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A [[Bordism|bordism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h0460102.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h0460103.png" /> is a compact manifold whose boundary is the disjoint union of closed manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h0460104.png" /> which are deformation retracts (cf. [[Deformation retract|Deformation retract]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h0460105.png" />. The simplest example is the trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h0460107.png" />-cobordism
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<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/h/h046/h046010/h0460108.png" /></td> </tr></table>
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Two manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h0460109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601010.png" /> are said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601012.png" />-cobordant if there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601013.png" />-cobordism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601014.png" /> joining them.
+
A [[Bordism|bordism]]  $  ( W;  M _ {0} , M _ {1} ) $,
 +
where  $  W $
 +
is a compact manifold whose boundary is the disjoint union of closed manifolds $  M _ {0} , M _ {1} $
 +
which are deformation retracts (cf. [[Deformation retract|Deformation retract]]) of  $  W $.  
 +
The simplest example is the trivial  $  h $-
 +
cobordism
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601015.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601016.png" />-cobordism such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601019.png" /> are simply-connected differentiable (or piecewise-linear) manifolds and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601021.png" /> is diffeomorphic (or piecewise-linearly isomorphic) to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601022.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601023.png" /> and therefore <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601024.png" /> (the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601026.png" />-cobordism theorem [[#References|[4]]]). Thus, proving the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601027.png" /> reduces to providing an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601028.png" />-cobordism, which can be achieved by methods of algebraic topology. For this reason, this theorem is basic in passing from the homotopy classification of simply-connected manifolds to their classification up to a diffeomorphism (or a piecewise-linear isomorphism). Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601030.png" />, is a compact differentiable manifold with simply-connected boundary, then it is diffeomorphic to the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601031.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601033.png" />, is a manifold that is homotopy equivalent to the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601034.png" />, then it is homeomorphic (and even piecewise-linearly isomorphic) to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601035.png" /> (the generalized Poincaré conjecture).
+
$$
 +
( M \times [ 0, 1];  M \times \{ 0 \} , M \times \{ 1 \} ).
 +
$$
  
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601036.png" />-cobordism theorem allows one to classify the differentiable structures on the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601038.png" /> [[#References|[6]]], and also on the homotopy type of an arbitrary closed simply-connected manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601040.png" /> [[#References|[1]]].
+
Two manifolds  $  M _ {0} $
 +
and  $  M _ {1} $
 +
are said to be  $  h $-
 +
cobordant if there is an  $  h $-
 +
cobordism  $  W $
 +
joining them.
  
In the case of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601041.png" />-cobordism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601042.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601043.png" /> there is, in general, no diffeomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601044.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601045.png" />.
+
If  $  ( W;  M _ {0} , M _ {1} ) $
 +
is an h $-
 +
cobordism such that  $  W $,
 +
$  M _ {0} $,
 +
$  M _ {1} $
 +
are simply-connected differentiable (or piecewise-linear) manifolds and  $  \mathop{\rm dim}  W \geq  6 $,
 +
then  $  W $
 +
is diffeomorphic (or piecewise-linearly isomorphic) to  $  M _ {0} \times [ 0, 1] $:  
 +
$  W \approx M _ {0} \times [ 0, 1] $
 +
and therefore  $  M _ {0} \approx M _ {1} $(
 +
the  $  h $-
 +
cobordism theorem [[#References|[4]]]). Thus, proving the isomorphism  $  M _ {0} \approx M _ {1} $
 +
reduces to providing an  $  h $-
 +
cobordism, which can be achieved by methods of algebraic topology. For this reason, this theorem is basic in passing from the homotopy classification of simply-connected manifolds to their classification up to a diffeomorphism (or a piecewise-linear isomorphism). Thus, if  $  W  ^ {n} $,
 +
$  n \geq  6 $,
 +
is a compact differentiable manifold with simply-connected boundary, then it is diffeomorphic to the disc  $  D  ^ {n} $.  
 +
If  $  M  ^ {n} $,
 +
$  n \geq  5 $,
 +
is a manifold that is homotopy equivalent to the sphere  $  S  ^ {n} $,
 +
then it is homeomorphic (and even piecewise-linearly isomorphic) to $  S  ^ {n} $(
 +
the generalized Poincaré conjecture).
  
