MORSE THEORY for "The Yang-Mills...MORSE THEORY (ref "Morse Theory by J.Milnor") Morse Lemma Let be a nondegenerated critical point of on , then there exists a chart such that , where

MORSE THEORY

(ref "Morse Theory by J.Milnor")

Morse Lemma Let be a nondegenerated critical point of on , then thereexists a chart such that ,where is called the index of at . We obtain a fact that non-degeneratedcirtical points are isolated.

Homotopy type in Term of Critical Values

Def*

Thm 1 Let be a smooth real value function on , and suppose iscompact and contains no critical point. Then is differeomorphic to .Moreover is a deformation retract of .

Sketch of proof: choose a Riemann metric on and then Let in . Choose in and

lies in . Let there exists a one parameter group associate to .Then we have =1. Now consider . It carries diffeomorphically to . And if ; else.

Thm 2 Let be a smooth fucntion on , and let be a nondegenerated criticalpoint with index . Setting , and suppose that iscompact and contains no cirtical point other than , for some . Then, forall sufficiently small , the set has the homotopy type of with attached.

sketch of proof: First choose a nghd of mentioned in Morse Lemma and asrequired and

in thisTheorem. We claim that there exists a function such that: 1,

; 2,The cirtical point of are the same as those of ;3,The region is a deformation retract of . Construction of

. Where . Denote as

. Moreover we can prove that is a defoemationretract of .

RMK: is homotopic to .

Thm 3 If is a diifferentiable function on a manifold with no critical points,and if is compact, then have a homotopy type of a CW-complex, withone cell of dimension for each critical point of index .

The Morse Inequalities

Def = th Betti number of =rank over of .

Lemma 4: is subadditive since we have the exact sequence:

pf: Since is a additive function, we have .

def 5 Euler characteristic

Lemma 6: is additive.

pf

.

Thm 7 Let be a compact manifold and be a differentiable function on Mwith isolated, nondegenerated, critical points, and let be suchthat contains exactly cirtical points (W.L.O.G. we can choose satisfyingthis condition by perturbation). Then

.Let denote the number of critical point of index . Then we conclude theweak morse inequalities . And

.

Thm 8 (the more subtle one) is sub additive, where . We obseve that

so we obtain . Applying this function on

. And hence . Henceforth we know

that .

Cor 9 Suppose that then we know that , by the inequalityabove we obtain . Also suppose that

so we can deduce that .

Cor 10 Let and

. We know that and hence we use power series of and the subtle morse

inequality we know that the coefficients of are non-negative.

Thm 11 Let . , where is a polynomial with non-negative coefficients.

pf: using the axioms for cohomology alike, we deduce that

. Define alike, by the same reasoning we obtain .