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The University of Southampton
Mathematical Sciences

Research project: Topological superrigidity

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Following our earlier work on Bass-Serre splittings for Poincaré duality groups, we discovered a new topological splitting theorem for high dimensional manifolds in the spirit of the classical torus theorem. In the following statement a (T)-manifold is any closed manifold such that the universal cover is an irreducible globally symmetric Riemannian manifold of non-compact type, and which is one of the following: of real rank at least 2; quaternionic hyperbolic space; the Cayley hyperbolic plane.

Project Overview

The topological superrigidity theorem: Let N be a closed (T)-manifold of dimension 2k>5. Given any aspherical topological manifold M of dimension 2k+1 and any π1-injective, continuous function j from N to M, there is a manifold M’ homotopy equivalent to M and an embedded, 2-sided, π1-injective codimension-1 submanifold N' in M’ such that up to homotopy, the map j factors through a finite degree covering p of N'.

When M, N satisfy the Borel conjecture, for example, following the recent work of Bartels and Leuck if M, N are CAT(0) manifolds (or more generally have (CAT(0) or hyperbolic fundamental groups), then the conclusion may be strengthened to deduce that M’=M.The relationship with Waldhausen’s torus theorem is as follows: starting with a π1-injective map one deduces the existence of a π1-injective embedding. However the conclusion in the topological superrigidity theorem is considerably stronger: in the torus theorem the embedded submanifold may be topologically unrelated to the original one whereas here the resulting embedding is homotopy equivalent to the original one. Whether or not M satisfies the Borel conjecture we obtain the following corollary: If M is a 2k+1 manifold with trivial first Betti number and N has a non-trivial Pontryagin number (for example it is quaternionic hyperbolic) then there are no π1--injective maps from N to M. This generalises the known fact that there are no such immersions. When the target manifold M is CAT(0) the hypotheses of the topological superrigidity theorem closely parallel those of the geometric superrigidity theorem and together they give strong constraints on the existence of non-trivial codimension-1 maps of T-manifolds into aspherical manifolds.

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