How different canh-cobordant manifolds be?

Abstract

We study the homeomorphism types of manifolds h‐cobordant to a fixed one. Our investigation is partly motivated by the notion of special manifolds introduced by Milnor in his study of lens spaces. In particular, we revisit and clarify some of the claims concerning h‐cobordisms of these manifolds.

A simplification problem in manifold theory

“…To generalize Theorem 4.1 we need to introduce the concept of inertial invertible cobordisms: a cobordism (W, In all cases where these sets are computed, they are equal, but it is not known whether I(M ) = I TOP (M ) in general for a smooth manifold M of dimension ≥ 5, contrary to the claim in [30]. However, there is a smaller set, SI(M ), of strongly inertial invertible cobordisms, which indeed is the same in the two categories.…”

“…But note that h may not itself be homotopic to a homeomorphism! A counterexample is given in [30,Example 6.4].…”

“…(1)). Using a recent result of Jahren-Kwasik [30,Theorem 1.2], we now know that part (i) is, in general, "infinitely false", i.e. there are manifolds having countably many homeomorphism classes within their R-diffeomorphism class (see Example 4.5.(5)).…”

Whitehead torsion of inertial h-cobordisms

“…Let (W ; M, M ′ ) and (W ; M ′ , M ) be dual h-cobordisms with M,M ′ of dimension n. Then τ (W ; M ) and τ (W ; M ′ ) are related by the basic duality formula (cf. [21], [12]) h * (τ (W ; M ′ )) = (−1) n τ (W ; M ).…”

“…The last step, however, is in general very difficult, and what makes the problem even more complicated, but at the same time more interesting, is that there exist h-cobordisms with non-zero torsion, but were the ends still are isomorphic (cf. [10], [11], [18], [12]). Such h-cobordisms we call inertial.…”