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differential geometry w notes from teacher_Part_91

differential geometry w notes from teacher_Part_91 - 181...

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7.3. SINGULAR HOMOLOGY GROUPS 181 3. Therefore, a multiple gp of a single point is not a boundary for any g 0. 4. Thus, any point p M is a 0-cycle that is not a boundary. 5. Moreover, for any g G and any p M the 0-chain gp is a 0-cycle that is not a boundary. In particular, H 0 ( M ; Z ) = { 0 , ± p , ± 2 p , . . . } and H 0 ( M ; R ) = R is a one-dimensional vector space. Corollary 7.3.3 Let M be a compact connected manifold. Then the zero Betti number is equal to B 0 ( M ) = 1 . Proof : Follows from above. Theorem 7.3.4 Let M be a compact manifold consisting of k con- nected pieces M 1 , . . . M k . Then H 0 ( M ; R ) = R p 1 + · · · + R p k , where p i M i , i = 1 , . . . , k, meaning R p 1 + · · · + R p k = k i = 1 a i p i a i R , p i M i , i = 1 , . . . , k . Proof : In this case H 0 ( M ; R ) = R k is a k -dimensional vector space. di ff geom.tex; April 12, 2006; 17:59; p. 179
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182 CHAPTER 7. HOMOLOGY THEORY Corollary 7.3.4 Let M be a compact manifold consisting of k con- nected pieces. Then the zero Betti number is equal to B 0 ( M ) = k .
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