Manifolds with non-stable fundamental groups at infinity, II

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

References

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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

Volume 7, issue 1 (2003)

Craig R Guilbault and Frederick C Tinsley