Math 2.0

Manifolds and cell complexes Cobordism and surgery Conjecture

Borel conjecture

Builds on Topological manifoldHomotopy equivalenceHomotopy groupUniversal coverHomeomorphism

Statement

A path-connected space XX is aspherical when its homotopy groups πk(X)\pi_k(X) vanish for all k≥2k \ge 2; equivalently, its universal cover is contractible. Let MM and NN be closed connected aspherical topological manifolds. Then every homotopy equivalence f ⁣:M→Nf \colon M \to N is homotopic to a homeomorphism. In particular, two closed aspherical manifolds with isomorphic fundamental groups are homeomorphic.

Remarks

The conjecture is open in general. It holds in dimensions at most 22 by the classification of surfaces, and in dimension 33 as a consequence of the geometrization theorem. Farrell and Jones proved it for closed Riemannian manifolds of nonpositive sectional curvature of dimension at least 55. Bartels and Lück proved it in dimensions at least 55 when the fundamental group is hyperbolic or CAT(0). In terms of the surgery exact sequence, it says that the structure set of a closed aspherical manifold of dimension at least 55 has one element. The analogue for diffeomorphisms is false: for n≥5n \ge 5 there are smooth manifolds homeomorphic to the nn-torus but not diffeomorphic to it.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Homeomorphism, concept, no evidence levelHomeomorphismconceptHomotopy equivalence, concept, no evidence levelHomotopy equivalenceconceptHomotopy group, concept, no evidence levelHomotopy groupconceptTopological manifold, concept, no evidence levelTopological manifoldconceptUniversal cover, concept, no evidence levelUniversal coverconceptBorel conjecture, conjecture, no evidence levelBorel conjectureconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Homeomorphism concept
  2. Homotopy equivalence concept
  3. Homotopy group concept
  4. Topological manifold concept
  5. Universal cover concept
  6. Borel conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Borel_conjecture
About this page
TagX057MN9YY9
URIm2://X057MN9YY9
Kindconjecture
WritingW0 stub
MSC57N99, 57R67, 20F65
StatusThis copy of the site has no build data. Run m2 status X057MN9YY9.
AboutTopological manifold
UsesTopological manifold, Homotopy equivalence, Homotopy group, Universal cover, Homeomorphism
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Changed2 November 2026

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