Topological superrigidty

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Description/Abstract

Waldhausen's celebrated torus theorem plays a central role in the classification of topological 3-manifolds. It also led to a number of algebraic splitting theorems for discrete groups including Kropholler's algebraic torus theorem for Poincar\'e duality groups and to the algebraic annulus theorems of Dunwoody/Sageev and Scott/Swarup. Here, in the same spirit, we offer topological and algebraic decomposition theorems in the context of high dimensional aspherical manifolds, providing an algebraic splitting theorem for Poincar\'e duality groups and exploiting Cappell's splitting theory to extract the required topological splittings. As a result we 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. As an application of this we show that under certain circumstances the vanishing of the first Betti number for $M$ is an obstruction to the existence of such maps.