Geometry & Topology

Manifolds with non-stable fundamental groups at infinity

Craig R Guilbault

Full-text: Open access

Abstract

The notion of an open collar is generalized to that of a pseudo-collar. Important properties and examples are discussed. The main result gives conditions which guarantee the existence of a pseudo-collar structure on the end of an open n–manifold (n7). This paper may be viewed as a generalization of Siebenmann’s famous collaring theorem to open manifolds with non-stable fundamental group systems at infinity.

Article information

Source
Geom. Topol., Volume 4, Number 1 (2000), 537-579.

Dates
Received: 30 July 1999
Revised: 8 December 2000
Accepted: 27 December 2000
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883296

Digital Object Identifier
doi:10.2140/gt.2000.4.537

Mathematical Reviews number (MathSciNet)
MR1800296

Zentralblatt MATH identifier
0958.57023

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
non-compact manifold ends collar homotopy collar pseudo-collar semistable Mittag–Leffler

Citation

Guilbault, Craig R. Manifolds with non-stable fundamental groups at infinity. Geom. Topol. 4 (2000), no. 1, 537--579. doi:10.2140/gt.2000.4.537. https://projecteuclid.org/euclid.gt/1513883296


Export citation

References

  • F D Ancel, C R Guilbault, Z–compactifications of open manifolds, Topology, 38 (1999) 1265–1280
  • M Bestvina, Local homology properties of boundaries of groups, Michigan Math. J. 43 (1996) 123–139
  • M G Brin, T L Thickstun, 3–manifolds which are end 1–movable, Mem. Amer. Math. Soc. 81 (1989) no. 411
  • M Brown, The monotone union of open n–cells is an open n–cell, Proc. Amer. Math. Soc. 12 (1961) 812–814
  • G Carlsson, E K Pedersen, Controlled algebra and the Novikov conjectures for $K$– and $L$–theory, Topology, 34 (1995) 731–758
  • T A Chapman, L C Siebenmann, Finding a boundary for a Hilbert cube manifold, Acta Math. 137 (1976) 171–208
  • M M Cohen, A Course in Simple-Homotopy Theory, Springer–Verlag, New York (1973)
  • R J Daverman, F Tinsley, Laminations, finitely generated perfect groups, and acyclic maps, Michigan Math. J. 33 (1986) 343–351
  • R J Daverman, F Tinsley, Controls on the plus construction, Michigan Math. J. 43 (1996) 389–416
  • M W Davis, Groups generated by reflections and aspherical manifolds not covered by Euclidean space, Ann. of Math. 117 (1983) 293–325
  • M Davis, T Januszkiewicz, Hyperbolization of polyhedra, J. Diff. Geom. 34 (1991) 347–388
  • S Ferry, A stable converse to the Vietoris–Smale theorem with applications to shape theory, Trans. Amer. Math. Soc. 261 (1980) 369–386
  • S Ferry, Topological Manifolds and Polyhedra, book for Oxford University Press, writing in progress
  • S Ferry S Weinberger, A coarse approach to the Novikov Conjecture, from: “Novikov conjectures, index theory, and rigidity”, LNS 226, Cambridge University Press, 147–163
  • M H Freedman, F Quinn, Topology of 4–manifolds, Princeton University Press, Princeton, New Jersey (1990)
  • L S Husch, T M Price, Finding a boundary for a 3–manifold, Ann. of Math. 91 (1970) 223–235
  • B Hughes, A Ranicki, Ends of Complexes, Cambridge tracts in mathematics, 123, Cambridge University Press (1996)
  • O Kakimizu, Finding boundary for the semistable ends of 3–manifolds, Hiroshima Math. J. 17 (1987) 395–403
  • S Kwasik, R Schultz, Desuspension of group actions and the ribbon theorem, Topology 27 (1988) 443–457
  • M Mihalik, Semistability at the end of a group extension, Trans. Amer. Math. Soc. 277 (1983) 307–321
  • D Quillen, Cohomology of groups, Actes Congres Int. Math. Tome 2 (1970) 47–51
  • C P Rourke, B J Sanderson, Introduction to Piecewise-Linear Topology, Springer–Verlag, New York (1982)
  • L C Siebenmann, The obstruction to finding a boundary for an open manifold of dimension greater than five, PhD thesis, Princeton University (1965)
  • L C Siebenmann, On detecting open collars, Trans. Amer. Math. Soc. 142 (1969) 201–227
  • E H Spanier, Algebraic Topology, Springer Verlag, New York (1966)
  • C T C Wall, Finiteness conditions for CW complexes, Ann. Math. 8 (1965) 55–69
  • C T C Wall, Finiteness conditions for CW complexes II, Proc. Royal Soc. Ser. A, 295 (1966) 129–125