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August 2018 A Geometric Reverse to the Plus Construction and Some Examples of Pseudocollars on High-Dimensional Manifolds
Jeffrey J. Rolland
Michigan Math. J. 67(3): 485-509 (August 2018). DOI: 10.1307/mmj/1522980163

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

Citation

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Jeffrey J. Rolland. "A Geometric Reverse to the Plus Construction and Some Examples of Pseudocollars on High-Dimensional Manifolds." Michigan Math. J. 67 (3) 485 - 509, August 2018. https://doi.org/10.1307/mmj/1522980163

Information

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

zbMATH: 06969982
MathSciNet: MR3835562
Digital Object Identifier: 10.1307/mmj/1522980163

Subjects:
Primary: 57R19, 57R65
Secondary: 57M07, 57S30

Rights: Copyright © 2018 The University of Michigan

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Vol.67 • No. 3 • August 2018
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