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 R P 2 m . (a) There is only one connected component, so H 0 ( T 2 n ) = Z . 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 Z . The n -torus has a single interior, so H 2 ( T 2 n ) = Z . (b) R P 2 m has only one path component, so H 0 ( R P 2 m ) = Z . Each time we attach a new copy of R P 2 , we add another copy of Z to the first homology group, H 1 ( R P 2 m ). H 1 ( R P 2 ) = Z 2 , H 1 ( R P 2 ] R P 2 ) = Z 2 Z ,. . . , H 1 ( R P 2 m ) = Z 2 Z ···⊕ Z . The first homology group is the direct sum of the groups Z 2 and ( m - 1) copies of Z . There is no interior to R P 2 ; nor does attaching more copies of R P 2 create an interior. Therefore,
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This note was uploaded on 04/01/2008 for the course MATH 527 taught by Professor Rotman,regina during the Fall '07 term at Penn State.

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

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