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.

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