Classification of Surfaces(Geometry).pdf

Classification of Surfaces, a presentation on an introduction to algebraic geometry

1. CLASSIFICATION OF SURFACES
By
FAROMINIYI Ayodeji Simeon
Matric No: 192302
Department of Mathematics
University of Ibadan
Supervisor:DR. H.P. ADEYEMO
June 27, 2023
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

2. Outline
Introduction
Topology of surfaces
Classification of 2-manifolds (surfaces)
Euler Characteristics and its applications
Conclusion
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

3. Introduction
In Mathematics, a surface is a generalization of a plane which
does not need to be flat- that is, the curvature is not necessar-
ily zero. This is analogous to a curve generalizing a straight line.
There are several more precise definitions, depending on the con-
text and the mathematical tools that are used for the study.
Surfaces may be considered in different fields of mathematics, in-
cluding Topology, Differential Geometry, Algebra and so on. But
for this presentation, consideration is given to Topological surfaces
where the classification of compact connected surfaces is discussed
as well as concepts like Euler characteristics and its applications.
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

4. Topology
Topology is the abstraction of certain geometrical ideas, such as
continuity and closeness. Roughly speaking, topology is the ex-
ploration of manifolds and of the properties that remain invariant
under continuous invertible transformation, known as homeomor-
phisms.
Definition
a. Xis a subsetTofP(X) subject to the
following requirements:
i.ϕ,X∈ T
ii. U,V∈ T, thenU∩V∈ T
iii. Iis a set and{Ui:i∈I} ⊆ T, then∪i∈IUi∈ T
The elements ofTare called the open sets ofX(with respect
to the topologyT).
b. X,T) whereXis a set andT
is a topology onX. Elements ofXare called points in the
topological space (X,T)
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5. Basic Definitions
Definition
Let (X,T) be a topological space, a subspaceVofXis said to
be closed inXifX\Vis open inX.
Definition
Letf:X−→Ybe a map of topological spaces (X,Tx) and
(Y,TY).fis said to be continuous ifU∈ TY=⇒f
−1
(U)∈ TX
i.e.fis continuous if the pullback of any open setTYis open in
TX.
Definition
Given spaces (X,TX), (Y,TY), and a functionf:X−→Y.fis
said to be open if it sends open sets inXto open sets inY. i.e. if
U∈ TX=⇒f(U)∈ TX, thenfis open.
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

6. Homeomorphism
Definition
A homeomorphism between topological spacesXandYis a
bijective mapf:X−→Ysuch thatfand its inverse function
f
−1
are both continuous and open.
Remark
A homeomorphism is a one-one correspondence that preserves all
the structure that exists in a topological space, namely, the open
sets. Spaces are said to be homeomorphic or just equivalent if a
homeomorphism exists between them. This map is simply a
deformation of the first space to get the second which explains
why the structure is preserved. Properties of spaces which are
preserved under all homeomorphisms are called topological
invariants. They include all cardinal invariant properties (e.g
finiteness, countability), Separability and Density properties.
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

7. Homeomorphism
Example
The letters of the English Alphabet can be grouped into
homeomorphism classes considering them as thin abstract
topological spaces.
Figure:
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8. Compact and Connected Spaces
Definition: Connected Space
A partition{A,B}of a topological spaceXis a pair of non-empty
subsets ofX, such thatX=A∪B,A∩B=ϕand bothAandB
are open inX. A topological space is connected if and only if it
admits no partition.
Definition: Compact Spaces
A topological spaceXis compact if every open cover ofX
contains a finite subcover. An open cover is a collection{Ui}
where{Ui} ∈ Tis open for alliand
S
{Ui}=X.
A subcover is a subset of{Ui}whose union still is the whole ofX.
A finite subcover is a subcover containing a finite number of
subsets.
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

