Ons-cobordisms of metacyclic prism manifolds (original) (raw)

1989, Inventiones Mathematicae

In geometric topology it is important to have useful criteria for recognizing special types of manifolds. For example, it is often necessary to determine whether a connected compact manifold with boundary W can be expressed as a product V• [0, 1]. If W is two-dimensional, then the classification of surfaces shows that some obvious necessary conditions are sufficient. Specifically, the boundary must split into two components, and the inclusion of either must be a homotopy equivalence. A manifold that satisfies these conditions is called an h-cobordism, if the inclusions are simple homotopy equivalences, the manifold is called an s-cobordism. Every product Vx [0, 1] is an s-cobordism, and the s-cobordism theorem of D. Barden, B. Mazur and J. Stallings shows that the converse is also true in all dimensions greater than or equal to six. This result holds in each of standard categories of manifolds. In other words, if the manifold W is taken to be smooth or PL, then the isomorphism with Wo x [0, 1] may also be taken to be smooth or PL. For many years it has been known that the s-cobordism theorem extends to certain specific situations in dimension 3, 4, and 5 (compare Lawson [27], Shaneson [48]) but it was also known that the s-cobordism theorem does not generalize completely in at least one of these dimensions (Siebenmann [49], Lawson [27]). In fact, it was known that failures occur in each of the three manifold categories (topological, PL, and smooth manifolds). Recent advances in low-dimensional topology have improved our understanding of the extent to which the s-cobordism theorem can and cannot be generalized. The results of M. Freedman yield a topological s-cobordism theorem in dimension five provided the fundamental group of W is relatively small [16, 39]. On the other hand, Freedman's results and earlier work of T. Matumoto and L. Siebenmann [32] (cf. [23]) show that the topological s-cobordism theorem fails in dimension four; one can even insist that both boundary components are homeomorphic (specifically to S 1 x RP2).