Finding Stable-Boundaries for Open Five-Dimensional Manifolds

Abstract

Introduction. In 1965 W. Browder, J. Levine and G. R. Livesay characterized those smooth open n-manifolds, n > 5, which are interiors of compact manifolds with simply-connected boundary. (See [1].) Shortly afterward L. Siebenmann identified an invariant for a tame end of any smooth open n-manifold W, n ? 5. He showed that for n > 5, W is the interior of a compact manifold if and only if W has finitely many ends, each tame with invariant zero. (See [13].) The failure in dimension four of the well-known Whitney Trick prevents one from extending Siebenmann's arguments in order to obtain this result for arbitrary open 5-manifolds. Although the failure of Whitney's Trick prevents us from solving many problems involving 4-manifolds, it is widely known that useful information can often be obtained after stabilization; i.e. upon adding copies of S2 X 52 to the 4-manifolds by connected-sum. A noteworthy example is the 5-dimensional Stable S-cobordism Theorem. (See Quinn [12] or Lawson [9].) For other examples consult Wall [17] or Cappell and Shaneson [2]. In this paper we prove that if W is any smooth open 5-manifold having finitely many ends, each tame with invariant zero, then after adding a certain number of copies of 52 X 52 X R by connected-sum along suitable smooth arcs, the resulting open 5-manifold is the interior of a smooth compact manifold. Using different techniques M. Freedman and F. Quinn [5] have recently shown that if W is an open topological 5-manifold such that H*(W; Z) is finitely-generated and each end of W is 1-connected then W is the interior of a compact manifold whose boundary is an F4-manifold. (An F4-manifold is a homology 4-manifold with isolated 1-LC non-manifold points.) At present there is no corresponding nonsimply-connected theorem.