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| In [[mathematics]], specifically [[geometric topology]], the '''Borel conjecture''' asserts that an [[Aspherical space|aspherical]] [[closed manifold]] is determined by its [[fundamental group]], up to [[homeomorphism]]. It is a [[Rigidity (mathematics)|rigidity]] conjecture, demanding that a weak, algebraic notion of equivalence (namely, a [[homotopy|homotopy equivalence]]) imply a stronger, topological notion (namely, a homeomorphism).
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| ==Precise formulation of the conjecture==
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| Let <math>M</math> and <math>N</math> be [[closed manifold|closed]] and [[Aspherical space|aspherical]] topological [[manifold]]s, and let
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| :<math>f : M \to N</math>
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| be a [[homotopy|homotopy equivalence]]. The '''Borel conjecture''' states that the map <math>f</math> 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.
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| This conjecture is false if [[topological manifold]]s and homeomorphisms are replaced by [[smooth manifold]]s and [[diffeomorphism]]s; counterexamples can be constructed by taking a [[connected sum]] with an [[exotic sphere]].
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| ==The origin of the conjecture==
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| In a May 1953 letter to [[Jean-Pierre Serre|Serre]] (web reference below), [[Armand Borel]] asked the question whether two aspherical manifolds with isomorphic fundamental groups are homeomorphic.
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| ==Motivation for the conjecture==
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| 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 space]]s which are not homeomorphic.
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| 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 manifold]]s 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.
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| ==Relationship to other conjectures==
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| * The Borel conjecture implies the [[Novikov conjecture]] for the special case in which the reference map <math>f : M \to BG</math> is a homotopy equivalence.
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| * The [[Poincaré conjecture]] asserts that a closed manifold homotopy equivalent to <math>S^3</math>, the [[3-sphere]], is homeomorphic to <math>S^3</math>. This is not a special case of the Borel conjecture, because <math>S^3</math> is not aspherical. Nevertheless, the Borel conjecture for the [[Torus|3-torus]] <math>T^3 = S^1 \times S^1 \times S^1</math> implies the Poincaré conjecture for <math>S^3</math>.
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| ==References==
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| * 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.
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| * M. Kreck, and W. Lück, ''The Novikov conjecture.'' Geometry and algebra. Oberwolfach Seminars, 33. Birkhäuser Verlag, Basel, 2005.
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| * [http://www.maths.ed.ac.uk/~aar/surgery/borel.pdf The birth of the Borel conjecture], Extract from letter from [[Armand Borel|Borel]] to [[Jean-Pierre Serre|Serre]], 2nd May, 1953
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| [[Category:Geometric topology]]
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| [[Category:Homeomorphisms]]
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| [[Category:Conjectures]]
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| [[Category:Surgery theory]]
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The name of the writer is Numbers. Managing individuals is his profession. It's not a common thing but what she likes doing is foundation leaping and now she is trying to earn cash with it. For a whilst I've been in South Dakota and my mothers and fathers live nearby.
Here is my site ... wixothek.com