A Geometric Reverse To The Plus Construction And Some Examples Of Pseudo-Collars On High-Dimensional Manifolds

TL;DR

This paper introduces a geometric method to reverse Quillen's plus construction, enabling the creation of uncountably many distinct high-dimensional manifold ends called pseudo-collars, with similar boundary and homology properties but different fundamental groups.

Contribution

It develops a new geometric procedure for the reverse of the plus construction and constructs diverse pseudo-collars with identical boundary and homology but distinct pro-fundamental groups.

Findings

Constructed uncountably many pseudo-collars with same boundary.

Pseudo-collars share pro-homology systems at infinity.

Pro-fundamental groups at infinity are all distinct.

Abstract

In this paper, we develop a geometric procedure for producing a reverse to Quillen's plus construction, a construction called a 1-sided h-cobordism or semi-h-cobordism. We then use this reverse to the plus construction to produce uncountably many distinct ends of manifolds called pseudo-collars, which are stackings of 1-sided h-cobordisms. Each of our pseudo-collars has the same boundary and pro-homology systems at infinity and similar group-theoretic properties for their pro-fundamental group systems at infinity. In particular, the kernel group of each group extension for each 1-sided h-cobordism in the pseudo-collars is the same group. Nevertheless, the pro-fundamental group systems at infinity are all distinct. A good deal of combinatorial group theory is needed to verify this fact, including an application of Thompson's group V. The notion of pseudo-collars originated in Hilbert cube manfold theory, where it was part of a necessary and suffcient condition for placing a Z-set as the boundary of an open Hilbert cube manifold. We are interested in pseudo-collars on finite-dimensional manifolds for the same reason, attempting to put a Z-set as the boundary of an open high-dimensional manifold.