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 ,
where denotes the Euler characteristic of .
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.