William Thurston's elliptization conjecture states that a closed

3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...

with finite

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

is spherical, i.e. has a Riemannian metric of constant positive sectional curvature.

Relation to other conjectures

A 3-manifold with a Riemannian metric of constant positive sectional curvature is covered by the 3-sphere, moreover the group of covering transformations are isometries of the 3-sphere. If the original 3-manifold had in fact a trivial fundamental group, then it is

homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...

to the

3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...

(via the covering map). Thus, proving the elliptization conjecture would prove the Poincaré conjecture as a corollary. In fact, the elliptization conjecture is logically equivalent to two simpler conjectures: the Poincaré conjecture and the spherical space form conjecture. The elliptization conjecture is a special case of Thurston's geometrization conjecture, which was proved in 2003 by G. Perelman.

References

For the proof of the conjectures, see the references in the articles on geometrization conjecture or Poincaré conjecture. * William Thurston. ''Three-dimensional geometry and topology. Vol. 1''. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. . * William Thurston
The Geometry and Topology of Three-Manifolds
1980 Princeton lecture notes on geometric structures on 3-manifolds, that states his elliptization conjecture near the beginning of section 3. Riemannian geometry 3-manifolds Conjectures that have been proved {{topology-stub