Lecture 9: The Homology Long Exact Sequence

9. THE HOMOLOGY LONG EXACT SEQUENCE 19 We have the identity map ιn : ∆n → ∆n , the universal n-simplex, in Sinn (∆n ) ⊆ Sn (∆n ). It is not a cycle; its boundary dιn ∈ Sn−1 (∆n ) is the alternating sum of the faces of the n-simplex. Each of these singular simplices lies in ∂∆n , so dιn ∈ Sn−1 (∂∆n ), and [ιn ] ∈ Sn (∆n , ∂∆n ) is a relative cycle. We will see that the relative homology Hn (∆n , ∂∆n ) is infinite cyclic, with generator [ιn ]. 9 The homology long exact sequence A pair of spaces (X, A) gives rise to a short exact sequence of chain complexes: 0 → S∗ (A) → S∗ (X) → S∗ (X, A) → 0 . In homology, this will relate H∗ (A), H∗ (X), and H∗ (X, A). To investigate what happens, let’s suppse we have a general short exact sequence of chain complexes, 0 → A∗ → B ∗ → C ∗ → 0 , and study what happens in homology. Clearly the composite H∗ (A) → H∗ (B) → H∗ (C) is trivial. Is this sequence exact? Let [b] ∈ Hn (B) such that g([b]) = 0. It’s determined by some b ∈ Bn such that d(b) = 0. If g([b]) = 0, then there is some c ∈ Cn+1 such that dc = gb. Now, g is surjective, so there is some b ∈ Bn+1 such that g(b) = c. Then we can consider db ∈ Bn , and g(d(b)) = d(c) ∈ Cn . What is b − db? This maps to zero in Cn , so by exactness there is some a ∈ An such that f (a) = b − db. Is a a cycle? Well, f (da) = d(f a) = d(b − db) = db − d2 b = db, but we assumed that db = 0, so f (da) = 0. This means that da is zero because f is an injection by 20 CHAPTER 1. SINGULAR HOMOLOGY exactness. Therefore a is a cycle. What is [a] ∈ Hn (A)? Well, f ([a]) = [b − db] = [b]. This proves exactness of Hn (A) → Hn (B) → Hn (C). On the other hand, H∗ (A) → H∗ (B) may fail to be injective, and H∗ (B) → H∗ (C) may fail to be surjective. Instead: Theorem 9.1 (The homology long exact sequence). Let 0 → A∗ → B∗ → C∗ → 0 be a short exact sequence of chain complexes. Then there is a natural homomorphism ∂ : Hn (C) → Hn−1 (A) such that the sequence / Hn+1 (C) ··· ∂ Hn (A) t / Hn (B) / Hn (C) ∂ Hn−1 (A) t / ··· is exact. Proof. We’ll construct ∂, and leave the rest as an exercise. Here’s an expanded part of this short exact sequence: 0 0 0 / An+1 f / Bn+1 d  / An f / An−1 f / Cn+1 d  / Bn d  g  g d / Bn−1 g /0 d  / Cn  /0 d / Cn−1 /0 Let c ∈ Cn be a cycle: dc = 0. The map g is surjective, so pick a b ∈ Bn such that g(b) = c, and consider db ∈ Bn−1 . Well, g(d(b)) = d(g(b)) = dc = 0. So by exactness, there is some a ∈ An−1 such that f (a) = db. How many choices are there of picking a? Only one, because f is injective. We need to check that a is a cycle. What is d(a)? Well, d2 b = 0, so da maps to 0 under f . But because f is injective, da = 0, i.e., a is a cycle. This means we can define ∂[c] = [a]. To make sure that this is well-defined, let’s make sure that this choice of homology class a didn’t depend on the b that we chose. Pick some other b0 such that g(b0 ) = c. Then there is a0 ∈ An−1 such that f (a0 ) = db0 . We want a − a0 to be a boundary, so that [a] = [a0 ]. We want a ∈ An such that da = a − a0 . Well, g(b − b0 ) = 0, so by exactness, there is a ∈ An such that f (a) = b − b0 . What is da? Well, da = d(b − b0 ) = db − db0 . But f (a − a0 ) = b − b0 , so because f is injective, da = a − a0 , i.e., [a] = [a0 ]. I leave the rest of what needs checking to the listener. Example 9.2. A pair of spaces (X, A) gives rise to a natural long exact sequence in homology: ··· / Hn+1 (X, A) . ∂ Hn (A) t / Hn (X) ∂ Hn−1 (A) t / ··· / Hn (X, A) 9. THE HOMOLOGY LONG EXACT SEQUENCE 21 Example 9.3. Let’s think again about the pair (Dn , S n−1 ). By homotopy invariance we know that Hq (Dn ) = 0 for q > 0, since Dn is contractible. So ∂ : Hq (Dn , S n−1 ) → Hq−1 (S n−1 ) is an isomorphism for q > 1. The bottom of the long exact sequence looks like this: H0 (S n−1 ) s 0 / H1 (D n , S n−1 ) / H0 (D n ) / H0 (D n , S n−1 ) /0 When n > 1, both S n−1 and Dn are