All <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601046.png" />-cobordisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601047.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601049.png" /> fixed are classified by a certain Abelian group, namely the [[Whitehead group|Whitehead group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601050.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601051.png" />. Corresponding to a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601052.png" />-cobordism is an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601053.png" /> that is an invariant of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601054.png" />; it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601055.png" /> and is called the torsion (sometimes the [[Whitehead torsion|Whitehead torsion]]) of the given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601057.png" />-cobordism. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601058.png" /> (or, equivalently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601059.png" />), then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601060.png" />-cobordism is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601062.png" />-cobordism. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601063.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601064.png" />-cobordism such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601065.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601066.png" /> vanishes if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601067.png" /> (the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601069.png" />-cobordism theorem). The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601070.png" />-cobordism theorem is a special case of this theorem in view of the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601071.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601072.png" />-cobordism theorem is also true for topological manifolds [[#References|[9]]].
+
The  $  h $-
 +
cobordism theorem allows one to classify the differentiable structures on the sphere  $  S  ^ {n} $,
 +
$  n \geq  5 $[[#References|[6]]], and also on the homotopy type of an arbitrary closed simply-connected manifold  $  M  ^ {n} $,  
 +
$  n \geq  5 $[[#References|[1]]].
  
For an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601073.png" />-cobordism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601074.png" />, the torsion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601075.png" /> is defined along with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601076.png" />; if the given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601077.png" />-cobordism is orientable, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601078.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601079.png" /> and the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601080.png" /> is conjugate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601081.png" /> in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601082.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601083.png" /> is finite and Abelian, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601084.png" />.
+
In the case of an h $-
 +
cobordism $  ( W;  M _ {0} , M _ {1} ) $
 +
with $  \pi _ {1} W \neq \{ 1 \} $
 +
there is, in general, no diffeomorphism from  $  W $
 +
to $  M _ {0} \times [ 0, 1] $.
  
If two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601085.png" />-cobordisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601087.png" /> are glued along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601088.png" /> to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601089.png" />-cobordism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601090.png" />, then
+
All  $  h $-
 +
cobordisms  $  ( W;  M _ {0} , M _ {1} ) $
 +
with  $  \mathop{\rm dim}  W \geq  6 $
 +
and  $  M _ {0} $
 +
fixed are classified by a certain Abelian group, namely the [[Whitehead group|Whitehead group]]  $  \mathop{\rm Wh}  \pi _ {1} $
 +
of the group  $  \pi _ {1} M _ {0} $.  
 +
Corresponding to a given  $  h $-
 +
cobordism is an element of  $  \mathop{\rm Wh}  \pi _ {1} $
 +
that is an invariant of the pair  $  ( W, M _ {0} ) $;
 +
it is denoted by  $  \tau ( W, M _ {0} ) $
 +
and is called the torsion (sometimes the [[Whitehead torsion|Whitehead torsion]]) of the given  $  h $-
 +
cobordism. If  $  \tau ( W, M _ {0} ) = 0 $(
 +
or, equivalently,  $  \tau ( W, M _ {1} ) = 0 $),
 +
then the  $  h $-
 +
cobordism is called an  $  s $-
 +
cobordism. If  $  ( W;  M _ {0} , M _ {1} ) $
 +
is an  $  h $-
 +
cobordism such that  $  \mathop{\rm dim}  W \geq  6 $,
 +
then  $  \tau ( W, M _ {0} ) $
 +
vanishes if and only if  $  W \approx M _ {0} \times [ 0, 1] $(
 +
the $  s $-
 +
cobordism theorem). The  $  h $-
 +
cobordism theorem is a special case of this theorem in view of the fact that  $  \mathop{\rm Wh}  \{ 1 \} = 0 $.  
 +
The  $  s $-
 +
cobordism theorem is also true for topological manifolds [[#References|[9]]].
  
