## Geometry & Topology

- Geom. Topol.
- Volume 7, Number 1 (2003), 255-286.

### Manifolds with non-stable fundamental groups at infinity, II

Craig R Guilbault and Frederick 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.

#### Article information

**Source**

Geom. Topol., Volume 7, Number 1 (2003), 255-286.

**Dates**

Received: 6 September 2002

Accepted: 12 March 2003

First available in Project Euclid: 21 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.gt/1513883098

**Digital Object Identifier**

doi:10.2140/gt.2003.7.255

**Mathematical Reviews number (MathSciNet)**

MR1988286

**Zentralblatt MATH identifier**

1032.57020

**Subjects**

Primary: 57N15: Topology of $E^n$ , $n$-manifolds ($4 \less n \less \infty$) 57Q12: Wall finiteness obstruction for CW-complexes

Secondary: 57R65: Surgery and handlebodies 57Q10: Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. [See also 19B28]

**Keywords**

end tame inward tame open collar pseudo-collar semistable Mittag-Leffler perfect group perfectly semistable Z-compactification

#### Citation

Guilbault, Craig R; Tinsley, Frederick C. Manifolds with non-stable fundamental groups at infinity, II. Geom. Topol. 7 (2003), no. 1, 255--286. doi:10.2140/gt.2003.7.255. https://projecteuclid.org/euclid.gt/1513883098