Precise Formulation of The Conjecture
Let and be closed and aspherical topological manifolds, and let
be a homotopy equivalence. The Borel conjecture states that the map 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.
This conjecture is false if topological manifolds and homeomorphisms are replaced by smooth manifolds and diffeomorphisms; counterexamples can be constructed by taking a connected sum with an exotic sphere.
Read more about this topic: Borel Conjecture
Famous quotes containing the words precise, formulation and/or conjecture:
“An imaginative book renders us much more service at first, by stimulating us through its tropes, than afterward, when we arrive at the precise sense of the author. I think nothing is of any value in books, excepting the transcendental and extraordinary.”
—Ralph Waldo Emerson (1803–1882)
“Art is an experience, not the formulation of a problem.”
—Lindsay Anderson (b. 1923)
“There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)