h-cobordism

In mathematics , h-cobordism is a term from the topology of manifolds .

definition

A -dimensional cobordism between -dimensional manifolds and is called h-cobordism if the inclusions and homotopy equivalences are. ${\displaystyle (n+1)}$ ${\displaystyle W}$ ${\displaystyle n}$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle M\to W}$ ${\displaystyle N\to W}$

The last condition can be replaced by the a priori weaker condition for the relative homology groups . ${\displaystyle H_{*}(W,M;\mathbb {Z} )=0}$

Important sentences

In dimensions , according to the h-cobordism theorem ( Stephen Smale ), every h-cobordism between simply connected manifolds is trivial, i.e. a product . (This applies both in the differentiable as well as in the piecewise linear or in the topological category.) ${\displaystyle n\geq 5}$ ${\displaystyle W\simeq M\times \left[0,1\right]}$

If the manifolds are not simply connected, then according to the s-cobordism theorem ( Barry Mazur , John Stallings , Dennis Barden ) the h-cobordisms are classified by the Whitehead group of the fundamental group . ${\displaystyle Wh(\pi _{1}M)}$

In the topological category, the h-cobordism theorem also applies in dimension 4, but not in the differentiable category. This is related to the failure of the Whitney trick in differentiable 4-manifolds.

literature

• John Milnor : Lectures on the h-cobordism theorem , Princeton University Press 1965
• A. Scorpan: The wild world of 4-manifolds , Amer. Math. Soc. 2005, ISBN 978-0-8218-3749-8

Web links

• Yu. Rudyak: h-cobordism in: Encyclopedia of Mathematics, Springer / Kluwer, ISBN 978-1-55608-010-4

Individual evidence

1. ↑ S. Smale, On the structure of manifolds , Amer. J. Math., Vol. 84, 1962, pp. 387-399