARY MAN AND AN EXTRAORDINARY
THEORY.
Beveêly continued his very singular narrative, saying:—“You have
already been informed of the singular doom that hangs over me—
that I am condemned to perpetual transmigrations, unless relieved
by a marriage with a woman in whom not one drop of the blood of
Adam circulates—and even then, the love must be perfect and
mutual. Thus my chance is about as one in three hundred and
ninety-six billions against, to a single one for me. This doom has
brought around me, as it did around others before me, certain
beings, powers, influences, and at length I became a voluntary
adept in the Rosicrucian mysteries and brotherhood. How, when, or
where I was found worthy of initiation, of course I am not at liberty
to tell; suffice it that I belong to the Order, and have been—by
renouncing certain things—admitted to the companionship of the
living, the dead, and those who never die; have been admitted to
the famous Derishavi-Laneh, and am familiar with the profoundest
secrets of the Fakie-Deeva Records; and through life have had ever
three great possibilities before me: one of these—I being a neutral
soul—is that of becoming after death a chief of a supreme order,
called the Light; or of its opposite, called the Shadow—to which I am
tempted by invisible, but potent agencies; and the third of which is
the one I dread most—the perpetuation of the doom to wander the
earth for ages, in various bodies, as the result of the curse
pronounced by a dying man ages ago, as you already have been

told, unless I be redeemed by a true marriage with a woman in
whom not one drop of the blood of Adam circulates. I desire to
avoid all three if possible, and to share the lot of other men.
“I have another mysterious thing to relate to you. Doubtless you
recollect that the curse was uttered by the young poet—and that the
mysterious voice heard in the dungeon where he was slain, declared
that thenceforth, until the doom was fully accomplished, this youth
during all his ages should be known as the Stranger. Well, in the
course of the centuries that rolled away, this Stranger became a
member of an august Fraternity in the Heavens, known as the Power
of the Light. You know, also, that I, who was the king, incurred the
penalty of wandering till relieved; and you are also aware that him
who was the Vizier was sentenced to a singular destiny under the
name of Dhoula Bel. Well, he also became an active member of a
vast Association in the Spaces, known as the Power of the Shadow.
This is but one half of the mystery, for it became the object of both
Dhoula Bel and the Stranger—who both knew that in my birth from
the woman Flora—years before I underwent my present incarnation
—that I would be in every respect a Neutral man; one having no
tendencies whatever, naturally, to either good or evil, but only
toward ATTAINMENT; and as such neutral man, it became possible to
forego my doom, and to become supreme chief of either of the
Orders named; hence both Dhoula Bel and the Stranger, beside their
original, have the strong additional motive of making me subservient
to their loftier views; and to achieve it, they frequently attend me in
visible and invisible shapes—tempting, nearly ruining, and as often
saving me from dangers worse than death itself—in what way has
already been partly told, and will be hereafter seen.
“In one of my frequent sojourns in Paris, I became acquainted with
a few reputed Rosicrucians, and after sounding their depths, found
the water very shallow, and very muddy—as had been the case with
those I met in London—Bulwer, Jennings, Wilson, Belfedt, Archer,
Socher, Corvaja, and other pretended adepts—like the Hitchcocks,
Kings, Scotts, and others of that ilk, on American soil. At length,
there came an invitation from Baron D——t, for me to attend, and

