Manifolds and cell complexes Morse theory and handles Theorem

Morse inequalities

Builds on Smooth manifold Morse function Singular homology Betti number Euler characteristic

Statement

Let $M$ be a compact smooth manifold without boundary, of dimension $n$ , and let $f \colon M \to \R$ be a Morse function. For $0 \le k \le n$ , let $c_k$ be the number of critical points of $f$ of index $k$ . Let $F$ be a field, and let $b_k = \dim_F H_k(M; F)$ be the $k$ -th Betti number of $M$ over $F$ , where $H_k$ is singular homology. Then:

(weak Morse inequalities) $c_k \ge b_k$ for every $k$ ;
(strong Morse inequalities) $\sum_{i=0}^{k} (-1)^{k-i} c_i \ge \sum_{i=0}^{k} (-1)^{k-i} b_i$ for every $k$ ;
$\sum_{k=0}^{n} (-1)^k c_k = \sum_{k=0}^{n} (-1)^k b_k = \chi(M)$ , where $\chi(M)$ is the Euler characteristic of $M$ .

Remarks

The idea of the proof: as $c$ increases, the sublevel set $\set{f \le c}$ changes up to homotopy only when $c$ crosses a critical value. At such a value it changes by attaching one cell of dimension $k$ for each critical point of index $k$ . So $M$ is homotopy equivalent to a CW complex with $c_k$ cells of dimension $k$ . The inequalities follow from the cellular chain complex of this CW complex. The strong inequalities imply the weak ones, by adding the inequalities for $k$ and $k - 1$ . Item 3 is the Euler characteristic of the CW complex, computed once from cells and once from homology. Example: the height function on an upright torus has $c_0 = 1$ , $c_1 = 2$ , $c_2 = 1$ , and the torus has $b_0 = 1$ , $b_1 = 2$ , $b_2 = 1$ , so all inequalities are equalities. Consequence: every Morse function on a closed orientable surface of genus $g$ has at least $2g + 2$ critical points.

Going deeper

For a student

The inequalities are the first quantitative link between the analysis of a function and the topology of its domain. They can be used in two directions. They bound the number of critical points of a function, as in Morse's results on geodesics. They also bound Betti numbers, once one constructs a function with few critical points. A Morse function for which all inequalities are equalities is called perfect. Morse homology refines the inequalities by building a chain complex on the critical points whose homology is $H_*(M)$ .

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Tag	`MH7MEB0MRR`
URI	`m2://MH7MEB0MRR`
Kind	statement, theorem
Writing	W0 stub
MSC	57R70, 58E05
Status	This copy of the site has no build data. Run `m2 status MH7MEB0MRR`.
About	Morse function
Uses	Smooth manifold, Morse function, Singular homology, Betti number, Euler characteristic
Changed	2 November 2026