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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Spherical alterations of handles: embedding the manifold plus construction

Craig R Guilbault and Frederick C Tinsley

Algebraic & Geometric Topology 13 (2013) 35–60

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.

spherical alteration, perfect group, plus construction, generalized plus construction
Mathematical Subject Classification 2010
Primary: 57N15, 57Q12
Secondary: 57Q10, 57R65
Received: 11 August 2011
Revised: 10 January 2012
Accepted: 17 August 2012
Published: 6 February 2013
Craig R Guilbault
Department of Mathematical Sciences
University of Wisconsin-Milwaukee
Milwaukee, WI 53201
Frederick C Tinsley
Department of Mathematics & Computer Science
Colorado College
14 East Cache La Poudre St
Colorado Springs, CO 80903