<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/h/h046/h046010/h04601091.png" /></td> </tr></table>
+
For an  $  h $-
 +
cobordism  $  ( W; M _ {0} , M _ {1} ) $,
 +
the torsion  $  \tau ( W, M _ {1} ) $
 +
is defined along with  $  \tau ( W, M _ {0} ) $;  
 +
if the given  $  h $-
 +
cobordism is orientable, then  $  \tau ( W, M _ {0} ) = (- 1) ^ {n - 1 } \tau  ^ {*} ( W, M _ {1} ) $,
 +
where  $  n = \mathop{\rm dim}  W $
 +
and the element  $  \tau  ^ {*} $
 +
is conjugate to  $  \tau $
 +
in the group  $  \mathop{\rm Wh}  \pi _ {1} $.
 +
In particular, if  $  \pi _ {1} $
 +
is finite and Abelian,  $  \tau  ^ {*} = \tau $.
  
If two copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601092.png" /> are glued along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601093.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601094.png" /> is odd and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601095.png" />, then one obtains an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601096.png" />-cobordism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601097.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601098.png" /> when there is no diffeomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h04601099.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010100.png" />, that is, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010101.png" /> does not imply that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010102.png" />-cobordism connecting them is trivial.
+
If two h $-
 +
cobordisms  $  ( W;  M _ {0} , M _ {1} ) $
 +
and $  ( W  ^  \prime  ;  M _ {1} , M _ {2} ) $
 +
are glued along  $  M _ {1} $
 +
to the  $  h $-
 +
cobordism $  ( W \cup W  ^  \prime  ;  M _ {0} , M _ {1} ) $,  
 +
then
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010103.png" /> is a closed connected manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010104.png" />, then there exists for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010105.png" /> an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010106.png" />-cobordism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010107.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010108.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010110.png" /> (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010111.png" />) have the same torsion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010112.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010113.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010114.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010115.png" /> is even and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010116.png" /> is finite, there is a finite set of distinct manifolds that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010117.png" />-cobordant with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010118.png" />. This is not the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010119.png" /> is odd.
+
$$
 +
\tau ( W \cup W  ^  \prime  , M _ {0} )  = \
 +
\tau ( W, M _ {0} ) + \tau ( W  ^  \prime  , M _ {1} ).
 +
$$
  
If two homotopy-equivalent manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010120.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010121.png" /> are imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010122.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010123.png" /> sufficiently large, and their normal bundles are trivial, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010124.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010125.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010126.png" />-cobordant. If, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010127.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010128.png" /> are of the same [[Simple homotopy type|simple homotopy type]], that is, if the torsion of this homotopy equivalence vanishes, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010129.png" />.
+
If two copies of  $  W $
 +
are glued along  $  M _ {1} $,  
 +
where  $  \mathop{\rm dim}  W $
 +
is odd and  $  \pi _ {1} = \mathbf Z _ {5} $,  
 +
then one obtains an  $  h $-
 +
cobordism  $  ( 2W;  M _ {0} , M _ {0}  ^  \prime  ) $,  
 +
where  $  M _ {0} = M _ {0}  ^  \prime  $
 +
when there is no diffeomorphism from  $  W $
 +
to  $  M _ {0} \times [ 0, 1] $,  
 +
that is, when  $  M _ {0} \approx M _ {1} $
 +
does not imply that the h $-
 +
cobordism connecting them is trivial.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010130.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010131.png" />-cobordism and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010132.png" /> is a closed manifold, then there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010133.png" />-cobordism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010134.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010135.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010136.png" /> is the [[Euler characteristic|Euler characteristic]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010137.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010138.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010139.png" />, then
+
If $  M _ {0} $
 +
is a closed connected manifold and  $  \mathop{\rm dim}  M _ {0} \geq  5 $,  
 +
then there exists for any  $  \tau \in  \mathop{\rm Wh}  \pi _ {1} M _ {0} $
 +
an h $-
 +
cobordism $  ( W;  M _ {0} , M _ {1} ) $
 +
with  $  \tau ( W, M _ {0} ) = \tau $.  
 +
If  $  ( W;  M _ {0} , M _ {1} ) $
 +
and  $  ( W  ^  \prime  ;  M _ {0} , M _ {1}  ^  \prime  ) $(
 +
with $  \mathop{\rm dim}  W \geq  6 $)
 +
have the same torsion  $  \tau ( W, M _ {0} ) = \tau ( W  ^  \prime  , M _ {0} ) $,  
 +
then  $  W \approx W  ^  \prime  $
 +
relative to  $  M _ {0} $.  
 +
When  $  \mathop{\rm dim}  M _ {0} $
 +
is even and  $  \pi _ {1} M _ {0} $
 +
is finite, there is a finite set of distinct manifolds that are  $  h $-
 +
cobordant with  $  M _ {0} $.  
 +
This is not the case when  $  \mathop{\rm dim}  M _ {0} $
 +
is odd.
  
