Algebraic Topology M382C Michael Starbird Fall 2007 2 Contents 1 Introduction 5 1.1 Basic Examples . 5 1.2 Simplices . 6 1.3 Simplicial Complexes . 7 1.4 2-manifolds . 9 1.4.1 2-manifolds as simplicial complexes . 11 1.4.2 2-manifolds as quotient spaces . 14 1.5 Questions . 17 2 2-manifolds 19 2.1 Classification of 2-manifolds . 19 2.1.1 Classification Proof I . 20 2.1.2 Classification Proof II . 23 2.2 PL Homeomorphism . 27 2.3 Invariants . 28 2.3.1 Euler characteristic . 28 2.3.2 Orientability . 29 2.4 CW complexes . 35 2.5 2-manifolds with boundary . 38 2.6 *Non-compact surfaces . 42 3 Fundamental group and covering spaces 49 3.1 Fundamental group . 50 3.1.1 Cartesian products . 56 3.1.2 Induced homomorphisms . 57 3.2 Retractions and fixed points . 58 3.3 Van Kampen's Theorem, I . 60 3.3.1 Van Kampen's Theorem: simply connected intersec- tion case . 61 3 4 CONTENTS 3.3.2 Van Kampen's Theorem: simply connected pieces case 63 3.4 Fundamental groups of surfaces . 63 3.5 Van Kampen's Theorem, II . 66 3.6 3-manifolds . 67 3.6.1 Lens spaces . 67 3 3.6.2 Knots in S ........................ 69 3.7 Homotopy equivalence of spaces . 72 3.8 Higher homotopy groups . 73 3.9 Covering spaces . 75 3.10 Theorems about groups . 82 4 Homology 83 4.1 Z2 homology . 84 4.1.1 Simplicial Z2 homology . 85 4.1.2 CW Z2-homology . 86 4.2 Homology from parts, special cases . 91 4.3 Chain groups and induced homomorphisms . 93 4.4 Applications of Z2 homology . 96 4.5 Z2 Mayer-Vietoris Theorem . 97 4.6 Introduction to simplicial Z-homology . 100 4.6.1 Chains, boundaries, and definition of simplicial Z- homology . 100 4.7 Chain groups and induced homomorphisms . 105 4.8 Relationship between fundamental group and first homology . 106 4.9 Mayer-Vietoris Theorem . 107 A Review of Point-Set Topology 111 B Review of Group Theory 117 C Review of Graph Theory 125 D The Jordan Curve Theorem 127 Chapter 1 Introduction Abstracting and generalizing essential features of familiar objects often lead to the development of important mathematical ideas. One goal of geomet- rical analysis is to describe the relationships and features that make up the essential qualities of what we perceive as our physical world. The strategy is to find ideas that we view as central and then to generalize those ideas and to explore those more abstract extensions of what we perceive directly. Much of topology is aimed at exploring abstract versions of geometrical objects in our world. The concept of geometrical abstraction dates back at least to the time of Euclid (c. 225 B.C.E.) The most famous and basic spaces are named for him, the Euclidean spaces. All of the objects that we will study in this course will be subsets of the Euclidean spaces. 1.1 Basic Examples n n Definition ( R ). We define real or Euclidean n-space, denoted by R , as the set n R := f(x1; x2; : : : ; xn)jxi 2 R for i = 1; : : : ; ng: n We begin by looking at some basic subspaces of R . n Definition (standard n-disk). The n-dimensional disk, denoted D is de- fined as n n D := f(x1; : : : ; xn) 2 R j0 ≤ xi ≤ 1 for i = 1; : : : ; n g n times ∼ z }| { n = [0; 1] × [0; 1] × · · · × [0; 1] ⊂ R : 5 6 CHAPTER 1. INTRODUCTION 1 1 For example, D = [0; 1]. D is also called the unit interval, sometimes denoted by I. Definition (standard n-ball, standard n-cell). The n-dimensional ball or n cell, denoted B , is defined as: n n 2 2 B := f(x1; : : : ; xn) 2 R jx1 + ::: + xn ≤ 1g: Fact 1.1. The standard n-ball and the standard n-disk are compact and homeomorphic. n Definition (standard n-sphere). The n-dimensional sphere, denoted S , is defined as n n+1 2 2 S := f(x0; : : : ; xn) 2 R jx0 + ::: + xn = 1g: n+1 n Note. Bd B = S As usual, the term n-sphere will apply to any space homeomorphic to the standard n-sphere. 