Surgery on \(\widetilde {\mathbb{SL}} \times \mathbb{E}^n\)-manifolds (Q2882474)

The Borel Conjecture, for closed manifolds, predicts that aspherical closed manifolds that have isomorphic fundamental groups (and thus are homotopy equivalent) are homeomorphic. Proofs of the conjecture for many classes of aspherical manifolds are in the work of Farrell-Hsiang and Farrell-Jones. In particular, Farrell-Jones proved the conjecture for closed non-positively curved manifolds [\textit{F. T. Farrell} and \textit{L. E. Jones}, ``Topological rigidity for compact nonpositively curved manifolds'', Proc. Symp. Pure Math. 54, Part 3, 229--274 (1993; Zbl 0796.53043)].NEWLINENEWLINE In the paper under review, the Borel Conjecture is proved for manifolds that are finite covers of manifolds that admit a codimension-2 effective torus action generalizing the result of Nicas-Stark [\textit{A. J. Nicas} and \textit{C. W. Stark}, ``\(K\)-theory and surgery of codimension-two torus actions on aspherical manifolds'', J. Lond. Math. Soc., II. Ser. 31, 173--183 (1985; Zbl 0573.57018)].NEWLINENEWLINEMore precisely, the main result of the paper is that the Borel Conjecture is proved for the following class of manifolds \(M\): manifolds that are the total space of an orbifold bundle with \(\mathbb{H}^2\)-fibers \(F\) that are flat \(m\)-manifolds of dimension \(\geq 3\) such that the Hirsch-Plotkin radical of their fundamental group (which is a free abelian group of rank \(m\)) is centralized by a subgroup of finite index in \({\pi}_1(M)\). Using hyperelementary induction, the proof is reduced to the case that the orbifold fundamental group of the base admits an epimorphism to \(\mathbb{Z}\). Then, the pull-back of the cover admits a natural filtration by orbifolds (manifolds) with boundary. The result is proved by doing surgery on doubles of such manifolds.NEWLINENEWLINEAs a Corollary, the authors prove the Borel Conjecture for \(\widetilde{\mathbb{SL}}{\times}\mathbb{E}^n\)-manifolds (\(n \geq 2\)). For the \(4\)-dimensional case, the authors prove that if a \(4\)-dimensional closed manifold has the same fundamental group and the same Euler characteristic as a \(\widetilde{\mathbb{SL}}{\times}\mathbb{E}^1\)-manifold, then they are \(s\)-cobordant.