Manifolds with non-stable fundamental groups at infinity, III

Abstract

We continue our study of ends non-compact manifolds. The over-arching aim is to provide an appropriate generalization of Siebenmann's famous collaring theorem that applies to manifolds having non-stable fundamental group systems at infinity. In this paper a primary goal is finally achieved; namely, a complete characterization of pseudo-collarability for manifolds of dimension at least 6. 57N15, 57Q12; 57R65, 57Q10

Noncompact manifolds that are inward tame

“…With all necessary definitions in place, we are ready to prove the second main theorem described in the introduction. We begin by stating a strong form of the theorem, written in the style of earlier characterization theorems from [Si] and [GT2].…”

“…The obvious next goal is attempting to improve the N i to generalized (n − 2)-neighborhoods of infinity, which by item viii) would necessarily be homotopy collars. In previous work [Si], [Gu1] and [GT2], that is the final (also the most difficult and interesting) step. The same is true here, where the weakened hypotheses create greater difficulties and the strategy and end goal must eventually be altered.…”

“…Under the π 1 -stability hypothesis of [Si], H n−2 ( R i , ∂ N i ) can now be killed by sliding the offending (n − 2)-handles h n−2 1 , · · · , h n−2 s off the (n − 3)-handles and carving out their interiors. Under the weaker P-semistability hypothesis of [GT2], a similar strategy works, but only after a significant preparatory step, made possible by perfect kernels. In [Gu1] an alternate strategy was employed.…”

“…, · · · , h n−2 s , the task was accomplished by introducing new (n − 3)handles, which became images of the h n−2 1 , · · · , h n−2 s under the resulting boundary map, thereby trivializing the kernel. Complete discussions of these approaches can be found in [GT2,§3] and [Gu1,§8]; the strategy employed here is based on the latter.…”

“…In [Gu1], [GT1] and [GT2] we carried out a program to generalize L.C. Siebenmann's famous Manifold Collaring Theorem [Si] in ways applicable to manifolds with non-stable fundamental group at infinity.…”