266:—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.
192:. 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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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
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Nevertheless, there are classes of manifolds for which homotopy equivalences between them can be homotoped to homeomorphisms. For instance, the
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raised the question whether two aspherical manifolds with isomorphic fundamental groups are homeomorphic. A positive answer to the question "
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This article is about Armand Borel's conjecture in geometric topology. For Émile Borel's conjecture in analysis/measure theory, see
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Is every homotopy equivalence between closed aspherical manifolds homotopic to a homeomorphism?
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595:(2002). "The Borel conjecture". In Farrell, F.T.; Goettshe, L.; Lueck, W. (eds.).
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Geometry and algebra. Oberwolfach
Seminars, 33. Birkhäuser Verlag, Basel, 2005.
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conjecture, asserting that a weak, algebraic notion of equivalence (namely,
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Topology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001)
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is not aspherical. Nevertheless, the Borel conjecture for the
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asserts that a closed manifold homotopy equivalent to
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468:{\displaystyle T^{3}=S^{1}\times S^{1}\times S^{1}}
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275:The Borel conjecture implies the
270:Relationship to other conjectures
307:{\displaystyle f\colon M\to BG}
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150:{\displaystyle f\colon M\to N}
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681:Unsolved problems in geometry
243:Motivation for the conjecture
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251:which are not homeomorphic.
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495:{\displaystyle S^{3}}
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181:{\displaystyle f}
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83:{\displaystyle M}
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54:. It is a
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122:, and let
116:aspherical
41:aspherical
453:×
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296:→
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120:manifolds
556:Topology
549:(1986).
350:3-sphere
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211:with an
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164:. The
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