Borel conjecture

In mathematics, specifically geometric topology, the Borel conjecture (named for Armand Borel) asserts that an aspherical closed manifold is determined by its fundamental group, up to homeomorphism. It is a rigidity conjecture, demanding that a weak, algebraic notion of equivalence (namely, a homotopy equivalence) imply a stronger, topological notion (namely, a homeomorphism). In mathematics, specifically geometric topology, the Borel conjecture (named for Armand Borel) asserts that an aspherical closed manifold is determined by its fundamental group, up to homeomorphism. It is a rigidity conjecture, demanding that a weak, algebraic notion of equivalence (namely, a homotopy equivalence) imply a stronger, topological notion (namely, a homeomorphism). There is a different Borel conjecture (named for Émile Borel) in set theory. It asserts that every strong measure zero set of reals is countable. Work of Nikolai Luzin and Richard Laver shows that this conjecture is independent of the ZFC axioms. This article is about the Borel conjecture in geometric topology. Let M {displaystyle M} and N {displaystyle N} be closed and aspherical topological manifolds, and let be a homotopy equivalence. The Borel conjecture states that the map f {displaystyle f} is homotopic to a homeomorphism. 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. This conjecture is false if topological manifolds and homeomorphisms are replaced by smooth manifolds and diffeomorphisms; counterexamples can be constructed by taking a connected sum with an exotic sphere. In a May 1953 letter to Jean-Pierre Serre (web reference below), Armand Borel asked the question whether two aspherical manifolds with isomorphic fundamental groups are homeomorphic. A basic question is the following: if two closed manifolds are homotopy equivalent, are they homeomorphic? This is not true in general: there are homotopy equivalent lens spaces which are not homeomorphic. Nevertheless, there are classes of manifolds for which homotopy equivalences between them can be homotoped to homeomorphisms. 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.