differential geometry w notes from teacher_Part_101

# differential geometry w notes from teacher_Part_101 - 7.6....

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

## This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

### Page1 / 2

differential geometry w notes from teacher_Part_101 - 7.6....

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online