Geometry & Topology, Vol. 7 (2003) Paper no. 7, pages 255--286.

Manifolds with non-stable fundamental groups at infinity, II

C R Guilbault and F C Tinsley

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

AMS subject classification. Primary: 57N15, 57Q12. Secondary: 57R65, 57Q10.

DOI: 10.2140/gt.2003.7.255

E-print: arXiv:math.GT/0304031

Submitted to GT on 6 September 2002. Paper accepted 12 March 2003. Paper published 31 March 2003.

Notes on file formats

C R Guilbault and F C Tinsley
Department of Mathematical Sciences, University of Wisconsin-Milwaukee
Milwaukee, Wisconsin 53201, USA
Department of Mathematics, The Colorado College
Colorado Springs, Colorado 80903, USA


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