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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. ==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 manifolds and homeomorphisms are replaced by smooth manifolds and diffeomorphisms; counterexamples can be constructed by taking a connected sum with an exotic sphere. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Borel conjecture」の詳細全文を読む スポンサード リンク
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