<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/h/h046/h046010/h046010140.png" /></td> </tr></table>
+
If two homotopy-equivalent manifolds  $  M _ {1} $
 +
and  $  M _ {2} $
 +
are imbedded in  $  \mathbf R  ^ {N} $,
 +
with  $  N $
 +
sufficiently large, and their normal bundles are trivial, then  $  M _ {1} \times S  ^ {N} $
 +
and  $  M _ {2} \times S  ^ {N} $
 +
are  $  h $-
 +
cobordant. If, moreover,  $  M _ {1} $
 +
and  $  M _ {2} $
 +
are of the same [[Simple homotopy type|simple homotopy type]], that is, if the torsion of this homotopy equivalence vanishes, then  $  M _ {1} \times S  ^ {N} \approx M _ {2} \times S  ^ {N} $.
  
In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010141.png" />; furthermore, two closed manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010143.png" /> of the same dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010144.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010145.png" />-cobordant if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010146.png" />.
+
If  $  ( W;  M _ {0} , M _ {1} ) $
 +
is an  $  h $-
 +
cobordism and  $  P $
 +
is a closed manifold, then there is an  $  h $-
 +
cobordism  $  ( W \times P; M _ {0} \times P, M _ {1} \times P) $
 +
with  $  \tau ( W \times P, M _ {0} \times P) = \tau ( W, M _ {0} ) \chi ( P) $,
 +
where  $  \chi ( P) $
 +
is the [[Euler characteristic|Euler characteristic]] of $  P $.  
 +
If  $  \mathop{\rm dim}  W \geq  5 $
 +
and $  P = S  ^ {1} $,
 +
then
  
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010147.png" />-cobordism structure has not been completely elucidated for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010148.png" /> (1978). Thus there is the following negative result [[#References|[8]]]: There exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010149.png" />-cobordism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010150.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010151.png" /> is the four-dimensional torus, for which there is no diffeomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010152.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010153.png" />; since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010154.png" />, this means that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010155.png" />-cobordism theorem fails for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010156.png" />.
+
$$
 +
W \times S  ^ {1}  \approx \
 +
M _ {0} \times S  ^ {1} \times [ 0, 1]  \approx \
 +
M _ {1} \times S  ^ {1} \times [ 0, 1].
 +
$$
 +
 
 +
In particular,  $  M _ {0} \times S  ^ {1} \approx M _ {1} \times S  ^ {1} $;
 +
furthermore, two closed manifolds  $  M _ {0} $
 +
and  $  M _ {1} $
 +
of the same dimension  $  \geq  5 $
 +
are  $  h $-
 +
cobordant if and only if  $  M _ {0} \times \mathbf R  ^ {1} \approx M _ {1} \times \mathbf R  ^ {1} $.
 +
 
