Namespaces
Variants
Actions

Difference between revisions of "Morse inequalities"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
 +
<!--
 +
m0649701.png
 +
$#A+1 = 57 n = 0
 +
$#C+1 = 57 : ~/encyclopedia/old_files/data/M064/M.0604970 Morse inequalities
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m0649701.png" /> be a [[Morse function|Morse function]] on a smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m0649702.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m0649703.png" /> (without boundary) having a finite number of critical points. Then the [[Homology group|homology group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m0649704.png" /> is finitely generated and is therefore determined by its rank, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m0649705.png" />, and its torsion rank, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m0649706.png" /> (the torsion rank of an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m0649707.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m0649708.png" /> can be imbedded). The Morse inequalities relate the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m0649709.png" /> of critical points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497010.png" /> with [[Morse index|Morse index]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497011.png" /> to these ranks, and have the form:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497012.png" /></td> </tr></table>
+
$$
 +
r _  \lambda  + t _  \lambda  + t _ {\lambda - 1 }  \leq  m _  \lambda  ,\ \
 +
\lambda = 0 \dots n ;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497013.png" /></td> </tr></table>
+
$$
 +
\sum _ { i= } 0 ^  \lambda  ( - 1 ) ^ {\lambda - i } r _ {i}  \leq  \sum _ { i= } 0 ^  \lambda  ( - 1 ) ^ {\lambda - i } m _ {i} ,\  \lambda = 0 \dots n .
 +
$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497014.png" /> the last Morse inequality is always an equality, so that
+
For $  \lambda = n $
 +
the last Morse inequality is always an equality, so that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497015.png" /></td> </tr></table>
+
$$
 +
\sum _ { i= } 0 ^ { n }  ( - 1 )  ^ {i} m _ {i}  = \chi ( M) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497016.png" /> is the [[Euler characteristic|Euler characteristic]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497017.png" />.
+
where $  \chi ( M) $
 +
is the [[Euler characteristic|Euler characteristic]] of $  M $.
  
The Morse inequalities also hold for Morse functions of a triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497018.png" />, it suffices to replace the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497019.png" /> by the relative homology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497020.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497021.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497022.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497023.png" />, there is a Morse function with two critical points; hence it follows immediately (see [[Morse theory|Morse theory]]) that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497024.png" /> is homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497025.png" /> (see [[Poincaré conjecture|Poincaré conjecture]]). A similar application of Smale's theorem allows one to prove theorems on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497027.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497029.png" />-cobordism.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497030.png" /> on an infinite-dimensional Hilbert manifold, and they relate (for any regular values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497032.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497033.png" />) the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497034.png" /> of critical points of finite index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497035.png" /> lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497036.png" />, with the rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497037.png" /> and torsion rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497038.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497039.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497040.png" />. Namely,
+
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,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497041.png" /></td> </tr></table>
+
$$
 +
r _  \lambda  ( a , b ) +
 +
t _  \lambda  ( a , b ) +
 +
t _ {\lambda - 1 }  ( a , b )  \leq  m _  \lambda  ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497042.png" /></td> </tr></table>
+
$$
 +
\sum _ { i= } 0 ^  \lambda  ( - 1 ) ^ {\lambda - i }
 +
r _ {i} ( a , b )  \leq  \sum _ { i= } 0 ^  \lambda  ( - 1 ) ^ {\lambda - i } m _ {i} ;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497043.png" /></td> </tr></table>
+
$$
 +
\lambda  = 0 , 1 ,\dots .
 +
$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497044.png" /> large enough the latter inequality becomes an equality.
+
For $  \lambda $
 +
large enough the latter inequality becomes an equality.
  
 
====References====
 
====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>
 
<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====
 
====Comments====
 
Another version of the Morse inequalities can be stated as follows, cf. [[#References|[a1]]].
 
Another version of the Morse inequalities can be stated as follows, cf. [[#References|[a1]]].
  
For a Morse function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497045.png" /> one introduces the quantity
+
For a Morse function $  f $
 +
one introduces the quantity
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497046.png" /></td> </tr></table>
+
$$
 +
M _ {t} ( f  )  = \sum _ { p } t ^ {\lambda ( p) } ,
 +
$$
  
where the sum is taken over the critical points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497047.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497049.png" /> is the index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497050.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497051.png" />. In the compact case this sum is finite, since the critical points are discrete. The polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497052.png" />, which is also called the Morse polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497053.png" />, has the Poincaré polynomial of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497054.png" /> as a lower bound in the following sense. Let
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497055.png" /></td> </tr></table>
+
$$
 +
P _ {t} ( W)  = \sum t  ^ {k}  \mathop{\rm dim}  H _ {k} ( W ; K) ,
 +
$$
  
where the homology is taken relative to some fixed coefficient field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497056.png" />. Then the following Morse inequality holds: For every non-degenerate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497057.png" /> there exists a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497058.png" /> with non-negative coefficients such that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064970/m06497059.png" /></td> </tr></table>
+
$$
 +
M _ {t} ( f  ) - P _ {t} ( f  )  = ( 1 + t ) \cdot Q _ {t} ( f  ) .
 +
$$
  
 
====References====
 
====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>
 
<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>

Revision as of 08:01, 6 June 2020


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
How to Cite This Entry:
Morse inequalities. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Morse_inequalities&oldid=17143
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