9. Hausdorff Spaces
Definition
A topological space (X,T) is Hausdorff if for any pointsu,v∈X,
there exist open subsetsU,V∈ T, such thatU∋uandV∋v
andU∩V=ϕ.
To paraphrase, any two points can be separated by open sets.
Remark
Almost all spaces encountered in analysis are Hausdorff; most
importantly, the real numbers (under the standard metric topology
on real numbers) are a Hausdorff space. More generally, all metric
spaces are Hausdorff. In fact, many spaces of use in analysis, such
as topological groups and topological manifolds, have the
Hausdorff condition explicitly stated in their definitions.
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

10. Topology of Surfaces
Surfaces are some of the simplest, yet most interesting topological
objects. They (surfaces) belong to a class of topological objects
that is formed by manifolds.
Loosely speaking, a surface is a topological space with the property
that around every point, there is an open subset that is homeomor-
phic to an open disc in the plane (the interior of a circle). We say
that a surface islocally Euclidean.
Informally, two surfacesX1andX2are equivalent if each one can
be continuously deformed into the other. More precisely, this means
that there is a homeomorphism. So, two surfaces are considered to
be equivalent if there is a homeomorphism between them.
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

11. Manifolds and surfaces
Definition
1. n-dimensional manifold is Hansdorff topological space
such that every point has a neighbourhood topologically
equivalent (homeomorphic) to an n-dimensional open disc
with centrexand radiusr. A 2-manifold is often called a
surface and a 1-manifold is a curve.
2. n-manifold with boundary is a topological space such that
every point has a neighbourhood homeomorphic to either an
open n-dimensional disc or the half-disc. Points with half-disc
neighbourhoods are called boundary point.
Figure:
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

12. Complexes, and Triangulations
Triangulations are a fundamental tool to obtain a deep un-
derstanding of the topology of surfaces. Roughly speaking, a
triangulation of a surface is a way of cutting up the surface
into triangular regions such that these triangles are the images
of triangles in the plane. In other words, Informally, a triangu-
lation is a collection of triangles satisfying certain adjacency
conditions.
Intuitively, a surface can be triangulated if it is homeomorphic
to a space obtained by pasting triangles together along edges.
A technical way to achieve this is to define the combinato-
rial notion of a two-dimensional complex, a formalization of a
polyhedron with triangular faces.
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13. Definition
A cell is a space whose interior is homeomorphic to the unit
n-dimensional ball of the Euclidean spaceR
n
. The boundary of a
cell must be composed of a finite number of lower-dimensional
cells, where 0, 1, 2, 3 - dimensional cells are as defined below:
(1)
(2)
at two 0-dimensional cells.
(3)
example, the quadrilateral defined by the line segments AB,
BC, CD, and DA.
(4)
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

14. Definition
A complexKis a finite set of cells with the following properties:
(1) Kare also cells ofK.
(2) σandτare cells inK, thenint(σ)∩int(τ) =ϕ. A
complex where all cells have dimension at most n is an
n-complex. For K a complex, we write|K|to denote the
geometry resulting from gluing cells together.
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

15. Definition: Planar diagrams
A planar diagram is a polygon with 2nedges where pairs of edges
are identified with either the same or opposite orientation. Given
the quotient topology, the labelled edges are identified or glued
together. Planar diagram code the gluing instructions needed for
the construction of 2-manifolds.
Example
The following are compact, connected surfaces along with planar
diagram representations:
1.S
2
, the sphere inR
3
represented by circle inR
2
with the
upper hemisphere edge identified with the lower hemisphere
edge.
2.T
2
, the torus inR
3
represented by the two pairs of opposite
edges identified together.
3.P
2
, the projective plane inR
4
represented by circle inR
2
with
the upper hemisphere edge identified opposingly with the
lower hemisphere edge
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

16. Planar diagrams
Figure:
surfaces, the sphere, the torus, the projective plane. The cylinder,
Mobius band, and Klein bottle are also included.
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

17. Remark
The neighbourhoods of points on the planar diagram can be used
to check whether a space is a surface as illustrated in the following
theorem;
Figure:
vertex point of a planar diagram.
Theorem
The topological space represented by a planar diagram is a
compact, connected surface.
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

