What is Borel conjecture?

In mathematics, specifically geometric topology, the Borel conjecture asserts that an aspherical closed manifold is determined by its fundamental group, up to homeomorphism. It is a rigidity conjecture, asserting that a weak, algebraic notion of equivalence should imply a stronger, topological notion.

There is a different Borel conjecture 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.