path-connected, so the map H0 (S n−1 ) → H0 (Dn ) is an isomorphism, and H1 (Dn , S n−1 ) = H0 (Dn , S n−1 ) = 0 . When n = 1, we discover that H1 (D1 , S 0 ) = Z and H0 (D1 , S 0 ) = 0 . The generator of H1 (D1 , S 0 ) is represented by any 1-simplex ι1 : ∆1 → D1 such that d0 ι = c01 and d1 ι = c00 (or vice versa). To go any further in this analysis, we’ll need another tool, known as “excision.” We can set this up for reduced homology (as in Lecture 5) as well. Note that any map induces an isomorphism in Se−1 , so to a pair (X, A) we can associate a short exact sequence 0 → Se∗ (A) → Se∗ (X) → S∗ (X, A) → 0 and hence a long exact sequence / Hn+1 (X, A) . ··· ∂ e n (A) H t e n (X) /H / Hn (X, A) ∂ e n−1 (A) H t / ··· e ∗ (Dn ) = 0 and so In the example (Dn , S n−1 ), H ∼ = e q−1 (S n−1 ) ∂ : Hq (Dn , S n−1 ) − →H e −1 (∅) = Z. This is for all n and q. This even works when n = 0; remember that S −1 = ∅ and H why I like this convention. The homology long exact sequence is often used in conjunction with an elementary fact about a map between exact sequences known as the five lemma. Suppose you have two exact sequences of abelian groups and a map between them – a “ladder”: A4  d f4 B4 d / A3  d f3 / B3 d / A2  d f2 / B2 d / A1  d f1 / B1 d / A0  f0 / B0 22 CHAPTER 1. SINGULAR HOMOLOGY When can we guarantee that the middle map f2 is an isomorphism? We’re going to “diagram chase.” Just follow your nose, making assumptions as necessary. Surjectivity: Let b2 ∈ B2 . We want to show that there is something in A2 mapping to b2 . We can consider db2 ∈ B1 . Let’s assume that f1 is surjective. Then there’s a1 ∈ A1 such that f1 (a1 ) = db2 . What is da1 ? Well, f0 (da1 ) = d(f1 (a1 )) = d(db) = 0. So we want f0 to be injective. Then da1 is zero, so by exactness of the top sequence, there is some a2 ∈ A2 such that da2 = a1 . What is f2 (a2 )? To answer this, begin by asking: What is d(f2 (a2 ))? By commutativity, d(f2 (a2 )) = f1 (d(a2 )) = f1 (a1 ) = db2 . Let’s consider b2 − f2 (a2 ). This maps to zero under d. So by exactness, there is b3 ∈ B3 such that d(b3 ) = b2 − f2 (a2 ). If we assume that f3 is surjective, then there is a3 ∈ A3 such that f3 (a3 ) = b3 . But now d(a3 ) ∈ A2 , and f2 (d(a3 )) = d(f3 (a3 )) = b2 −f2 (a2 ). This means that b2 = f (a2 + d(a3 )), verifying surjectivity of f2 . This proves the first half of the following important fact. The second half is “dual” to the first. Proposition 9.4 (Five lemma). In the map of exact sequences above, • If f0 is injective and f1 and f3 are surjective, then f2 is surjective. • If f4 is surjective and f3 and f1 are injective, then f2 is injective. Very commonly one knows that f0 , f1 , f3 , and f4 are all isomorphisms, and concludes that f2 is also an isomorphism. For example: Corollary 9.5. Let 0 0 / A0 ∗  f / A∗ / B0 ∗ / C0 ∗ g   / B∗ /0 h / C∗ /0 be a map of short exact sequences of chain complexes. If two of the three maps induced in homology by f, g, and h are isomorphisms, then so is the third. Here’s an application. Proposition 9.6. Let (A, X) → (B, Y ) be a map of pairs, and assume that two of A → B, X → Y , and (X, A) → (Y, B) induce isomorphims in homology. Then the third one does as well. Proof. Just apply the five lemma to the map between the two homology long exact sequences. 10 Excision and applications We have found two general properties of singular homology: homotopy invariance and the long exact sequence of a pair. We also claimed that H∗ (X, A) “depends only on X − A.” You have to be careful about this. The following definition gives conditions that will capture the sense in which the relative homology of a pair (X, A) depends only on the complement of A in X. Definition 10.1. A triple (X, A, U ) where U ⊆ A ⊆ X, is excisive if U ⊆ Int(A). The inclusion (X − U, A − U ) ⊆ (X, A) is then called an excision. Theorem 10.2. An excision induces an isomorphism in homology, ∼ = H∗ (X − U, A − U ) − → H∗ (X, A) .