 +
The  $  h $-
 +
cobordism structure has not been completely elucidated for $  n < 6 $(
 +
1978). Thus there is the following negative result [[#References|[8]]]: There exists an h $-
 +
cobordism $  ( W;  T  ^ {4} , T  ^ {4} ) $,  
 +
where $  T  ^ {4} $
 +
is the four-dimensional torus, for which there is no diffeomorphism from $  W $
 +
to $  T  ^ {4} \times [ 0, 1] $;  
 +
since $  \mathop{\rm Wh}  \pi _ {1} T  ^ {4} = 0 $,  
 +
this means that the $  s $-
 +
cobordism theorem fails for $  n = 5 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.P. Novikov,  "Homotopy-equivalent smooth manifolds I"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''28''' :  2  (1964)  pp. 365–474  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Milnor,  "Lectures on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010157.png" />-cobordism theorem" , Princeton Univ. Press  (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Milnor,  "Whitehead torsion"  ''Bull. Amer. Math. Soc.'' , '''72'''  (1966)  pp. 358–462</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Smale,  "On the structure of manifolds"  ''Amer. J. Math.'' , '''84'''  (1962)  pp. 387–399</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Milnor,  "Sommes des variétés différentiables et structures différentiables des sphères"  ''Bull. Soc. Math. France'' , '''87'''  (1959)  pp. 439–444</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M. Kervaire,  J. Milnor,  "Groups of homotopy spheres I"  ''Ann. of Math. (2)'' , '''77'''  (1963)  pp. 504–537</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  B. Mazur,  "Relative neighbourhoods and the theorems of Smale"  ''Ann. of Math.'' , '''77'''  (1963)  pp. 232–249</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  L.C. Siebenmann,  "Disruption of low-dimensional handlebody theory by Rohlin's theorem"  J.C. Cantrell (ed.)  C.H. Edwards jr. (ed.) , ''Topology of manifolds'' , Markham  (1969)  pp. 57–76</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  R. Kirby,  L. Siebenmann,  "On the triangulation of manifolds and the Hauptvermutung"  ''Bull. Amer. Math. Soc.'' , '''75'''  (1969)  pp. 742–749</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M.A. Kervaire,  "Le théorème de Barden–Mazur–Stallings"  M.A. Kervaire (ed.)  G. de Rham (ed.)  S. Maumary (ed.) , ''Torsion et type simple d'homotopie'' , ''Lect. notes in math.'' , '''48''' , Springer  (1967)  pp. 83–95</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  R. Thom,  "Les classes caractéristiques de Pontryagin des variétés triangulées" , ''Symp. Internac. Topol. Algebr.'' , Univ. Nac. Aut. Mexico &amp; UNESCO  (1958)  pp. 54–67</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  C.P. Rourke,  B.J. Sanderson,  "Introduction to piecewise-linear topology" , Springer  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.P. Novikov,  "Homotopy-equivalent smooth manifolds I"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''28''' :  2  (1964)  pp. 365–474  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Milnor,  "Lectures on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046010/h046010157.png" />-cobordism theorem" , Princeton Univ. Press  (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Milnor,  "Whitehead torsion"  ''Bull. Amer. Math. Soc.'' , '''72'''  (1966)  pp. 358–462</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Smale,  "On the structure of manifolds"  ''Amer. J. Math.'' , '''84'''  (1962)  pp. 387–399</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Milnor,  "Sommes des variétés différentiables et structures différentiables des sphères"  ''Bull. Soc. Math. France'' , '''87'''  (1959)  pp. 439–444</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M. Kervaire,  J. Milnor,  "Groups of homotopy spheres I"  ''Ann. of Math. (2)'' , '''77'''  (1963)  pp. 504–537</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  B. Mazur,  "Relative neighbourhoods and the theorems of Smale"  ''Ann. of Math.'' , '''77'''  (1963)  pp. 232–249</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  L.C. Siebenmann,  "Disruption of low-dimensional handlebody theory by Rohlin's theorem"  J.C. Cantrell (ed.)  C.H. Edwards jr. (ed.) , ''Topology of manifolds'' , Markham  (1969)  pp. 57–76</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  R. Kirby,  L. Siebenmann,  "On the triangulation of manifolds and the Hauptvermutung"  ''Bull. Amer. Math. Soc.'' , '''75'''  (1969)  pp. 742–749</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M.A. Kervaire,  "Le théorème de Barden–Mazur–Stallings"  M.A. Kervaire (ed.)  G. de Rham (ed.)  S. Maumary (ed.) , ''Torsion et type simple d'homotopie'' , ''Lect. notes in math.'' , '''48''' , Springer  (1967)  pp. 83–95</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  R. Thom,  "Les classes caractéristiques de Pontryagin des variétés triangulées" , ''Symp. Internac. Topol. Algebr.'' , Univ. Nac. Aut. Mexico &amp; UNESCO  (1958)  pp. 54–67</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  C.P. Rourke,  B.J. Sanderson,  "Introduction to piecewise-linear topology" , Springer  (1972)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 19:42, 5 June 2020


