The Borel conjecture for closed aspherical four-manifolds (homeomorphism form)

Theorem 1.1: there are closed connected aspherical topological four-manifolds $M$ and $N$, with the same word-hyperbolic fundamental group, that are homotopy equivalent but not homeomorphic. Theorem 1.2: some such $M$ has a self-homotopy equivalence not homotopic to any homeomorphism, so the prescribed-class form also fails. The examples come from Davis reflection applied to a Poincaré chamber whose odd cyclic covers have no manifold model. This disproves the Borel conjecture as an all-dimension statement and in dimension four specifically. Not shown: anything in dimensions five or higher, or smooth-category statements; it also shows that Khan's rigidity up to s-cobordism cannot be upgraded to homeomorphism for such groups (curator's reading of the introduction).

Theorem 1.1: there are closed connected aspherical topological four-manifolds $M$ and $N$, with the same word-hyperbolic fundamental group, that are homotopy equivalent but not homeomorphic. Theorem 1.2: some such $M$ has a self-homotopy equivalence not homotopic to any homeomorphism, so the prescribed-class form also fails. The examples come from Davis reflection applied to a Poincaré chamber whose odd cyclic covers have no manifold model. This disproves the Borel conjecture as an all-dimension statement and in dimension four specifically. Not shown: anything in dimensions five or higher, or smooth-category statements; it also shows that Khan's rigidity up to s-cobordism cannot be upgraded to homeomorphism for such groups (curator's reading of the introduction).