Morse Inequality

Additivity

Let $S$ be a function from certain pairs of spaces to the integers. $S$ is subadditive if whenever $Z\subset Y\subset X$ we have $S(X,Y)\le S(X,Y)+S(Y,Z)$. If equality holds, $S$ is called additive.

Definition

Given any field $F$ as coefficient group, let $R_{\lambda }(X,Y)=\lambda \text{th Betti number of}(X,Y)=\text{rank over}F\text{of}{H}_{\lambda }(X,Y;F)$

$R_{\lambda }$ is subadditive, as is easily seen by examining the following portion of the exact sequence for $(X,Y,Z)$: $\cdots \to H_{\lambda }(Y,Z)\to H_{\lambda }(X,Z)\to H_{\lambda }(X,Y)\to \dots$

Lemma

Let $S$ be subadditive and let $X_0\subset \cdots \subset X_n$. Then $S(X_n,X_0)\le {\sum }_{i=1}^{n}S(X_i,X_{i-1})$. If $S$ is additive then equality holds.

Weak Morse Inequalities

If $c_{\lambda }$ denotes the number of critical points of index $\lambda$ on the compact manifold $M$ then

• $R_{\lambda }(M)\le c_{\lambda }$ and
• $\sum (-1{)}^{\lambda }{R}_{\lambda }(M)=\sum (-1{)}^{\lambda }{c}_{\lambda }$.

Lemma

The function $S_{\lambda }$ is subadditive, where $S_{\lambda }(X,Y)=R_{\lambda }(X,Y)-R_{\lambda -1}(X,Y)+R_{\lambda -2}(X,Y)-\cdots \pm R_0(X,Y).$

Morse Inequality

$S_{\lambda }(M)\le {\sum }_{i=1}^k{S}_{\lambda }(M^{a_i},M^{a_{i-1}})=c_{\lambda }-c_{\lambda -1}+\cdots \pm c_0$ or equivalently $R_{\lambda }(X,Y)-R_{\lambda -1}(X,Y)+R_{\lambda -2}(X,Y)-\cdots \pm R_0(X,Y)\le c_{\lambda }-c_{\lambda -1}+\cdots \pm c_0$