Spherical alterations of handles: embedding the manifold plus construction

TL;DR

This paper introduces a spherical modification technique that generalizes the Quillen Plus Construction, enabling more flexible embeddings of cobordisms into high-dimensional manifolds with specific fundamental group properties.

Contribution

It develops a constructive spherical modification approach to embed cobordisms, extending the plus construction to broader classes of manifolds.

Findings

Generalized plus construction for manifold embeddings

Constructive spherical modification method

Enhanced embedding results for noncompact manifolds

Abstract

A key tool in our earlier work on ends of manifolds high-dimensional manifolds was an ability to embed cobordisms provided by the Quillen Plus Construction into those ends. Here we develop a `spherical modification' trick which provides a constructive approach to obtaining such embeddings. More importantly, this approach allows for more general embedding results. In this paper we develop generalizations of the plus construction and show how the corresponding cobordisms can be embedded in manifolds satisfying appropriate fundamental group properties. Results obtained here play an important role in our ongoing study of noncompact manifolds.