MATH527_HW10

# MATH527_HW10 - Serge Ballif MATH 527 Homework 10 Problem 1...

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Serge Ballif MATH 527 Homework 10 November 28, 2007 Problem 1. (a) Compute the homology groups of the n -fold torus T 2 n . (b) Compute the homology groups of the m -fold projective plane P 2 m . (a) There is only one connected component, so H 0 ( T 2 n ) = . The fundamental group of T 2 n has 2 n generators, and hence, the first homology group (which is just the abelianization of the fundamental group) is the direct sum of 2 n copies of . The n -torus has a single interior, so H 2 ( T 2 n ) = . (b) P 2 m has only one path component, so H 0 ( P 2 m ) = . Each time we attach a new copy of P 2 , we add another copy of to the first homology group, H 1 ( P 2 m ). H 1 ( P 2 ) = 2 , H 1 ( P 2 ] P 2 ) = 2 ,. . . , H 1 ( P 2 m ) = 2 · · · ⊕ . The first homology group is the direct sum of the groups 2 and ( m - 1) copies of . There is no interior to P 2 ; nor does attaching more copies of P 2 create an interior. Therefore, H 2 ( P 2 m ) = 0. Problem 2. Find path-connected spaces X and Y , and a continuous map f : X Y , such that f induces an isomor- phism between H 1 ( X ) and H

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