The Michigan Mathematical Journal

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

Jeffrey J. Rolland

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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 Z-set as the boundary of an open Hilbert cube manifold.

Article information

Michigan Math. J., Volume 67, Issue 3 (2018), 485-509.

Received: 14 October 2016
Revised: 4 October 2017
First available in Project Euclid: 6 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R65: Surgery and handlebodies 57R19: Algebraic topology on manifolds
Secondary: 57S30: Discontinuous groups of transformations 57M07: Topological methods in group theory


Rolland, Jeffrey J. A Geometric Reverse to the Plus Construction and Some Examples of Pseudocollars on High-Dimensional Manifolds. Michigan Math. J. 67 (2018), no. 3, 485--509. doi:10.1307/mmj/1522980163.

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