Some Results on Pseudo-Collar Structures on High-Dimensional Manifolds

TL;DR

This paper explores the geometric construction of pseudo-collars on high-dimensional manifolds using a reverse to Quillen's plus construction, introducing new methods for creating and analyzing one-ended open manifolds.

Contribution

It develops a geometric procedure for a reverse plus construction, enabling the creation of uncountably many pseudo-collars with specific properties.

Findings

Constructed uncountably many pseudo-collars.

Introduced a technique for producing 'nice' one-ended open manifolds.

Identified conditions for pseudo-collarability that are not all simultaneously satisfied.

Abstract

In this paper, we recall Quillen's plus construction for high-dimensional smooth manifolds and the solution to the group extension problem. We then develop a geometric procedure due for producing a "reverse" to the plus construction, a one-sided s-cobordism called a semi-s-cobordism, when the total group of the group extension is a semi-direct product. We then use this reverse to the plus construction to produce uncountably many different one-ended manifolds called pseudo-collars, which are stackings of semi-s- cobordisms. We finally display a technique for producing "nice" one-ended open manifolds which satisfy two of the necessary and suffcient conditions for being pseudo-collarable but not the third.