series10 - Homology and cohomology Prof Kathryn Hess Series...

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Homology and cohomology Spring 2009 Prof. Kathryn Hess Series 10 The purpose of the first part of this series is to show that the integral homology groups of a space X determine the homology and cohomology groups of X with arbitary coefficients. The statements are known as the universal coefficient theorems; one is for homology and involves the Tor functor, the other is for cohomology and involves the Ext functor. Recall from lecture the following theorem. Theorem 1 (Universal Coefficient Theorem) . Let C be a chain complex of free abelian groups C n and let G be any abelian group. Then for each n there is an exact sequence 0 H n ( C ) G α -→ H n ( C G ) β -→ Tor( H n 1 ( C ) , G ) 0 with homomorphisms α, β natural in C, G . This sequence splits, by a homo- morphism which is natural in G but not in C . Here, α ([ c ] g ) = [ c g ] . The singular homology version of this theorem follows immediately. Exercise 1. Observe that Theorem 1 implies Theorem 2. Theorem 2 (Universal Coefficient Theorem) . Let X be a topological space and let G be any abelian group. Then for each n there is an exact sequence 0 H n ( X ) G α -→ H n ( X ; G ) β -→ Tor( H n 1 ( X ) , G ) 0 with homomorphisms α, β natural in X, G . This sequence splits, by a homo- morphism which is natural in G but not in X , and hence H n ( X ; G ) = ( H n ( X ) G ) Tor( H n 1 ( X ) , G ) . Here, α ([ c ] g ) = [ c g ] . Let f : X -→ Y be a map of topological spaces. The purpose of the following exercise is to prove that if f induces an isomorphism on integral homology, then f induces an isomorphism on homology with coefficients in any abelian group G . Exercise 2. Use Theorem 2 together with the Five Lemma to prove Propo- sition 3. Proposition 3. Let f : X -→ Y be a map of topological spaces and let G be any abelian group. If the induced map f : H n ( X ) = -→ H n ( Y ) is an isomorphism for all n , then the induced map f : H n ( X ; G ) = -→ H n ( Y ; G ) is an isomorphism for all n . 1
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2 The following characterization [2, V.6.2] of flat abelian groups will be useful in Exercise 3. Proposition 4. Let G be an abelian group. Then G is flat if and only if G is torsion-free. Exercise 3. Let X be a topological space and let G be a torsion-free abelian group. Use Theorem 2 to prove the following. (a) H n ( X ; G ) = H n ( X ) G for all n 0. (b) H n ( X ; Q ) = H n ( X ) Q for all n 0. The purpose of the next part of this series is to introduce a universal coef- ficient theorem for cohomology. A first step is to recall from [Series 1] the following exactness property of the Hom functor.
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