View source for H-cobordism

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

<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>

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.

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).

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]]].

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" />.

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]]].

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" />.

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

<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>

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 <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.

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 <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

<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>

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" />.

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" />.

====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>



====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>