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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.

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
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