A topological splitting theorem for Poincaré duality groups and high-dimensional manifolds
The authors show that, ``for a wide class of manifold pairs, \(N, M\) with \(\dim(M) = \dim (N) + 1\), every \(\pi_1\)--injective map \(f: N \rightarrow M\) factorises up to homotopy as a finite cover of an embedding.'' This main result is reformulated subject to the Borel conjecture. As a corollary the authors obtain a topological rigidity theorem which they compare to the geometric superrigidity theorem by Mok, Siu and Yeung. The proof of the main result proceeds in two stages, the first one group theoretic and the second one topological in nature. The first stage is a splitting theorem for Poincaré duality groups. It involves Kazdhan's property \((T)\), is of independent interest, and provides a splitting of the fundamental group of \(M\). The second stage uses Cappell's surgery methods. These are summarised and require restriction to dimension \(N \geq 5\). The paper under review discusses the case of dimension \(N\) even and the authors intend to treat the odd dimensional case in a forthcoming paper. Finally, the main result yields an obstruction to the existence of \(\pi_1\)--injective continuous maps. The paper is clearly structured and well written.
scientific article; zbMATH DE number 6198084