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Unformatted text preview: 7.6. RELATIVE HOMOLOGY AND MORSE THEORY 201 • A relative boundary (mod ∂ M ) is a sum of an absolute boundary and a chain that lies on ∂ M . • Example. • The relative homology group is the quotient group of relative cycles mod- ulo the relative boundaries H p ( M , ∂ M ; G ) = Z p ( M , ∂ M ; G ) / B p ( M , ∂ M ; G ) . • Theorem 7.6.3 Let M be a compact Riemannian manifold with smooth boundary. Let k = B p ( M ) be the p-th Betti number. Let z (1) p , . . . , z ( k ) p , be a basis of real relative p-cycles of M (mod ∂ M ) in the real homology groups H p ( M , ∂ M ; R ) , that is, H p ( M , ∂ M ; R ) = k X i = 1 R c i . Let π 1 , . . . , π k be arbitrary real numbers. Then there is a unique normal harmonic p-form h p on M such that dh = δ h = , and D h p , z ( i ) p E = π i , i = 1 , 2 , . . . , k . Proof : 1. 7.6.2 Morse Theory • Let M be a closed manifold and f : M → R be a smooth function on M ....
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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