---
tag: X057MN9YY9
kind: conjecture
name: Borel conjecture
msc: [57N99, 57R67, 20F65]
uses: [QQQ9B1DK4P, XZG6MTSWCN, P976WYEFMB, Y1YPE6PM52, 0QT00PB5G2]
about: [QQQ9B1DK4P]
sources:
  - url: https://en.wikipedia.org/wiki/Borel_conjecture
same_as:
  wikidata: Q4138792
  wikipedia: Borel conjecture
history:
  posed_by: "Armand Borel"
  first_stated: "1953"
---

## Statement

A path-connected space $X$ is *aspherical* when its [[homotopy groups|P976WYEFMB]] $\pi_k(X)$ vanish for all $k \ge 2$; equivalently, its [[universal cover|Y1YPE6PM52]] is contractible.
Let $M$ and $N$ be closed connected aspherical [[topological manifolds|QQQ9B1DK4P]].
Then every [[homotopy equivalence|XZG6MTSWCN]] $f \colon M \to N$ is homotopic to a [[homeomorphism|0QT00PB5G2]].
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 $2$ by the classification of surfaces, and in dimension $3$ as a consequence of the geometrization theorem.
Farrell and Jones proved it for closed Riemannian manifolds of nonpositive sectional curvature of dimension at least $5$.
Bartels and Lück proved it in dimensions at least $5$ when the fundamental group is hyperbolic or CAT(0).
In terms of the [[surgery exact sequence|Q7C480HQ9J]], it says that the structure set of a closed aspherical manifold of dimension at least $5$ has one element.
The analogue for diffeomorphisms is false: for $n \ge 5$ there are smooth manifolds homeomorphic to the $n$-torus but not diffeomorphic to it.

## Explanations

### L1 human

An aspherical manifold is one whose shape is completely recorded by its loops: all higher-dimensional holes are absent.
The torus and closed hyperbolic manifolds are examples.
The conjecture says that for such manifolds the loops determine everything: if two of them have the same homotopy type, they are the same manifold up to homeomorphism.