take part in, a Mesmeric Séance. I attended; and from the
reputation I gained on that occasion, but a few days elapsed ere I
was summoned to the Tuilleriés, by command of his majesty,
Napoleon III.,
[3]
who for thirty-four years had been a True
Rosicrucian, and whom I had before met at the same place, but on a
different errand than the present. What then and there transpired,
so far as myself was an actor, it is not for me to say, further than
that certain experiments in clairvoyance were regarded as very
successful, even for Paris, which is the centre of the Mesmeric world,
and where there are hundreds who will read you a book blindfold;
and two—Alexis, and Adolph Didiér—who will do the same, though
the page be inclosed in the centre of a dozen boxes of metal or
wood, one within the other.
“On this occasion I had played and conquered at both chess and
écarte, no word being spoken, the games simultaneous, and the
players in three separate rooms. There was present, also, an Italian
gentleman with an unpronounceable name; a Russian Count
Tsovinski, and a Madame Dablin—a mesmerist and operatic singer.
After awhile his majesty asked the empress, and the general
(Pellisier), who afterwards became the Duke de Malakoff, if they
would submit to a trial of mesmerism by either of the three
professors of the art, named. They declined; whereupon the
Emperor, speaking aloud, asked ‘if any of the company were willing
to test, in their own persons, the vaunted powers of his excellency,
the Italian Count?’ whose methods of inducing his magnetic marvels
differed altogether from those usually adopted; inasmuch as he, like
Boucicault, the actor, in his famous play—‘The Phantom’—makes no
passes, scarcely glances for an instant at his subjects, and invariably
looks away from, not toward, them. Now, it is a well-known fact that
everybody believes everybody else, save themselves, subject to
mesmeric influence, as is often demonstrated at the weekly séances
of the Magnetic Society, held in the Rue Grenelle St. Honore.
“At the date of this Imperial Séance, spiritualism had not yet made
public pretensions in France, and although the Scotch trickster,

Daniel Hume, had crossed the Atlantic, and was at that time living at
Cox’s, in Jermyn street, Picadilly, London—yet he had not then
obtained the notoriety that subsequently became his, nor had half
Europe ran after those in whose presence tables tipped by heel, toe,
and genuine spirit power. Of course, then, spiritual phenomena, so
called, being then under bann, it could not be, and was not
depended on as a means of explaining what there and then took
place.
“ ‘With great pleasure,’ said the Count, in reply to a request to exhibit
his power. ‘With great pleasure, your majesty,’ and forthwith he
turned and looked straight into a massive mirror that occupied the
entire space between two windows of the saloon. As he spoke it
struck me that, somewhere, at some time, I had met this Italian
Rosicrucian, but where, for the life of me, I could not tell; yet I was
certain that I had heard that voice, and still more certain that I had
beheld that strange, sweet smile.
“The Count’s position before the mirror was such that, supposing his
eye had been a flame, the reflected rays would strike the forehead
of one of the company fairly in the centre. The person upon whom it
struck had not the least suspicion of what was being done. He did
not make the discovery until it was too late, for no sooner did the
operator get him fairly in focus, then he clenched his hands, looked
with ten-fold earnestness at the mirror, muttered to himself a few
unintelligible words, and the gentleman fell to the floor as if his
heart had been perforated by a bullet, or as if he had been struck
down with a club. In an instant all was confusion, everybody
thinking it a fit of apoplexy, except the Emperor, the operator, myself
and the Russian.
“Several went to raise him, but before they could do so he sprung to
his feet, began to sing and dance—the truth, at the same time,
flashed upon the company, that the phenomenon was mesmeric—
and in another minute to plead for his life, as if before his judges,
with the prison and the axe before him. The scene was solemn to
the last degree.

“Suddenly, and without a word from the Count, the pleading
changed to a musical scena; and although, at other times totally
incapable of singing or playing in the least degree, he performed
several difficult pieces in magnificent style, on the harp and piano,
accompanying the performances vocally, and in a manner that drew
involuntary plaudits from every person present.
“This part of the performance was suddenly terminated; for the
sleeping subject placed himself in the exact spot in which the Italian
had stood, and, like him, gazed steadily at the mirror, and in twenty
seconds the man who stood in the line of reflection fell to the floor,
and a lady who, in going to his assistance, chanced to strike that
line, instantly seized, raised him as easily as if he had been a doll,
and with him commenced a dance unique, wild and perfectly
indescribable. It was infectious, for in less than half a minute
seventeen persons, high lords and stately dames, were wheeling,
whirling, leaping, flying about the room in wilder measures than
were ever performed by mad Bachantes. They had all been
magnetized by proxy.
“Astonished beyond measure at this extraordinary display, I retired,
the better to watch the progress of the strange scene, to the
opposite side of the saloon, and leaned carelessly against one of two
colossal Japanese josses that stood there. No person was anywhere
near me, and in my surprise I murmured below my breath: ‘What
astonishing power!’ and am certain that a person standing close at
my side could not have discerned what I said, yet nevertheless the
thought was scarcely framed before the Count turned square upon
his heel, advanced straight toward me, smiled sweetly, strangely, as
he did so, and said: ‘All this power is yours—and much that is still
more mysterious—if you but say the word!’
“ ‘What word?’ asked I, surprised that a man should so readily read
my thought—for it is impossible that he could have heard my
exclamation.
“ ‘That you will voluntarily join the most august fraternity that ever
earth contained! Think of it! We shall meet again.’

