View source for H-cobordism

<!--This article has been texified automatically. Since there was no Nroff source code for this article, 
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
was used.
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.

Out of 1 formulas, 1 were replaced by TEX code.-->

{{TEX|semi-auto}}{{TEX|done}}
{{TEX|auto}}
{{TEX|done}}

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

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

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 [[#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).

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]]].

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|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]]].

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|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|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 [[#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====
<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 $h$-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====
For the generalized Poincaré conjecture see also [[#References|[a1]]].

====References====
<table>
<tr><td valign="top">[a1]</td> <td valign="top"> S. Smale, "Generalized Poincaré's conjecture in dimensions greater than four"  ''Ann. of Math. (2)'' , '''74'''  (1961)  pp. 391–406</td></tr>
</table>