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differential geometry w notes from teacher_Part_93

differential geometry w notes from teacher_Part_93 - 1 ∈...

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7.3. SINGULAR HOMOLOGY GROUPS 185 where A is the basic 1-cycle, and H 0 ( R P 2 ; R ) = R , H 2 ( R P 2 , R ) = 0 , H 1 ( R P 2 ; R ) = 0 , B 0 ( R P 2 ) = 1 , B 1 ( R P 2 ) = 0 , B 2 ( R P 2 ) = 0 . Torus T 3 . T 3 is a connected closed orientable manifold. H 0 ( T 3 ; Z ) = H 3 ( T 3 , Z ) = Z , H 1 ( T 3 ; Z ) = Z A + Z B + Z C , H 2 ( T 3 ; Z ) = Z D + Z E + Z F , where A , B and C are basic 1-cycles, and D , E and F are basic 2-cyles, B 0 ( T 3 ) = 1 , B 1 ( T 3 ) = B 2 ( T 3 ) = 3 , B 2 ( T 3 ) = 1 . Real Projective Space R P 3 . R P 3 is connected closed orientable. H 0 ( R P 3 ; R ) = H 3 ( R P 3 , R ) = R , H 1 ( R P 3 ; R ) = H 2 ( R P 3 ; R ) = 0 , B 0 ( R P 3 ) = B 3 ( R P 3 ) = 1 , B 1 ( R P 3 ) = B 2 ( R P 3 ) = 0 . di geom.tex; April 12, 2006; 17:59; p. 183
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186 CHAPTER 7. HOMOLOGY THEORY 7.4 de Rham Cohomology Groups Let M be a manifold and G = R be the coe cient group. Then C p ( M ; R ), Z p ( M ; R ), B p ( M ; R ) and H p ( M ; R ) are vector spaces. For simplicity we will denote them in this section simply by C p ( M ), Z p ( M ), B p ( M ) and H p ( M ). Let C p ( M ) = C ( Λ p ( M )) be the space of smooth p -forms on M . We will call the closed p -form p -cocyles and the space Z p ( M ) = { α p C p ( M ) | d α p = 0 } of all closed p -forms on M , the cocycle group . The exact p -forms on M are called the p -coboundaries and the space B p ( M ) = { α p Z p ( M ) | α p = d β p + 1 for some β p +
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Unformatted text preview: 1 ∈ C p + 1 ( M ) } of all exact p-forms on M is called the coboundary group. • Both Z p ( M ) and B p ( M ) are vector spaces with real coe ffi cients. • Recall that the exterior derivative is a map d p : C p ( M ) → C p + 1 ( M ) such that Ker d p = Z p ( M ) and Im d p-1 = B p ( M ) . • Two closed forms are said to be equivalent (or cohomologous ) if they di ff er by an exact form. • The collection of all equivalence classes of closed forms is the quotient vector space H p ( M ) = Z p ( M ) / B p ( M ) called the p-th de Rham cohomology group . di ff geom.tex; April 12, 2006; 17:59; p. 184...
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