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

Let  $  f $
be a [[Morse function|Morse function]] on a smooth  $  n $-dimensional manifold  $  M $ (without boundary) having a finite number of critical points. Then the [[Homology group|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|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|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 [[#References|[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|Morse theory]]) that  $  M $
is homeomorphic to  $  S  ^ {n} $ (see [[Poincaré conjecture|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Morse,   "The calculus of variations in the large" , Amer. Math. Soc.  (1934)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Smale,   "Generalized Poincaré's conjecture in dimensions greater than four"  ''Ann. of Math.'' , '''74'''  (1961)  pp. 391–466</TD></TR></table>

====Comments====
Another version of the Morse inequalities can be stated as follows, cf. [[#References|[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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Bott,   "Lectures on Morse theory, old and new"  ''Bull. Amer. Math. Soc.'' , '''7''' :  2  (1982)  pp. 331–358</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.W. Milnor,   "Morse theory" , Princeton Univ. Press  (1963)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.S. Palais,   "Morse theory on Hilbert manifolds"  ''Topology'' , '''2'''  (1963)  pp. 299–340</TD></TR></table>