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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 1cycles. • Klein Bottle K 2 . • Since K 2 is connected closed nonorientable 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 1cycles, 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 nonorientable. 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.
 Spring '10
 Wong
 Geometry

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