The Michigan Mathematical Journal
- Michigan Math. J.
- Volume 67, Issue 3 (2018), 485-509.
A Geometric Reverse to the Plus Construction and Some Examples of Pseudocollars on High-Dimensional Manifolds
In this paper, we develop a geometric procedure for producing a “reverse” to Quillen’s plus construction, a construction called a -sided -cobordism or semi--cobordism. We then use this reverse to the plus construction to produce uncountably many distinct ends of manifolds called pseudocollars, which are stackings of -sided -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 -sided -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 .
The notion of pseudocollars originated in Hilbert cube manifold theory, where it was part of a necessary and sufficient condition for placing a -set as the boundary of an open Hilbert cube manifold.
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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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. https://projecteuclid.org/euclid.mmj/1522980163