Lecture 2: Homology

4 2 CHAPTER 1. SINGULAR HOMOLOGY Homology In the last lecture we introduced the standard n-simplex ∆n ⊆ Rn+1 . Singular simplices in a space X are maps σ : ∆n → X and constitute the set Sinn (X). For example, Sin0 (X) consists of points of X. We also described the face inclusions di : ∆n−1 → ∆n , and the induced “face maps” di : Sinn (X) → Sinn−1 (X) , 0 ≤ i ≤ n , given by precomposing with face inclusions: di σ = σ ◦ di . For homework you established some quadratic relations satisfied by these maps. A collection of sets Kn , n ≥ 0, together with maps di : Kn → Kn−1 related to each other in this way, is a semi-simplicial set. So we have assigned to any space X a semi-simplicial set S∗ (X). To the semi-simplicial set {Sinn (X), di } we then applied the free abelian group functor, obtaining a semi-simplicial abelian group. Using the di s, we constructed a boundary map d which makes S∗ (X) a chain complex – that is, d2 = 0. We capture this process in a diagram: H∗ {spaces} / {graded abelian groups} O Sin∗  {semi-simplicial sets}  take homology Z(−) / {chain complexes} {semi-simplicial abelian groups} Example 2.1. Suppose we have σ : ∆1 → X. Define φ : ∆1 → ∆1 by sending (t, 1 − t) to (1 − t, t). Precomposing σ with φ gives another singular simplex σ which reverses the orientation of σ. It is not true that σ = −σ in S1 (X). However, we claim that σ ≡ −σ mod B1 (X). This means that there is a 2-chain in X whose boundary is σ + σ. If d0 σ = d1 σ, so that σ ∈ Z1 (X), then σ and −σ are homologous: [σ] = −[σ] in H1 (X). To construct an appropriate boundary, consider the projection map π : ∆2 → ∆1 that is the affine extension of the map sending e0 and e2 to e0 and e1 to e1 . 2 1 1 0 0 2. HOMOLOGY cnx 5 We’ll compute d(σ ◦ π). Some of the terms will be constant singular simplices. Let’s write : ∆n → X for the constant map with value x ∈ X. Then d(σ ◦ π) = σπd0 − σπd1 + σπd2 = σ − c1σ(0) + σ . The constant simplex c1σ(0) is an “error term,” and we wish to eliminate it. To achieve this we can use the constant 2-simplex c2σ(0) at σ(0); its boundary is c1σ(0) − c1σ(0) + c1σ(0) = c1σ(0) . So σ + σ = d(σ ◦ π + c2σ(0) ) , and σ ≡ −σ mod B1 (X) as claimed. Some more language: two cycles that differ by a boundary dc are said to be homologous, and the chain c is a homology between them. Let’s compute the homology of the very simplest spaces, ∅ and ∗. For the first, Sinn (∅) = ∅, so S∗ (∅) = 0. Hence · · · → S2 → S1 → S0 is the zero chain complex. This means that Z∗ (∅) = B∗ (∅) = 0. The homology in all dimensions is therefore 0. For ∗, we have Sinn (∗) = {cn∗ } for all n ≥ 0. Consequently Sn (∗) = Z for n ≥ 0 and 0 for n < 0. For each i, di cn∗ = cn−1 , so the boundary maps d : Sn (∗) → Sn−1 (∗) in the chain complex depend ∗ on the parity of n as follows: ( n X cn−1 for n even, and ∗ d(cn∗ ) = (−1)i cn−1 = ∗ 0 for n odd. i=0 This means that our chain complex is: 0 1 0 1 0←Z← −Z← −Z← −Z← − ··· . The boundaries coincide with the cycles except in dimension zero, where B0 (∗) = 0 while Z0 (∗) = Z. Therefore H0 (∗) = Z and Hi (∗) = 0 for i 6= 0. We’ve defined homology groups for each space, but haven’t yet considered what happens to maps between spaces. A continuous map f : X → Y induces a map f∗ : Sinn (X) → Sinn (Y ) by composition: f∗ : σ 7→ f ◦ σ . For f∗ to be a map of semi-simplicial sets, it needs to commute with face maps: We need f∗ ◦ di = di ◦ f∗ . A diagram is said to be commutative if all composites with the same source and target are equal, so this equation is equivalent to commutativity of the diagram Sinn (X)  f∗ di Sinn−1 (X) f∗ / Sinn (Y )  di / Sinn−1 (Y ) . Well, di f∗ σ = (f∗ σ) ◦ di = f ◦ σ ◦ di , and f∗ (di σ) = f∗ (σ ◦ di ) = f ◦ σ ◦ di as well. The diagram remains commutative when we pass to the free abelian groups of chains. 6 CHAPTER 1. SINGULAR HOMOLOGY If C∗ and D∗ are chain complexes, a chain map f : C∗ → D∗ is a collection of maps fn : Cn → Dn such that the following diagram commutes for every n: Cn  fn dC Cn−1 fn−1 / Dn  dD / Dn−1 For example, if f : X → Y is a continuous map, then f∗ : S∗ (X) → S∗ (Y ) is a chain map as discussed above. A chain map induces a map in homology f∗ : Hn (C) → Hn (D). The method of proof is a socalled “diagram chase” and it will be the first of many. We check that we get a map Zn (C) → Zn (D). Let c ∈ Zn (C), so that dC c = 0. Then dD fn (c) = fn−1 dC c = fn−1 (0) = 0, because f is a chain map. This means that fn (c) is also an n-cycle, i.e., f gives a map Zn (C) → Zn (D). Similarly, we get a map Bn (C) → Bn (D). Let c ∈ Bn (C), so that there exists c0 ∈ Cn+1 such that dC c0 = c. Then fn (c) = fn dC c0 = dD fn+1 (c0 ). Thus fn (c) is the boundary of fn+1 (c0 ), and f gives a map Bn (C) → Bn (D). The two maps Zn (C) → Zn (D) and Bn (C) → Bn (D) quotient to give a map on homology f∗ : Hn (X) → Hn (Y ). 3 Categories, functors, natural transformations From spaces and continuous maps, we constructed graded abelian groups and homomorphisms. We now cast this construction in the more general language of category theory. Our discussion of category theory will be interspersed throughout the text, introducing new concepts as they are needed. Here we begin by introducing the basic definitions. Definition 3.1. A category C consists of the following data.