0 1 2 Question 1.2. Describe S , S , and S . Are they homeomorphic? If not, are there any properties that would help you distinguish between them? 1.2 Simplices n One class of spaces in R we will be studying will be manifolds or k- k manifolds, which are made up of pieces that locally look like R , put to- gether in a \nice" way. In particular, we will be studying manifolds that use triangles (or their higher-dimensional equivalents) as the basic building blocks. n Since k-dimensional \triangles" in R (called simplices) are the basic building blocks we will be using, we begin by giving a vector description of them. n Definition (1-simplex). Let v0, v1 be two points in R . If we consider v0 1 and v1 as vectors from the origin, then σ = fµv1 + (1 − µ)v0 j 0 ≤ µ ≤ 1g 1 is the straight line segment between v0 and v1. σ can be denoted by fv0v1g 1 or fv1v0g (the order the vertices are listed in doesn't matter). The set σ is called a 1-simplex or edge with vertices (or 0-simplices) v0 and v1. 1.3. SIMPLICIAL COMPLEXES 7 Definition (2-simplex). Let v0, v1, and v2 be three non-collinear points in n R . Then 2 σ = fλ0v0 + λ1v1 + λ2v2 j λ0 + λ1 + λ2 = 1 and 0 ≤ λi ≤ 18i = 0; 1; 2g is a triangle with edges fv0v1g, fv1v2g, fv0v2g and vertices v0, v1, and v2. 2 The set σ is a 2-simplex with vertices v0, v1, and v2 and edges fv0v1g, 2 fv1v2g, and fv0v2g. fv0v2v2g denotes the 2-simplex σ (where the order the vertices are listed in doesn't matter). Note that the plural of simplex is simplices. Definition (n-simplex and face of a simplex). Let fv0; v2; : : : ; vng be a set N n affine independent points in R . Then an n-simplex σ (of dimension n), N denoted fv0v1v2 : : : vng, is defined to be the following subset of R : ( ) n n X σ = λ0v0 + λ1v1 + ::: + λnvn λi = 1 ; 0 ≤ λi ≤ 1; i = 0; 1; 2; : : : ; n : i=0 An i-simplex whose vertices are any subset of i+1 of the vertices of σn is an n (i-dimensional) face of σ . The face obtained by deleting the vm vertex from n the list of vertices of σ is often denoted by fv0v1v2 ::: vcm : : : vng. (Note that it is an (n − 1)-simplex.) Exercise 1.3. Show that the faces of a simplex are indeed simplices. Fact 1.4. The standard n-ball, standard n-disk and the standard n-simplex are compact and homeomorphic. We will use the terms n-disk, n-cell, n-ball interchangeably to refer to any topological space homeomorphic to the standard n-ball. 1.3 Simplicial Complexes n Simplices can be assembled to create polyhedral subsets of R known as complexes. These simplicial complexes are the principal objects of study for this course. Definition (finite simplicial complex). Let T be a finite collection of sim- n j j plices in R such that for every simplex σi in T , each face of σi is also a simplex in T and any two simplices in T are either disjoint or their inter- n S j section is a face of each. Then the subset K of R defined by K = σi 8 CHAPTER 1. INTRODUCTION j running over all simplices σi in T is a finite simplicial complex with trian- gulation T , denoted (K; T ). The set K is often called the underlying space of the simplicial complex. If n is the maximum dimension of all simplices in T , then we say (K; T ) is of dimension n. Example 1. Consider (K; T ) to be the simplicial complex in the plane where T = f f(0; 0)(0; 1)(1; 0)g; f(0; 0)(0; −1)g; f(0; −1)(1; 0)g; f(0; 0)(0; 1)g; f(0; 1)(1; 0)g; f(1; 0)(0; 0)g; f(0; 0)g; f(0; 1)g; f(1; 0)g; f(0; −1)gg : So K is a filled in triangle and a hollow triangle as pictured. (0,1) (0,0) (1,0) (0,−1) Exercise 1.5. Draw a space made of triangles that is not a simplicial com- plex, and explain why it is not a simplicial complex. We have started by making spaces using simplices as building blocks. But what if we have a space, and we want to break it up into simplices? If J is a topological space homeomorphic to K where K is a the underlying space m of a simplicial complex (K; T ) in R , then we say that J is triangulable. Exercise 1.6. Show that the following space is triangulable: by giving a triangulation of the space. 1.4. 2-MANIFOLDS 9 Definition (subdivision). Let (K; T ) be a finite simplicial complex. Then T 0 is a subdivision of T if (K; T 0) is a finite simplicial complex, and each simplex in T 0 is a subset of a simplex in T. Example 2. The following picture illustrates a finite simplicial complex and a subdivision of it. (K,T) (K,T') There is a standard subdivision of a triangulation that later will be useful: Definition (derived subdivision).