“ ‘When? where?’ I asked hurriedly, for the august company were
observing us, especially the Emperor, who, beneath his heavy brows,
was evidently paying quite as much attention to us as to the
wonderful things then occurring across the room.
“He did not reply directly, but, by a continuation of his breach of
etiquette resumed, saying: ‘By the exercise of the power I possess,
and will impart to you, conditionally; you shall be capable of
depriving any man of speech, and make man, woman or child
perfectly subservient to your silent command, as the people yonder
are to mine. There is Jean Boyard, in this Paris, who merely looks at
any small object, and makes it dance toward him. You shall exceed
him fifty-fold! On the Boulevart du Temple M. Hector produces a full-
blown rose from a green bud, in seven minutes; you shall be able to
do it in one.
“ ‘In the Rue de Bruxelles lives a girl—Julie Vimart—who exceeds
Alexis and all the other sleepers, for she beats you at chess, tells
you all you know, and much that you have forgotten; you shall do all
that and more. In the street Grand Père, lives a boy who brings
messages from the living, in their sleep; meets and converses with
your friends—when they slumber, and describes them as perfectly as
the sun paint their portraits in the cameras of Talbot and Dagguerre;
you shall have that power.
“ ‘In the Rue du Jour, is a Sage Femme, who cures all diseases that
are curable, by a simple touch and prayer: you shall have that power
greater than she can ever hope to. It is only necessary to say ‘I will
have these powers!’ and they shall be yours. They all are well worth
having. I learned my secret among the magi of the East—men not
half so civilized as are we of the West; but who, nevertheless, know
a great deal more than the sapient men of Christendom—that is,
less of machinery, politics, and finance; but a great deal more of the
human soul, its nature, its powers, and the methods of their
developement. Instead of being surprised at modern scientific
revelations, we of the True Temple——’ ‘What Temple?’ I interrupted
him to ask. ‘Of the Supreme Dome of the Rosie Cross,’ said he.

“The Emperor must have heard this question and its answer, for he
directly crossed over to us, and actually joined this curious tête-à-
tête. The Count bowed; did not seem at all embarrassed by the
presence of the son of Admiral Verhuiel, the great Dutch founder of
the Second Empire—or Emperor ——.
“ ‘As I was saying,’ the Count resumed, ‘instead of being elated at
what Western science has done, we are ashamed of the tardy steps
of “Progress”—Progress indeed! Where is it, save in wretchedness,
poverty, crime, selfishness, and in the accrement of misery. Progress
is more fancied than real. Civilization is a misnomer, utilitarianism a
desecration of man’s soul, Philosophy an imposture, and learning
altogether false!’
“I was pleased to see the Emperor join the conversation at this
point, for two reasons: first, to hear what he had to say; and
secondly, to observe whether the subjects on the floor could be kept
under the Count’s influence while his mind was abstracted from
them and centered on matters entirely different.
“ ‘Do not be disturbed at what he says,’ said his majesty, ‘for these
Mesmerists are all slightly mad.’ And he smiled, while the Count
shrugged his shoulders, and exclaimed:
“ ‘With a method, however!’
“Then turning his attention toward the company, by some
inscrutable power he stopped the dance, restored the subjects to
their normal state, and almost instantly thereafter exercised it upon
Madame Dablin, who straightway, with closed eyes, approached a
grand piano, swept its keys with matchless skill, as a prelude, and
then launched forth into one of the strangest, most brilliant, yet wild
and weird fantasias, that genius ever dreamed of. I cannot now stop
to describe its effect upon the company, nor upon myself, for my
whole being was absorbed at that moment in matters far more
important to me than a mesmeric experiment, however interesting
and successful it might be; for at best, its effect and memory would
be transient and ephemeral, while, on the contrary, the things I

