Algebraic & Geometric Topology

Spherical alterations of handles: embedding the manifold plus construction

Craig R Guilbault and Frederick C Tinsley

Full-text: Open access

Abstract

Quillen’s famous plus construction plays an important role in many aspects of manifold topology. In our own work [Geometry and Topology 7 (2006) 541–556] on ends of open manifolds, an ability to embed cobordisms provided by the plus construction into the manifolds being studied was a key to completing the main structure theorem. In this paper we develop a “spherical modification” trick that allows for a constructive approach to obtaining those embeddings. More importantly, this approach can be used to obtain more general embedding results. In this paper we develop generalizations of the plus construction (together with the corresponding group-theoretic notions) and show how those cobordisms can be embedded in manifolds satisfying appropriate fundamental group properties. Results obtained here are motivated by, and play an important role in, our ongoing study of noncompact manifolds.

Article information

Source
Algebr. Geom. Topol., Volume 13, Number 1 (2013), 35-60.

Dates
Received: 11 August 2011
Revised: 10 January 2012
Accepted: 17 August 2012
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715491

Digital Object Identifier
doi:10.2140/agt.2013.13.35

Mathematical Reviews number (MathSciNet)
MR3031636

Zentralblatt MATH identifier
06138571

Subjects
Primary: 57N15: Topology of $E^n$ , $n$-manifolds ($4 \less n \less \infty$) 57Q12: Wall finiteness obstruction for CW-complexes
Secondary: 57Q10: Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. [See also 19B28] 57R65: Surgery and handlebodies

Keywords
spherical alteration perfect group plus construction generalized plus construction

Citation

Guilbault, Craig R; Tinsley, Frederick C. Spherical alterations of handles: embedding the manifold plus construction. Algebr. Geom. Topol. 13 (2013), no. 1, 35--60. doi:10.2140/agt.2013.13.35. https://projecteuclid.org/euclid.agt/1513715491


Export citation

References

  • M H Freedman, F Quinn, Topology of $4$–manifolds, Princeton Mathematical Series 39, Princeton Univ. Press (1990)
  • C R Guilbault, F C Tinsley, Manifolds that are inward tame at infinity, In progress
  • C R Guilbault, F C Tinsley, Manifolds with non-stable fundamental groups at infinity, II, Geom. Topol. 7 (2003) 255–286
  • C R Guilbault, F C Tinsley, Manifolds with non-stable fundamental groups at infinity, III, Geom. Topol. 10 (2006) 541–556
  • A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)
  • C P Rourke, B J Sanderson, Introduction to piecewise-linear topology, Ergeb. Math. Grenzgeb. 69, Springer, New York (1972)
  • J Stallings, Homology and central series of groups, J. Algebra 2 (1965) 170–181
  • U Stammbach, Anwendungen der holomogietheorie der gruppen auf zentralreihen und auf invarianten von präsentierungen, Math. Z. 94 (1966) 157–177
  • C T C Wall, Surgery on compact manifolds, 2nd edition, Mathematical Surveys and Monographs 69, Amer. Math. Soc. (1999) Edited and with a foreword by A A Ranicki