A bordism $ ( W; M _ {0} , M _ {1} ) $, where $ W $ is a compact manifold whose boundary is the disjoint union of closed manifolds $ M _ {0} , M _ {1} $ which are deformation retracts (cf. Deformation retract) of $ W $. The simplest example is the trivial $ h $- cobordism

$$ ( M \times [ 0, 1]; M \times \{ 0 \} , M \times \{ 1 \} ). $$

Two manifolds $ M _ {0} $ and $ M _ {1} $ are said to be $ h $- cobordant if there is an $ h $- cobordism $ W $ joining them.

If $ ( W; M _ {0} , M _ {1} ) $ is an $ h $- cobordism such that $ W $, $ M _ {0} $, $ M _ {1} $ are simply-connected differentiable (or piecewise-linear) manifolds and $ \mathop{\rm dim} W \geq 6 $, then $ W $ is diffeomorphic (or piecewise-linearly isomorphic) to $ M _ {0} \times [ 0, 1] $: $ W \approx M _ {0} \times [ 0, 1] $ and therefore $ M _ {0} \approx M _ {1} $( the $ h $- cobordism theorem [4]). Thus, proving the isomorphism $ M _ {0} \approx M _ {1} $ reduces to providing an $ h $- cobordism, which can be achieved by methods of algebraic topology. For this reason, this theorem is basic in passing from the homotopy classification of simply-connected manifolds to their classification up to a diffeomorphism (or a piecewise-linear isomorphism). Thus, if $ W ^ {n} $, $ n \geq 6 $, is a compact differentiable manifold with simply-connected boundary, then it is diffeomorphic to the disc $ D ^ {n} $. If $ M ^ {n} $, $ n \geq 5 $, is a manifold that is homotopy equivalent to the sphere $ S ^ {n} $, then it is homeomorphic (and even piecewise-linearly isomorphic) to $ S ^ {n} $( the generalized Poincaré conjecture).

The $ h $- cobordism theorem allows one to classify the differentiable structures on the sphere $ S ^ {n} $, $ n \geq 5 $[6], and also on the homotopy type of an arbitrary closed simply-connected manifold $ M ^ {n} $, $ n \geq 5 $[1].

In the case of an $ h $- cobordism $ ( W; M _ {0} , M _ {1} ) $ with $ \pi _ {1} W \neq \{ 1 \} $ there is, in general, no diffeomorphism from $ W $ to $ M _ {0} \times [ 0, 1] $.

All $ h $- cobordisms $ ( W; M _ {0} , M _ {1} ) $ with $ \mathop{\rm dim} W \geq 6 $ and $ M _ {0} $ fixed are classified by a certain Abelian group, namely the Whitehead group $ \mathop{\rm Wh} \pi _ {1} $ of the group $ \pi _ {1} M _ {0} $. Corresponding to a given $ h $- cobordism is an element of $ \mathop{\rm Wh} \pi _ {1} $ that is an invariant of the pair $ ( W, M _ {0} ) $; it is denoted by $ \tau ( W, M _ {0} ) $ and is called the torsion (sometimes the Whitehead torsion) of the given $ h $- cobordism. If $ \tau ( W, M _ {0} ) = 0 $( or, equivalently, $ \tau ( W, M _ {1} ) = 0 $), then the $ h $- cobordism is called an $ s $- cobordism. If $ ( W; M _ {0} , M _ {1} ) $ is an $ h $- cobordism such that $ \mathop{\rm dim} W \geq 6 $, then $ \tau ( W, M _ {0} ) $ vanishes if and only if $ W \approx M _ {0} \times [ 0, 1] $( the $ s $- cobordism theorem). The $ h $- cobordism theorem is a special case of this theorem in view of the fact that $ \mathop{\rm Wh} \{ 1 \} = 0 $. The $ s $- cobordism theorem is also true for topological manifolds [9].