18. Orientable surfaces
An important decision one must make when seeking to decide how
a surface fits into the classification of surfaces is whether or not
the surface is orientable.
Definition
A topological surface is said to be orientable if it does not contain
any subset homeomorphic to the Mobius band. In other words, an
orientable surface has 2 distinct sides.
Remark
Both the Klein bottleKand the projection planePare
non-orientable (because the real projection plane with one
point removed is homeomorphic to the open Mobius strip).
The sphere and the torus are orientable.
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

19. Connected Sums
Definition
The connected sum of two surfacesMandN, denoted byM#N,
is the surface obtained by removing a disk from each of them and
gluing them along the boundary components that results. The
boundary of a disk is a circle, so these boundary components are
circles.
Figure:
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

20. Remark
i.
meaning thatS#M=M.
ii. Tis also described as
attaching a “handle” to the other summandM. IfMis
orientable, then so isT#M.
iii.
P#P, is the Klein bottleK. Any connected sum involving a
real projective plane is non-orientable.
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

21. Classification of surfaces
In this section, all compact(Combinatorial) surfaces without bound-
ary are completely categorized up to topological equivalence.
Theorem
Every compact connected surface is homeomorphic to the sphere,
a connected sum of Tori, or a connected sum of projective planes.
That is, every closed surface is one of the following:
i.S
2
ii.T
2
#T
2
#· · ·#T
2
(nT
2
)
iii.RP
2
#RP
2
#· · ·#RP
2
(nP
2
)
Remark
The surfaces in the first two families are orientable and it is
convenient to combine the two families by regarding the sphere as
the connected sum of 0 tori. However, the surfaces in the third
family are non-orientable.
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

22. The Euler characteristic of 2-manifolds (surfaces)
The Euler characteristic is a significant topological invariant (a
property of a topological space which is invariant/preserved under
homeomorphisms) which is commonly dented by X.
Definition
A subdivision of a compact surfaceXis a partition ofXinto
finitely any cells of dimension 0,1 or 2, where ani-cell is a subset
homeomorphic to thei-discD
i
={x∈R
i
:∥x∥<1}.R
0
=D
0
is
a point.D
1
= (−1,1). So 0-cells are points or vertices. 1-cells are
edges. Each 1-cell must have a 0-cell at each of its end points.
The 2-cells are called faces.
Theorem
Every surface has a subdivision.
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

23. Euler Characteristic
Definition
Given a subdivision of a surfaceS, letVbe the number of
vertices,Et he number of edges andFthe number of faces. The
Euler characteristic ofSwith the given subdivision is given by
χ(S) =V−E+F.
Theorem
The Euler characteristic is a topological invariant for compact
connected surfaces and so does not depend on the representation.
In other words, if|K|and|L|are compact, connected surfaces and
|K|=|L|, thenχ(K) =χ(L)
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

24. Example
i.
χ(S
2
) = 2−1 + 1 = 2
ii.
χ(T) = 1−2 + 1 = 0
iii.
χ(P
2
) = 1−1 + 1 = 1
iv. V= 1,E= 2,F= 1,thus
χ(Klein) = 1−2 + 1 = 0
v. V= 2,E= 1,F= 1,thus
χ(Mobius) = 2−1 + 1 = 2
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

25. Figure:
bottle and Mobius band.
Remark
i.
i.χ(S
2
) = 2
ii.χ(nT
2
) = 2−2n
iii.χ(nP
2
) = 2−n
2.
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

26. Genus
The genus of a surface is a related invariant which counts the
number of tori or handles in a surface for the orientable case,
and the number of twisted pairs or projective planes for the non-
orientable case.
Definition
LetSbe a compact surface. The genus ofSis:
g(S) =



1
2
(2−χ) if S is orientable
2−χ if S is non-orientable
Remark
The Klein bottle is thus referred to as the non-orientable surface
with genus 2, andn-handles torus as the orientable surface with
genusn. The genus provides no additional information but is a
way of converting the Euler characteristic into a number which is
directly related to the physical properties of the surfaces.
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

