# Morse Inequalities

Let $X$ be a CW complex (with a fixed CW decomposition) with $c_d$ cells of dimension $d$. Let $\mathbb{F}$ be a field and let $b_d=\dim(H_d(X;\mathbb{F}))$.
(i) (The Weak Morse Inequalities) For each $d$,

$\displaystyle c_d\geq b_d.$
(ii)

$\chi(X)=b_0-b_1+b_2-\dots=c_0-c_1+c_2-\dots$,
where $\chi(X)$ denotes the Euler characteristic of $X$.

Proof:

The proof is by linear algebra (see Hatcher pg. 147).

By rank-nullity theorem (秩-零化度定理), $\dim C_d=\dim Z_d+\dim B_{d-1}$.

By definition of homology, $\dim Z_d=\dim B_d+\dim H_d$.

$\therefore c_d=\dim B_d+\dim B_{d-1}+b_d$.

In particular, $c_d\geq b_d$.

Taking alternating sum gives $\displaystyle \sum_d(-1)^d c_d=\sum_d(-1)^d b_d.$

Reference: A user’s guide to discrete Morse theory by R. Forman.

## Author: mathtuition88

https://mathtuition88.com/

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