For an $ h $- cobordism $ ( W; M _ {0} , M _ {1} ) $, the torsion $ \tau ( W, M _ {1} ) $ is defined along with $ \tau ( W, M _ {0} ) $; if the given $ h $- cobordism is orientable, then $ \tau ( W, M _ {0} ) = (- 1) ^ {n - 1 } \tau ^ {*} ( W, M _ {1} ) $, where $ n = \mathop{\rm dim} W $ and the element $ \tau ^ {*} $ is conjugate to $ \tau $ in the group $ \mathop{\rm Wh} \pi _ {1} $. In particular, if $ \pi _ {1} $ is finite and Abelian, $ \tau ^ {*} = \tau $.

If two $ h $- cobordisms $ ( W; M _ {0} , M _ {1} ) $ and $ ( W ^ \prime ; M _ {1} , M _ {2} ) $ are glued along $ M _ {1} $ to the $ h $- cobordism $ ( W \cup W ^ \prime ; M _ {0} , M _ {1} ) $, then

$$ \tau ( W \cup W ^ \prime , M _ {0} ) = \ \tau ( W, M _ {0} ) + \tau ( W ^ \prime , M _ {1} ). $$

If two copies of $ W $ are glued along $ M _ {1} $, where $ \mathop{\rm dim} W $ is odd and $ \pi _ {1} = \mathbf Z _ {5} $, then one obtains an $ h $- cobordism $ ( 2W; M _ {0} , M _ {0} ^ \prime ) $, where $ M _ {0} = M _ {0} ^ \prime $ when there is no diffeomorphism from $ W $ to $ M _ {0} \times [ 0, 1] $, that is, when $ M _ {0} \approx M _ {1} $ does not imply that the $ h $- cobordism connecting them is trivial.

If $ M _ {0} $ is a closed connected manifold and $ \mathop{\rm dim} M _ {0} \geq 5 $, then there exists for any $ \tau \in \mathop{\rm Wh} \pi _ {1} M _ {0} $ an $ h $- cobordism $ ( W; M _ {0} , M _ {1} ) $ with $ \tau ( W, M _ {0} ) = \tau $. If $ ( W; M _ {0} , M _ {1} ) $ and $ ( W ^ \prime ; M _ {0} , M _ {1} ^ \prime ) $( with $ \mathop{\rm dim} W \geq 6 $) have the same torsion $ \tau ( W, M _ {0} ) = \tau ( W ^ \prime , M _ {0} ) $, then $ W \approx W ^ \prime $ relative to $ M _ {0} $. When $ \mathop{\rm dim} M _ {0} $ is even and $ \pi _ {1} M _ {0} $ is finite, there is a finite set of distinct manifolds that are $ h $- cobordant with $ M _ {0} $. This is not the case when $ \mathop{\rm dim} M _ {0} $ is odd.

If two homotopy-equivalent manifolds $ M _ {1} $ and $ M _ {2} $ are imbedded in $ \mathbf R ^ {N} $, with $ N $ sufficiently large, and their normal bundles are trivial, then $ M _ {1} \times S ^ {N} $ and $ M _ {2} \times S ^ {N} $ are $ h $- cobordant. If, moreover, $ M _ {1} $ and $ M _ {2} $ are of the same simple homotopy type, that is, if the torsion of this homotopy equivalence vanishes, then $ M _ {1} \times S ^ {N} \approx M _ {2} \times S ^ {N} $.

