Volume 7, issue 1 (2003)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Manifolds with non-stable fundamental groups at infinity, II

Craig R Guilbault and Frederick C Tinsley

Geometry & Topology 7 (2003) 255–286

arXiv: math.GT/0304031

Abstract

In this paper we continue an earlier study of ends non-compact manifolds. The over-arching goal is to investigate and obtain generalizations of Siebenmann’s famous collaring theorem that may be applied to manifolds having non-stable fundamental group systems at infinity. In this paper we show that, for manifolds with compact boundary, the condition of inward tameness has substatial implications for the algebraic topology at infinity. In particular, every inward tame manifold with compact boundary has stable homology (in all dimensions) and semistable fundamental group at each of its ends. In contrast, we also construct examples of this sort which fail to have perfectly semistable fundamental group at infinity. In doing so, we exhibit the first known examples of open manifolds that are inward tame and have vanishing Wall finiteness obstruction at infinity, but are not pseudo-collarable.

Keywords
end, tame, inward tame, open collar, pseudo-collar, semistable, Mittag-Leffler, perfect group, perfectly semistable, Z-compactification
Mathematical Subject Classification 2000
Primary: 57N15, 57Q12
Secondary: 57R65, 57Q10
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Publication
Received: 6 September 2002
Accepted: 12 March 2003
Published: 31 March 2003
Proposed: Steve Ferry
Seconded: Benson Farb, Robion Kirby
Authors
Craig R Guilbault
Department of Mathematical Sciences
University of Wisconsin at Milwaukee
Milwaukee
Wisconsin 53201
USA
Frederick C Tinsley
Department of Mathematics
The Colorado College
Colorado Springs
Colorado 80903
USA