Elliptization conjecture: Difference between revisions
m (3 revisions) |
|||
| (2 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
{{topology theorem}} | |||
{{universal cover prediction|topological manifold}} | |||
{{in dimension|3}} | |||
==Statement== | |||
===Verbal statement=== | |||
Any closed 3-manifold with finite fundamental group is [[spherical manifold|spherical]], viz it has a Riemannian metirc with positive sectional curvature (or equivalently, its universal cover is the 3-sphere). | |||
==Relation with other results== | |||
===Weaker results=== | |||
* [[Poincare conjecture]]: In the particular case that the fundamental group is trivial, this actually tells us that the manifold is homeomorphic to the 3-sphere | |||
Latest revision as of 19:39, 18 May 2008
This article makes a prediction about the universal cover of a manifold based on given data at the level of a:topological manifold
This result is about manifolds in dimension:3
Statement
Verbal statement
Any closed 3-manifold with finite fundamental group is spherical, viz it has a Riemannian metirc with positive sectional curvature (or equivalently, its universal cover is the 3-sphere).
Relation with other results
Weaker results
- Poincare conjecture: In the particular case that the fundamental group is trivial, this actually tells us that the manifold is homeomorphic to the 3-sphere