might learn from the Italian might last so long as my conscious soul
endured. I was not, therefore, disappointed when he resumed his
talk. I cannot now repeat the ipsissima verba of what he said, but
the substance, in reply to questions by the Emperor and myself, was
in effect this:
“ ‘The soul and its qualities, passions and volume are all clearly
marked upon the physique, and are apparent to all who possess the
proper key; to all others, the difficulty lies in correctly reading these
signs, and a still greater in assigning to each faculty its actual, its
possible, and its relative strength and value. Every act that a man
does has an effect upon both his body and soul, and the imprints
thereof are indelibly stamped upon his features; therefore his past—
even his most secret act or thought—can be read by the adept with
as much ease as if his face were a printed page, the type being
large, smooth and clear. Every man is susceptible of being controlled
mesmerically by another, because no man is collectively stronger
than his weakest faculty; a chain is no stronger than its most
defective link. Now I control men because I know at a glance which
is the most vulnerable portion of their nature. Self-love, Emulation
and Will are the trinity in unity around which the Psychal Republic
revolves. One of these is always vulnerable; subdue that, and you
subdue the man. Now, when I perform such experiments as those
now being exhibited, I first mesmerize, not the entire brain, but a
single faculty, which in turn speedily subdues all the rest. The mind
of man is a mirror! Conceded. Well, then, I forthwith, by an effort of
will, entirely vacate my own mind, thinking of nothing but a
revolving wheel. The subject reflects my action; then in fancy I sing,
dance, play, and the subject reflects my thought by appropriate
action.’
“ ‘But,’ said one, ‘suppose your subject understands nothing about
these accomplishments. How then?’
“ ‘All souls understand them. Bodies may not; and I bring the soul
under subjection, not the body merely.’

“ ‘This is a dangerous power to possess,’ said the Emperor, ‘and none
but a good man ought to have it.’
“ ‘A bad man cannot become a true Rosicrucian, although men have
turned their arms against the race, and the secrets of the fraternity,
like all things else, have been trifled with and abused. Thus it is
possible for an expert to cure a diseased man by the exercise of the
power alluded to. But the rule is dual: it is also possible to kill a
healthy man by the same mysterious means; and indeed it has often
been done, especially by the natives of Africa.
“ ‘I persuade my soul that you are sick and will die, and if I keep up
the will and wish, nothing is more certain than that both will be
accomplished. Some men naturally possess enormous powers of will,
and are able to project visible images, like those of a
phantasmagoria—images of whatever they choose to fancy—a
flower, a hand, arm, or a human form—and these spectra will be
visible to scores of startled observers, who, in their utter ignorance
of the human mind and body, and their respective and conjoined
powers, believe them to be the veritable ghosts of dead men, and
objects produced by them. I learned recently that in London is at
this moment a young Scotchman, named Hume, who possesses this
power to a remarkable degree, and also that of levitation, and who
is coining fame and fortune by pretending that the psychical
phenomenon is really and truly spiritual—which is not the case. I
learned this great secret in the Punjaub, of Naumsavi Chitty, the
chief of the Rosicrucians of India, and the greatest reformer since
Budha.’
“At this point the Emperor asked the Count to exhibit a specimen of
his spectre-producing power, to which the latter assented. First he
walked rapidly several times up and down the saloon, gave
directions to lower the lights, which was done, and then, as before,
he stood still directly in front of the mirror for a minute or two, and
then, in a sharp, cracked tone, repeated thrice the word ‘Look!’ We
did so, and as I live, there flashed the semblance of a thousand
chains of vivid lightning across the face of the mirror, along the floor,

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Algebraic And Geometric Surgery Andrew Ranicki

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