Introduction to Algebraic Topology

Mathew GeorgeU.G. 4th year

Indian Institute of Science

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Contents

0.1 Homotopic functions . . . . . . . . . . . . . . . . . . . . . . . . . 30.2 Fundamental group . . . . . . . . . . . . . . . . . . . . . . . . . . 30.3 Homotopy Type . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.4 Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 50.5 The Brouwer fixed-point theorem . . . . . . . . . . . . . . . . . . 5

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Introduction

Algebraic topology is a branch of topology which uses algebraic objects andmethods to prove topological results. The idea stems from the observationthat there are algebraic objects associated with each topological space whichare invariant under homeomorphisms of the space. So, a knowledge of theseassociated objects can be used to classify topological spaces.

0.1 Homotopic functions

Def : Two functions f and g from X to Y are said to be homotopic if thereexists a continuous function F : X × [0, 1]→ Y such that,

F [x, 0] = f(x)

F [x, 1] = g(x)

The homotopy is relative to a subspace A ,where f and g agree, if in additionF [x, t] = f(x) = g(x) for all x ∈ A. Two continuous functions f and g on [0, 1]which are homotopic relative to the set {0, 1} will be referred to as homotopiccurves (f'g). If f and g are two homotopic curves with f(1) = g(0) then,

f.g =

{f(2t) , 0 6 t 6 1

2

g(2t− 1) , 12 6 t 6 1

It can be easily verified that homotopy is an equivalence relation.

0.2 Fundamental group

The set of all closed curves on a topological space gives us some informationabout the ’shape’ of the space. For instance, on R2 − {(0, 0)} all the closedcurves can be classified based on the number of times it winds around origin.So knowing the different ’types’ of curves can give us an idea about the ’hole’on the space. This idea can be made mathematically precise by classifyingclosed curves on a topological space based on their homotopy equivalenceclasses.

Def : The set of all homotopy equivalence classes of closed curves at a point phas a group structure with group operation being < α > . < β >=< α.β >.This is called the fundamental group based at p (π1(X, p)).

Def : If f : X → Y is a continuous function, then the homomorphism inducedby f at a point p is the function f∗ : π1(X, p)→ π1(Y, f(p)) where,

f∗(α) =< f ◦ α >

This can be easily verified to be a homomorphism.It can be proved that fundamental groups are topological invariants by usingthe induced homomorphism between the fundamental groups of thehomeomorphic spaces.

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0.3 Homotopy Type

The fundamental group is in fact left invariant by a larger class of maps thanthe class of homeomorphisms.

Def : Two spaces X and Y have the same homotopy type, or are homotopyequivalent, if there exist maps f : X → Y and g : Y → X such that f ◦ g ' 1Yand g ◦ f ' 1X .Remarks:

1. Homeomorphic spaces have same homotopy type.

2. Any convex subset of a euclidean space is homotopy equivalent of apoint. In particular the unit disk is same homotopy type as a point.

Def : Let A be a subspace of X. A homotopy G : X × I → X which is relativeto A and for which G(x, 0) = x and G(x, 1) ∈ A for all x ∈ X will be called adeformation retraction of X onto A.If there is a deformation retraction of X onto A, then X and A have the samehomotopy type (take f : A→ X to be inclusion and g : X → A to bex 7→ G(x, 1)). Few examples of deformation retractions are given below,

The figures show the deformation retraction of a disk with two holes onto theone-point union of two circles (figure of eight); and onto a space which lookslike the letter θ

Theorem: If two path-connected spaces are of the same homotopy type, theyhave isomorphic fundamental groups.This theorem finds immediate applications. The Mobius strip, the cylinder,the punctured plane R2 − {0}, and the solid torus, all have the homotopy typeof a circle, and consequently have the same fundamental group. This theoremalong with remark 2 shows that the fundamental group of the disk is trivial.

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0.4 Covering Spaces

Def : A space X is locally connected if for each point x ∈ X and each open setv containing x, there exists a connected open set U such that x ∈ U ⊂ V . Aspace X is locally arcwise connected if for each point x ∈ X and each open setV containing x,there exists an open set U , with x ∈ U ⊂ V , such thatwhenever x1, x2 ∈ U , there exists a path α from α1 to α2 with α(I) ⊂ P

Def : Let X and X̃ be arcwise connected, locally arcwise connected spaces,and let p : X̃ → X be continuous. The pair (X̃, p) is called a covering space ofX if p is surjective, and for each x ∈ X, there exists an open set U in Xcontaining x such that p−1(U) is the disjoint union of open sets, each of whichis mapped homeomorphically onto U by p.

Theorem: Let (X̃, p) be a covering space of X. Let x̃ ∈ X̃ with p(x̃) = x. then

there exists an isomorphism between p−1({x}) and the coset space π1(X,x)

p∗π1(X̃,x̃).

Now we can use this theorem to calculate the fundamental group of the circle.consider the line R1 as the covering space for the circle S1, with p(r) = e2πir.Take as the base point in S1 the point z = 1. Then p−1({1}) is the set ofintegers. Since π1(R1, 0) is trivial, p∗π1(R1, 0) = (e), so π1(S1, 0) is isomorphicto Z.

0.5 The Brouwer fixed-point theorem

The 2 dimensional case of L.E.J Brouwer’s fixed-point theorem is the mostcelebrated application of the machinery developed so far.

Theorem: A continuous function from a ball to itself must leave atleast onepoint fixed.

Proof for dimension 2: Assume the converse is true. So for the unit disk Dwe have the map f : D → D which has no fixed points. Now for each point xin D draw a line segment joining f(x) to x and extend it to meet the boundarycircle. Sending x to the intersection of the line segment with the boundarygives us the map g(x). The continuity of g(x) is ensured by that of f(x).

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Now g(x) is identity on the boundary by construction. This is where thedirection we chose to extend the line segment becomes important. Take thepoint p = (1, 0) as base point for both C and D, and denote the inclusion of C

in D by i : C → D. The spaces and maps Ci−→ D

g−→ C give rise to groups andhomomorphisms

π1(C, p)i∗−→ π1(D, p)

g∗−→ π1(C, p)

Now g ◦ i(x) = x for all x in C, therefore g∗ ◦ i∗ is the identity homomorphismand g∗ must be onto. But π1(D, p) is the trivial group and π1(C, p) ∼= Z. Thisgives the contradiction.The above argument shows the interplay between algebra and topology at itsbest.

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Conclusion

Algebraic topology is a vast sub-field of mathematics which requires a lot ofgroundwork before progressing onto the more deeper theorems. The aim ofthis project was to develop a strong foundation for the subject which wasachieved to some extend due to the continuous help and support of Prof.Harish Seshadri.

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References

1. Basic Topology - M.A. Armstrong

2. Lecture Notes on Elementary Topology and Geometry - I.M. Singer, J.A.Thorpe

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