arXiv research

New examples show manifolds with noncompact boundaries that can't be pseudo-collarable.

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 …

In a previous paper, we developed general techniques for constructing a variety of pseudo-collars, as defined by Guilbault and Tinsley, with roots in earlier work by Chapman and Siebenmann. As an application of our techniques, we exhibited an uncountable collection of pseudo-collars, all with the same boundary and simi…

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 e…

The notion of an open collar is generalized to that of a pseudo-collar. Important properties and examples are discussed. The main result gives conditions which guarantee the existence of a pseudo-collar structure on the end of an open n-manifold (n > 6). This paper may be viewed as a generalization of Siebenmann's famo…

In this paper we obtain a complete characterization of pseudo-collarable $n$-manifolds for $n\geq 6$. This extends earlier work by Guilbault and Tinsley to allow for manifolds with noncompact boundary. In the same way that their work can be viewed as an extension of Siebenmann's dissertation that can be applied to mani…

We continue our study of ends non-compact manifolds. The over-arching aim is to provide an appropriate generalization of Siebenmann's famous collaring theorem that applies to manifolds having non-stable fundamental group systems at infinity. In this paper a primary goal is finally achieved; namely, a complete character…

We prove that an open manifold $M$ of dimension at least $5$ which admits a complete CAT(0) polyhedral metric is pseudo-collarable, its fundamental group at infinity is strongly perfectly semistable and has vanishing Chapman-Siebenmann obstruction $τ_{\infty}(M)$. Moreover, this implies that $M$ is topologically collap…

Guilbault and author prove sufficient conditions for $\mathcal{Z}$-compactifiability of certain manifolds.

We continue our study of ends of non-compact manifolds, with a focus on the inward tameness condition. For manifolds with compact boundary, inward tameness, has significant implications. For example, such manifolds have stable homology at infinity in all dimensions. We show that these manifolds have 'almost perfectly s…

In this paper we continue an earlier study of ends non-compact manifolds. The over-arching goal is to investigate and obtain generalizations of Siebenmann's famous collaring theorem that may be applied to manifolds having non-stable fundamental group systems at infinity. In this paper we show that, for manifolds with c…