A Geometric Reverse to the Plus Construction and Some Examples of Pseudocollars on High-Dimensional Manifolds

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 pseudocollars, which are stackings of $1$-sided $h$-cobordisms. Each of our pseudocollars has the same boundary and prohomology systems at infinity and similar group-theoretic properties for their profundamental group systems at infinity. In particular, the kernel group of each group extension for each $1$-sided $h$-cobordism in the pseudocollars is the same group. Nevertheless, the profundamental 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 pseudocollars originated in Hilbert cube manifold theory, where it was part of a necessary and sufficient condition for placing a $\mathcal{Z}$-set as the boundary of an open Hilbert cube manifold.