Borel topological rigidity conjecture

The Borel conjecture asserts that closed aspherical manifolds are topologically rigid: if $M$ and $N$ are closed connected topological $n$-manifolds, $n\ge5$, whose universal covers are contractible, then every homotopy equivalence $N\to M$ should be homotopic to a homeomorphism. Since a closed aspherical manifold is a $K(\pi,1)$, this says that such a manifold is determined up to homeomorphism by its fundamental group. The problem is traditionally attributed to Borel, in connection with his 1953 work on locally homogeneous spaces [Borel1953CompactCliffordKlein].

The modern approach runs through surgery theory: rigidity follows once the relevant assembly maps in algebraic $K$- and $L$-theory are isomorphisms. Farrell and Jones established topological rigidity for compact non-positively curved manifolds [FarrellJones1993TopologicalRigidity], and the Farrell–Jones conjecture, now verified for large classes of groups, yields the Borel conjecture whenever the fundamental group lies in one of those classes [Lueck2025FJSurvey].

Despite this extensive coverage, no argument handles an arbitrary closed aspherical manifold, so the conjecture remains open in general; a resolution would require assembly-map information for all possible fundamental groups of such manifolds.