Topological rigidity theorems (Q2776347)

Let \(X\) and \(Y\) be two homotopically equivalent topological spaces. A basic question is if this implies that \(X\) and \(Y\) are also homeomorphic. Clearly this is not true in general. So we need more conditions. Borel's conjecture gives one such sufficient condition. It says that if two compact aspherical manifolds with boundary are homotopically equivalent by a homotopy which, restricted to the boundary, is a homeomorphism, the homotopy equivalence is homotopic to a homeomorphism and during the homotopy the boundary is preserved. The Farrell and Jones topological rigidity theorem [\textit{F. T. Farrell} and \textit{L. E. Jones}, Proc. Symp. Pure Math. 54, Part 3, 229-274 (1993; Zbl 0796.53043)] shows that Borel's conjecture is true if the manifolds are closed, of dimension not equal to \(3\) and \(4\) and one of them supports a nonpositively curved Riemannian metric. This is one profound example of rigidity theorems in high dimensional manifold topology. NEWLINENEWLINENEWLINEThe article under review recalls statements of rigidity theorems and related results starting from the very beginning of this subject to the recent progress. A brief outline of background material and explanations of terms used in the theorems are given. Whenever necessary, main ideas behind the proofs of theorems are pointed out. NEWLINENEWLINENEWLINESection 11 is of particular importance for locating general sources of articles and books for a detailed study on this subject. The reference lists almost exhaustively the literature on rigidity results in geometry and topology.NEWLINENEWLINEFor the entire collection see [Zbl 0977.00029].