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Sets 2 1.1. Motivation 2 1.2. Triangulated Spaces 3 1.3

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Sets 2 1.1. Motivation 2 1.2. Triangulated Spaces 3 1.3 LECTURES ON HOMOLOGY THEORY DRAFT VERSION SERGEY MOZGOVOY Contents 1. Triangulated spaces and ∆-sets2 1.1. Motivation2 1.2. Triangulated spaces3 1.3. Geometric realization8 1.4. Simplicial complexes 10 1.5. Product triangulation 13 1.6. Barycentric subdivision 14 1.7. Simplicial homology 15 2. Homological algebra 19 3. Singular homology 23 3.1. Definition and basic properties 23 3.2. Homotopy invariance 25 3.3. Relative homology groups 27 3.4. Excision and Mayer-Vietoris 28 3.5. Long exact sequence for good pairs 30 3.6. Equivalence of simplicial and singular homologies 32 3.7. CW-complexes and cellular homology 33 4. Applications of Homology theory 35 4.1. Degree 35 4.2. Hedgehog theorem 36 4.3. Jordan curve theorem 36 4.4. Invariance of domain 38 4.5. Algebraic applications 38 4.6. Borsuk-Ulam theorem 39 4.7. Lefschetz fixed point theorem 41 Appendix A. Preliminaries 42 A.1. Relations 42 A.2. Quotient topology 43 Appendix B. Categories and functors 45 Appendix C. Simplicial approximation 47 Index 48 Date: March 13, 2019. 2 HOMOLOGY THEORY 1. Triangulated spaces and ∆-sets 1.1. Motivation. Algebraic topology relates problems of topology and algebra. At the first level which is the content of this course one reduces topological problems to algebra and in particular to linear algebra. At the next level one reduces algebraic problems to topology. The main approach is the following. Given a topological space X, one constructs an algebraic object H(X). This can be a group (like a fundamental group), a vector space, an algebra, etc. This construction is usually functorial, meaning that given a continuous map f: X ! Y , there is a homomorphism f∗: H(X) ! H(Y ). In this way we obtain a functor (see AppendixB) from the category of topological spaces to the category of groups, vector spaces, algebras, etc. The objective is to relate topological properties of X to algebraic properties of H(X). We will study just one construction of this type, called the homology theory. As applications of the theory that we will develop, we will later prove the following statements: (1) Rn are not homeomorphic to each other for different n 3.35. (2) A continuous map f: Dn ! Dn has a fixed point (Brouwer theorem) 3.37. (3) One can not comb a hedgehog smoothly, meaning that there is no continuous non- vanishing tangent vector field on S2 4.4. (4) If f: S1 ! R2 is injective and continuous, then R2nf(S1) consists of exactly two connected components (Jordan curve theorem) 4.7. (5) One can not embed Sn in Rn 4.10. Generally, if f: M ! N is an embedding of topological n-manifolds, with compact M and connected N, then f is a homeomorphism 4.13. (6) The field C is algebraically closed (Fundamental theorem of algebra) 4.17. (7) If f: Sn ! Rn is continuous, then there exists x 2 Sn with f(x) = f(−x) (Borsuk-Ulam theorem) 4.18. 2 (8) If F1;F2;F3 is a closed covering of S , then at least one Fi contains antipodal points n (x; −x 2 Fi). Generally, if F1;:::;Fn+1 is a closed covering of S , then at least one Fi contains antipodal points (Borsuk-Ulam theorem) 4.18. Example 1.1. Let us consider a finite connected graph on a plane. It consists of several points (called vertices) connected by line segments (called edges), without intersections. Connected components of the complement are called faces (we consider also the unbounded component). We assume that all bounded faces are homeomorphic to an open disc. Let v; e; f be the numbers of vertices, edges and faces respectively. Then Euler's formula states that v − e + f = 2: For example, consider a graph consisting of one point. Then v = 1; e = 0; f = 1 and the formula is satisfied. For a triangle on a plane, we have v = e = 3; f = 2 and the formula is again satisfied. The above decomposition of R2 can be interpreted as a decomposition (also called a triangu- lation if all faces are triangles) of the two-dimensional sphere S2 { the sphere is obtained from R2 by adding one point at infinity and we consider the unbounded component as containing this additional point. The left hand side of Euler's formula can be associated with any triangulation of S2. According