nLab h-cobordism theorem

Idea

The h-cobordism theorem (due to Smale 1962) and the s-cobordism theorem provide sufficient conditions for an h-cobordism to be isomorphic to a cylinder.

There are famous counterexamples where the h-cobordism theorem fails (in the smooth category) and h-cobordisms exist which are not isomorphic to a cylinder. These counterexamples arise specifically in the case of 5-dimensional cobordisms between 4-dimensional boundaries. The most prominent example involves the topological manifold known as the E₈ manifold, constructed by Freedman 1982. While this manifold exists topologically, the work of Donaldson 1983 proves that it cannot admit any smooth structure. This discrepancy allows for the construction of a smooth h-cobordism between a standard smooth 4-manifold (such as the K3 surface) and a “fake” topological version containing the E₈ manifold; this cobordism exists topologically but cannot carry a smooth product structure, thereby demonstrating the existence of exotic smooth structures on $\mathbb{R}^4$ and the failure of the smooth h-cobordism theorem in dimension 5.

In consequence, in dimensions $n$ where there exist h-cobordisms which are not isomorphic to cylinders (such as for $n=5$ or when non-trivial Whitehead torsion exists), the Segal space $PreCob(n)$ modeling the $(\infty,1)$-category $Cob(n)$ of cobordisms fails to be a complete Segal space. This is because the $\infty$-groupoid of invertible morphisms (in degree 1) contains components corresponding to these non-trivial h-cobordisms, which are not in the image of the degeneracy map from the $\infty$-groupoid of objects (in degree 0, which represents the classifying spaces of the diffeomorphism groups of closed $(n-1)$-manifolds). Consequently, the completion of this Segal space has a space of objects strictly “larger” than the classifying space of the diffeomorphism groups (Lurie 2009, Warning 2.2.8).

Statement

By an isomorphism of manifolds relative to a joint submanifold (usually a boundary component), we shall mean that the isomorphism restricts to the identity map on that submanifold.

H-Cobordism theorem

Consider the category either of topological manifolds, smooth manifolds, or piecewise linear manifolds.

Proposition

For $n \geq 6$, a compact simply connected $n$-dimensional h-cobordism $W$ between simply connected $(n-1)$-manifolds $W_{in}$ and $W_{out}$ is isomorphic, relative to $W_{in}$, to the cylinder $W_{in} \times [0,1]$.

S-Cobordism theorem

Consider the category either of topological manifolds, smooth manifolds, or piecewise linear manifolds.

Proposition

For $n \geq 6$, a compact $n$-dimensional h-cobordism $W$ between $(n-1)$-manifolds $W_{in}$ and $W_{out}$ is isomorphic, relative to $W_{in}$, to the cylinder $W_{in} \times [0,1]$ if and only if its Whitehead torsion $\tau(W, W_{in})$ in the Whitehead group? $Wh\big(\pi_1(W_{in})\big)$ vanishes.

Related entries

• h-cobordism

• exotic smooth structure

• Poincaré conjecture

References

The original h-cobordism theorem:

• Stephen Smale, On the structure of manifolds, Amer. J. of Math. 84 (1962) 387-399 [doi:10.2307/2372978, jstor:2372978, pdf]

Review:

• John Milnor, Lectures on the h-cobordism theorem, Princeton Univ. Press (1965) [jstor:j.ctt183psc9, pdf]

See also:

• Wikipedia: H-cobordism

Generalization to the s-cobordism theorem and counterexamples:

• Michael H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 3 (1982) 357-453 [doi:10.4310/jdg/1214437136]

• Simon Donaldson, An application of gauge theory to four-dimensional topology, Journal of Differential Geometry. 18 2 (1983) 279-315 [doi:10.4310/jdg/1214437665]

Relevance for the $(\infty,n)$-category of cobordisms:

• Jacob Lurie, Warning 2.2.8 in: On the Classification of Topological Field Theories, Current Developments in Mathematics 2008 (2009) 129-280 [doi:10.4310/CDM.2008.v2008.n1.a3, arXiv:0905.0465]

Generalization to equivariant cobordism:

• Shôrô Araki, Katsuo Kawakubo: Equivariant $s$-cobordism theorems, J. Math. Soc. Japan 40 2 (1988) 349-367 [doi:10.2969/jmsj/04020349, pdf]