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**A geometric reverse to the plus construction and some examples of pseudocollars on high-dimensional manifolds.**
*(English)*
Zbl 1448.57035

Let \(N^{n}\) be a compact smooth manifold. A one-sided \(h\)-cobordism (resp. \(s\)-cobordism) \((W,N,M)\) is a
cobordism for which either \(N \hookrightarrow W\)
or \(M \hookrightarrow W\) is a homotopy equivalence (resp. simple homotopy equivalence).
The main result of this paper shows the existence of
one-sided \(s\)-cobordims starting from a an exact sequence \(1 \rightarrow S \rightarrow G \rightarrow Q \rightarrow 1\) where
\(S\) is a finitely presented superperfect group, \(G\) is a semidirect product of \(Q\) by \(S\),
\(N\) is any \(n\)-manifold with \(n \geq 6\), and \(\pi_{1}(M) \cong Q\). This is a form of reverse to the plus construction,
a method for simplifying the fundamental group of a space without changing its homology and cohomology groups.
The author also shows that this allows
uncountably many pseudocollars on some closed manifolds with the same boundary.
These are obtained by using the first result and Thompson’s group.

Reviewer: Jonathan Hodgson (Swarthmore)

### MSC:

57R80 | \(h\)- and \(s\)-cobordism |

57R65 | Surgery and handlebodies |

57R19 | Algebraic topology on manifolds and differential topology |

57S30 | Discontinuous groups of transformations |

57M07 | Topological methods in group theory |

### Keywords:

pseudocollar; one-sided \(s\)-cobordism; reverse plus problem; Thompson group; handlebody; fundamental group at infinity
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\textit{J. J. Rolland}, Mich. Math. J. 67, No. 3, 485--509 (2018; Zbl 1448.57035)

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