differential geometry w notes from teacher_Part_92

differential geometry w notes from teacher_Part_92 - H ( S...

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7.3. SINGULAR HOMOLOGY GROUPS 183 Theorem 7.3.6 Let M be an n-dimensional compact manifold. Every real p-cycle z p in H p ( M ; R ) is homologous to a finite formal sum z p r X k = 1 a k V k of closed oriented p-dimensional submanifolds V k of M with real coef- ficients a k . Proof : Nontrivial. ± Theorem 7.3.7 Let M be an n-dimensional manifold and G be an Abelian group. Let z p and z 0 p be two cycles in H p ( M ; G ) that can be deformed into each other. Then they are homologous to each other z p z 0 p . Proof : 1. Since the deformation defines a deformation chain c p + 1 such that c p + 1 = z 0 p - z p . ± Proposition 7.3.1 Let M be an n-dimensional closed manifold and G be an Abelian group. Then for p > n the singular homology groups H p ( M ; G ) are trivial H p ( M ; G ) = 0 . Proof : 1. Singular homology groups are isomorphic to the simplicial homology groups. 2. Since there are no simplicial complexes of dimension p > n then all simplicial homology groups are trivial for p > n . ± di geom.tex; April 12, 2006; 17:59; p. 181
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184 CHAPTER 7. HOMOLOGY THEORY 7.3.5 Examples. Sphere S n . From the facts that S n is connected, orientable and closed it follows that
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Unformatted text preview: H ( S n ; G ) = H n ( S n ; G ) = G , H p ( S n ; G ) = , for p , , n , B ( S n ) = B n ( S n ) = 1 , B p ( S n ) = , for p , , n . • Torus T 2 . H ( T 2 ; G ) = H 2 ( T 2 ; G ) = G , H 1 ( T 2 ; G ) = GA + GB , B ( T 2 ) = 1 , B 1 ( T 2 ) = 2 , B 2 ( T 2 ) = 1 , where A and B are the basic 1-cycles. • Klein Bottle K 2 . • Since K 2 is connected closed non-orientable it follows that H ( K 2 ; Z ) = Z , H 2 ( K 2 , Z ) = , H 1 ( K 2 ; Z ) = Z A + Z 2 B , where A and B are the basic 1-cycles, and H ( K 2 ; R ) = R , H 2 ( K 2 , R ) = , H 1 ( K 2 ; R ) = R A , B ( K 2 ) = 1 , B 1 ( K 2 ) = 1 , B 2 ( K 2 ) = . • Real Projective Plane R P 2 . • R P 2 is connected closed non-orientable. H ( R P 2 ; Z ) = Z , H 2 ( R P 2 , Z ) = , H 1 ( R P 2 ; Z ) = Z 2 A , di ff geom.tex; April 12, 2006; 17:59; p. 182...
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

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differential geometry w notes from teacher_Part_92 - H ( S...

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