to the formula, this number is independent of the triangulation, hence it is an invariant of S2, called the Euler characteristic of S2. Homology theory in a nutshell is a generalization of Euler's formula to other topological spaces. HOMOLOGY THEORY 3 1.2. Triangulated spaces. By a space we will always mean a topological space. By a map between spaces we will always mean a continuous map, unless otherwise stated. We will study spaces that can be obtained by gluing together points, segments, triangles and higher-dimensional building blocks, called simplices. The structure that one obtains is called a triangulated space. Its combinatorial counterpart is called a ∆-set or a semi-simplicial set. Having this combinatorial structure, one can apply linearization to it and get an algebraic structure (an abelian group or a vector space), called the homology of the original space. Definition 1.2. Define the standard n-dimensional simplex (or n-simplex) to be the topological space n n n+1 X o ∆ = (t0; : : : ; tn) 2 R ti = 1; ti ≥ 0 8i : Example 1.3. ∆0 is a point, ∆1 is a line segment (edge), ∆2 is a triangle, ∆3 is a tetrahedron. 1 2 e1 ∆ e2 ∆ e0 e1 e0 Remark 1.4. Let us define an n-dimensional disc n n D = fx 2 R j kxk ≤ 1g ; an n-dimensional sphere n n+1 S = x 2 R kxk = 1 ; and an n-dimensional cube In = I × ::: × I;I = [0; 1]: | {z } n times Then ∆n ' Dn ' In n n n n−1 n n and @∆ ' @I ' @D = S . We can obtain S by gluing two hemispheres D± (both homeomorphic to Dn) along their boundary Sn−1. For every n ≥ 0, define [n] = f0; : : : ; ng. We will consider it as an ordered set. Definition 1.5. n n (1) For every i 2 [n], define the i-th vertex of ∆ to be the point ei 2 ∆ with ti = 1. n Pn Note that every point t 2 ∆ can be uniquely written in the form t = i=0 tiei, where P ti ≥ 0 and ti = 1. (2) More generally, for every subset I ⊂ [n], define the I-th face ∆I ⊂ ∆n to be I n ∆ = ft 2 ∆ j ti = 0 for i2 = Ig : This face is a simplex of dimension #I − 1. (3) For any map f:[m] ! [n], consider a map f m n X f∗ = ∆ : ∆ ! ∆ ; (s0; : : : ; sm) 7! (t0; : : : ; tn); tj = si f(i)=j P P or equivalently, i siei 7! i sief(i). Every subset I ⊂ [n] can be identified with a m n (strictly) increasing map f:[m] ! [n] having the image I. Then f∗: ∆ ! ∆ is injective m I n and f∗(∆ ) = ∆ . We call it the f-face of ∆ . 4 HOMOLOGY THEORY (4) A dimension n−1 face of ∆n is called a facet. There are n+1 facets in ∆n, corresponding to coface maps δi:[n − 1] ! [n] which are increasing maps that miss i 2 [n]. (5) Define the boundary @∆n to be the union of all facets. It consists of t 2 ∆n with at least one ti = 0. ◦ (6) Define the open simplex ∆n to be the interior of ∆n ◦ n n n n ∆ = ft 2 ∆ j ti > 0 8ig = ∆ n@∆ : ◦ Note that @∆0 = ? and ∆0 = ∆0. Example 1.6. Consider all faces of ∆2. Observe how they correspond to subsets of [2] or to increasing maps f:[m] ! [2] for m ≤ 2. Consider f: [0] ! [2] for 0 ≤ i ≤ 2 and the 0 2 corresponding maps f∗: ∆ ! ∆ . Definition 1.7. A triangulation K of a space X is a collection of maps n (φσ = σ: ∆ ! X)σ2K ; where n depends on σ which is called a simplex of dimension n = dim σ and ◦ n (1) The restriction σj ◦ is injective and X is the disjoint union of cells e = σ(∆ ). ∆n σ (2) The restriction of σ to a face of ∆n is one of the maps τ: ∆m ! X with τ 2 K. (3) A subset A ⊂ X is open () σ−1(A) is open in ∆n for each σ. The pair (X; K), consisting of a space X with a triangulation K, is called a triangulated space (or a ∆-complex). Remark 1.8. The map σ: ∆n ! X is not necessarily injective even though its restriction to the ◦ open simplex ∆n is injective. Nevertheless, we will often identify σ with its image σ(∆n) ⊂ X (especially on the drawings of triangulations). Example 1.9. (1) Circle S1 as a space homeomorphic to @∆2. There are 3 0-simplices and 3 1-simplices. v1 a b v0 v2 c (2) Circle obtained by gluing one point and one interval. a v0 1 We have K = fv0; ag, where v0 is a 0-simplex and a is a 1-simplex. Consider S = 0 1 fz 2 C j jzj = 1g and φv0 : ∆ ! S , pt 7! 1, 1 1 2πit φa: ∆ ' [0; 1] ! S ; t 7! e : (3) Circle obtained by gluing two points and two intervals. a v1 v0 b HOMOLOGY THEORY 5 (4) Sphere S2 as a space homeomorphic to @∆3 (boundary of a tetrahedron). There are 4 0-simplices, 6 1-simplices and 4 2-simplices.
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