If $ ( W; M _ {0} , M _ {1} ) $ is an $ h $- cobordism and $ P $ is a closed manifold, then there is an $ h $- cobordism $ ( W \times P; M _ {0} \times P, M _ {1} \times P) $ with $ \tau ( W \times P, M _ {0} \times P) = \tau ( W, M _ {0} ) \chi ( P) $, where $ \chi ( P) $ is the Euler characteristic of $ P $. If $ \mathop{\rm dim} W \geq 5 $ and $ P = S ^ {1} $, then

$$ W \times S ^ {1} \approx \ M _ {0} \times S ^ {1} \times [ 0, 1] \approx \ M _ {1} \times S ^ {1} \times [ 0, 1]. $$

In particular, $ M _ {0} \times S ^ {1} \approx M _ {1} \times S ^ {1} $; furthermore, two closed manifolds $ M _ {0} $ and $ M _ {1} $ of the same dimension $ \geq 5 $ are $ h $- cobordant if and only if $ M _ {0} \times \mathbf R ^ {1} \approx M _ {1} \times \mathbf R ^ {1} $.

The $ h $- cobordism structure has not been completely elucidated for $ n < 6 $( 1978). Thus there is the following negative result [8]: There exists an $ h $- cobordism $ ( W; T ^ {4} , T ^ {4} ) $, where $ T ^ {4} $ is the four-dimensional torus, for which there is no diffeomorphism from $ W $ to $ T ^ {4} \times [ 0, 1] $; since $ \mathop{\rm Wh} \pi _ {1} T ^ {4} = 0 $, this means that the $ s $- cobordism theorem fails for $ n = 5 $.

References

[1] S.P. Novikov, "Homotopy-equivalent smooth manifolds I" Izv. Akad. Nauk SSSR Ser. Mat. , 28 : 2 (1964) pp. 365–474 (In Russian)
[2] J. Milnor, "Lectures on the -cobordism theorem" , Princeton Univ. Press (1965)
[3] J. Milnor, "Whitehead torsion" Bull. Amer. Math. Soc. , 72 (1966) pp. 358–462
[4] S. Smale, "On the structure of manifolds" Amer. J. Math. , 84 (1962) pp. 387–399
[5] J. Milnor, "Sommes des variétés différentiables et structures différentiables des sphères" Bull. Soc. Math. France , 87 (1959) pp. 439–444
[6] M. Kervaire, J. Milnor, "Groups of homotopy spheres I" Ann. of Math. (2) , 77 (1963) pp. 504–537
[7] B. Mazur, "Relative neighbourhoods and the theorems of Smale" Ann. of Math. , 77 (1963) pp. 232–249
[8] L.C. Siebenmann, "Disruption of low-dimensional handlebody theory by Rohlin's theorem" J.C. Cantrell (ed.) C.H. Edwards jr. (ed.) , Topology of manifolds , Markham (1969) pp. 57–76
[9] R. Kirby, L. Siebenmann, "On the triangulation of manifolds and the Hauptvermutung" Bull. Amer. Math. Soc. , 75 (1969) pp. 742–749
[10] M.A. Kervaire, "Le théorème de Barden–Mazur–Stallings" M.A. Kervaire (ed.) G. de Rham (ed.) S. Maumary (ed.) , Torsion et type simple d'homotopie , Lect. notes in math. , 48 , Springer (1967) pp. 83–95
[11] R. Thom, "Les classes caractéristiques de Pontryagin des variétés triangulées" , Symp. Internac. Topol. Algebr. , Univ. Nac. Aut. Mexico & UNESCO (1958) pp. 54–67
[12] C.P. Rourke, B.J. Sanderson, "Introduction to piecewise-linear topology" , Springer (1972)

Comments

For the generalized Poincaré conjecture see also [a1].

References

[a1] S. Smale, "Generalized Poincaré's conjecture in dimensions greater than four" Ann. of Math. (2) , 74 (1961) pp. 391–406
How to Cite This Entry:
H-cobordism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=H-cobordism&oldid=47154
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article