- Borel conjecture
In
mathematics , specificallygeometric topology , the Borel conjecture asserts that an asphericalclosed manifold is determined by itsfundamental group , up tohomeomorphism . 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).Precise formulation of the conjecture
Let and be closed and aspherical topological manifolds, and let
:
be a homotopy equivalence. The Borel conjecture states that the map 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 manifold s and homeomorphisms are replaced by smooth manifolds and diffeomorphisms; counterexamples can be constructed by taking aconnected sum with anexotic sphere .Motivation for the conjecture
A basic question is the following: if two 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 anisometry —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.Relationship to other conjectures
* The Borel conjecture implies the
Novikov conjecture for the special case in which the reference map is a homotopy equivalence.
* ThePoincaré conjecture asserts that a closed manifold homotopy equivalent to , the3-sphere , is homeomorphic to . This is not a special case of the Borel conjecture, because is not aspherical. Nevertheless, the Borel conjecture for the 3-torus implies the Poincaré conjecture for .References
* F.T. Farrell, "The Borel conjecture. Topology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001)," 225–298, ICTP Lect. Notes, 9, "Abdus Salam Int. Cent. Theoret. Phys., Trieste," 2002.
* M. Kreck, and W. Lück, "The Novikov conjecture." Geometry and algebra. Oberwolfach Seminars, 33. Birkhäuser Verlag, Basel, 2005.
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