Let be a CW complex (with a fixed CW decomposition) with cells of dimension . Let be a field and let .

(i) (The Weak Morse Inequalities) For each ,

(ii)

,

where denotes the Euler characteristic of .

Proof:

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

By rank-nullity theorem (秩-零化度定理), .

By definition of homology, .

.

In particular, .

Taking alternating sum gives

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

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