Since aspherical manifolds with isomorphic fundamental groups are homotopy equivalent, the Borel conjecture implies that aspherical closed manifolds are determined, up to homeomorphism, by their fundamental groups.
In a May 1953 letter to Jean-Pierre Serre,[1] Armand Borel raised the question whether two aspherical manifolds with isomorphic fundamental groups are homeomorphic.
A positive answer to the question "Is every homotopy equivalence between closed aspherical manifolds homotopic to a homeomorphism?"
For instance, the Mostow rigidity theorem states that a homotopy equivalence between closed hyperbolic manifolds is homotopic to an isometry—in particular, to a homeomorphism.
The Borel conjecture is a topological reformulation of Mostow rigidity, weakening the hypothesis from hyperbolic manifolds to aspherical manifolds, and similarly weakening the conclusion from an isometry to a homeomorphism.