27. Application of the Euler Characteristic
Polyhedra
The Euler characteristic was originally defined for Polyhedra and
used to prove various theorems about them including the clas-
sification of the platonic solids. Generally, a polyhedron (plural:
polyhedra) is a solid or surface in three dimensions that can be
described by its vertices (corner points), edges (line segments con-
necting pair of vertices), faces (two dimensional polygons).
Definition
A regular polyhedron is a polyhedron whose faces all have the
same number of sides, and which also has the same number of
faces meeting at each vertex.
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

28. Remark
The regular Polyhedra which are topologically equivalent to the
sphere are known as the platonic solids and have been studied for
over 2000 years. They are named after the ancient Greek philoso-
pher Plato. The platonic solids include:
1.
vertex.
2.
at each vertex.
3.
vertex.
4.
vertex.
5.
each vertex.
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

29. Platonic Solids
Theorem
The platonic solids are the only regular Polyhedra topologically
equivalent to a sphere. In other words, there are only 5 platonic
solids.
Figure:
Note:
Characteristic
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

30. The Soccer Ball
It is common to construct soccer balls by stitching together pen-
tagonal and hexagonal pieces, with three pieces meeting at each
vertex and with every edge belonging to two polygons. It can be
shown that a soccer ball constructed in this way always has twelve
pentagons but with an unconstrained number of hexagons.
soccerball1.png
Figure:
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

31. Proof
Let the surface of the soccer ball be denoted byS. IfPpentagons
andHhexagons are used, then there areF=P+Hfaces,V=
(5P+ 6H)
3
vertices andE=
5P+ 6H
2
Then because the soccer ball is a sphere whose Euler characteristic
is 2, we have that:
χ(S) =V−E+F=
5P+ 6h
3
−
5P+ 6h
2
+P+H= 2
⇒
`
5P
3
−
5P
2
+P
´
+
`
6H
3
−
6H
2
+H
´
= 2
⇒
`
5P
3
−
5P
2
+P
´
+ 0 =
10P−15P+ 6P
6
= 2
⇒
P
6
= 2,P= 12
That is, a soccer ball constructed in this way always has 12 pen-
tagons.
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

32. Remark
This result is applicable in chemistry to fullerenes (an allotrope of
carbon with a closed mesh topology which are informally denoted
by by their empirical formulaCn, wherenis the number of carbon
atoms). The closed fullerenes, especiallyC60, are also informally
called buckyballs for their resemblance to the standard soccer ball.
Figure: C60fullerene (buckminsterfullerine)
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

33. Conclusion
Algebraically, surfaces are defined using varieties, whether affine
varieties or projective varieties which are just the zero locus for a
set of polynomials. Quadric surfaces which are algebraic hypersur-
faces are classified based on invariants such as rank and determi-
nants of the associated matrices of the defining equation.
Finally, we have seen that all compact connected surfaces can be
classified into three main groups: A sphere, a connected sum of
Tori(handles), or a connected sum of projective planes(cross caps).
To further classify these topological surfaces, we saw the impor-
tance of other topological invariants such as the Euler Characteris-
tics and the Genus and how they are applicable in different aspects
of day to day lives.
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

34. References
W. Fulton, Algebraic Topology, A first course, Graduate Texts
in Mathematics, vol. 153, 1st edn. (Springer, New York,
1995)
L. Christine Kinsey, Topology of Surfaces, Undergraduate
Texts in Mathematics, 1st edn. (Springer, New York, 1993)
W.S. Massey, Algebraic Topology: An Introduction, Graduate
Texts in Mathematics, vol. 56, 2nd edn. (Springer, New York,
1987)
J.R. Munkres, Topology, 2nd edn. (Prentice Hall, New Jersey,
2000)
J. Gallier and D. Xu, A Guide to the Classification Theorem
for Compact Surfaces, Geometry and Computing 9, DOI
10.1007/978-3-642-34364-3 3,©Springer-Verlag Berlin
Heidelberg 2013
M. Henle, A Combinatorial Introduction to Topology, 1st edn.
(Dover, New York, 1994)
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By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES

35. THANK YOU FOR LISTENING!
By FAROMINIYI Ayodeji Simeon Matric No: 192302 CLASSIFICATION OF SURFACES