# Morse inequalities

Inequalities following from Morse theory and relating the number of critical points (cf. Critical point) of a Morse function on a manifold to its homology invariants.

Let $f$ be a Morse function on a smooth $n$-dimensional manifold $M$ (without boundary) having a finite number of critical points. Then the homology group $H _ \lambda ( M)$ is finitely generated and is therefore determined by its rank, $r _ \lambda = \mathop{\rm rk} ( H _ \lambda ( M) )$, and its torsion rank, $t _ \lambda = t ( H _ \lambda ( M) )$ (the torsion rank of an Abelian group $A$ with a finite number of generators is the minimal number of cyclic groups in a direct-sum decomposition of which a maximal torsion subgroup of $A$ can be imbedded). The Morse inequalities relate the number $m _ \lambda$ of critical points of $f$ with Morse index $\lambda$ to these ranks, and have the form:

$$r _ \lambda + t _ \lambda + t _ {\lambda - 1 } \leq m _ \lambda ,\ \ \lambda = 0 \dots n ;$$

$$\sum _ { i= 0} ^ \lambda ( - 1 ) ^ {\lambda - i } r _ {i} \leq \sum _ { i= 0} ^ \lambda ( - 1 ) ^ {\lambda - i } m _ {i} ,\ \lambda = 0 \dots n .$$

For $\lambda = n$ the last Morse inequality is always an equality, so that

$$\sum _ { i= 0} ^ { n } ( - 1 ) ^ {i} m _ {i} = \chi ( M) ,$$

where $\chi ( M)$ is the Euler characteristic of $M$.

The Morse inequalities also hold for Morse functions of a triple $( W , V _ {0} , V _ {1} )$, it suffices to replace the groups $H _ \lambda ( M)$ by the relative homology groups $H _ \lambda ( W , V _ {0} )$.

According to the Morse inequalities, a manifold having "large" homology groups does not admit a Morse function with a small number of critical points. It is remarkable that the estimates in the Morse inequalities are sharp: On a closed simply-connected manifold of dimension $n \geq 6$ there is a Morse function for which the Morse inequalities are equalities (Smale's theorem, see [2]). In particular, on any closed manifold that is homotopically equivalent to the sphere $S ^ {n}$, with $n \geq 6$, there is a Morse function with two critical points; hence it follows immediately (see Morse theory) that $M$ is homeomorphic to $S ^ {n}$ (see Poincaré conjecture). A similar application of Smale's theorem allows one to prove theorems on $h$- and $s$-cobordism.

An analogue of the Morse inequalities holds for a Morse function $f : X \rightarrow \mathbf R$ on an infinite-dimensional Hilbert manifold, and they relate (for any regular values $a , b \in \mathbf R$, $a < b$, of $f$) the numbers $m _ \lambda ( a , b )$ of critical points of finite index $\lambda$ lying in $f ^ { - 1 } [ a , b]$, with the rank $r _ \lambda ( a , b )$ and torsion rank $t _ \lambda ( a , b )$ of the group $H _ \lambda ( M ^ {b} , M ^ {a} )$, where $M ^ {c} = f ^ { - 1 } ( - \infty , c ]$. Namely,

$$r _ \lambda ( a , b ) + t _ \lambda ( a , b ) + t _ {\lambda - 1 } ( a , b ) \leq m _ \lambda ,$$

$$\sum _ { i= 0} ^ \lambda ( - 1 ) ^ {\lambda - i } r _ {i} ( a , b ) \leq \sum _ { i= 0} ^ \lambda ( - 1 ) ^ {\lambda - i } m _ {i} ;$$

$$\lambda = 0 , 1 ,\dots .$$

For $\lambda$ large enough the latter inequality becomes an equality.

#### References

 [1] M. Morse, "The calculus of variations in the large" , Amer. Math. Soc. (1934) [2] S. Smale, "Generalized Poincaré's conjecture in dimensions greater than four" Ann. of Math. , 74 (1961) pp. 391–466

Another version of the Morse inequalities can be stated as follows, cf. [a1].

For a Morse function $f$ one introduces the quantity

$$M _ {t} ( f ) = \sum _ { p } t ^ {\lambda ( p) } ,$$

where the sum is taken over the critical points $p$ of $f$ and $\lambda ( p)$ is the index of $p$ relative to $f$. In the compact case this sum is finite, since the critical points are discrete. The polynomial $M _ {t} ( f )$, which is also called the Morse polynomial of $f$, has the Poincaré polynomial of the manifold $W$ as a lower bound in the following sense. Let

$$P _ {t} ( W) = \sum t ^ {k} \mathop{\rm dim} H _ {k} ( W ; K) ,$$

where the homology is taken relative to some fixed coefficient field $K$. Then the following Morse inequality holds: For every non-degenerate $f$ there exists a polynomial $Q _ {t} ( f ) = q _ {0} + q _ {1} t + \dots$ with non-negative coefficients such that

$$M _ {t} ( f ) - P _ {t} ( f ) = ( 1 + t ) \cdot Q _ {t} ( f ) .$$

#### References

 [a1] R. Bott, "Lectures on Morse theory, old and new" Bull. Amer. Math. Soc. , 7 : 2 (1982) pp. 331–358 [a2] J.W. Milnor, "Morse theory" , Princeton Univ. Press (1963) [a3] R.S. Palais, "Morse theory on Hilbert manifolds" Topology , 2 (1963) pp. 299–340
How to Cite This Entry:
Morse inequalities. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Morse_inequalities&oldid=52190
This article was adapted from an original article by M.M. PostnikovYu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article