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

Comments

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