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

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Homology and cohomology Spring 2009 Prof. Kathryn Hess Series 10 The purpose of the Frst 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 coe±cients. The statements are known as the universal coe±cient 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 Coe±cient 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 Coe±cient 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 coe±cients in any abelian group G . Exercise 2. Use Theorem 2 together with the ²ive 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 Fat abelian groups will be useful in Exercise 3. Proposition 4. Let G be an abelian group. Then G is fat 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)
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This note was uploaded on 01/18/2012 for the course INFORMATIK 2011 taught by Professor Phanthuongcang during the Winter '11 term at Cornell University (Engineering School).

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

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