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.
Keywords
spherical alteration, perfect group, plus construction,
generalized plus construction
