Aspherical manifolds
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* A closed, oriented $3$-manifold $M$ is aspherical if and only if it is [[irreducible]] and $\pi_1(M)$ is torsion free. | * A closed, oriented $3$-manifold $M$ is aspherical if and only if it is [[irreducible]] and $\pi_1(M)$ is torsion free. | ||
* In any dimension, if $M$ admits a metric of non-positive [[Wikipedia:Sectional_curvature|sectional curvature]] then $M$ is aspherical. | * In any dimension, if $M$ admits a metric of non-positive [[Wikipedia:Sectional_curvature|sectional curvature]] then $M$ is aspherical. | ||
− | * If $L$ is a Lie group with $\ | + | * If $L$ is a Lie group with $\pi_0(L)$ finite, $K$ is a maximal compact subgroup of $L$ and $G$ is a discrete torsion free lattice in $L$ then |
− | $$ | + | $$G \backslash L/K$$ |
is aspherical. | is aspherical. | ||
</wikitex> | </wikitex> |
Revision as of 12:03, 30 November 2009
An earlier version of this page was published in the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 12:23, 27 September 2012 and the changes since publication. |
Contents |
1 Introduction
A path-connected space is called aspherical is its higher homotopy groups vanish: for all . This article is about closed, aspherical manifolds which are connected manifolds with contractible universal cover .
2 Construction and examples
- is aspherical.
- Any surface , not homeomorphic to or is aspherical.
- A closed, oriented -manifold is aspherical if and only if it is irreducible and is torsion free.
- In any dimension, if admits a metric of non-positive sectional curvature then is aspherical.
- If is a Lie group with finite, is a maximal compact subgroup of and is a discrete torsion free lattice in then
is aspherical.
3 Invariants
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4 Classification/Characterization (if available)
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5 Further discussion
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6 References
This page has not been refereed. The information given here might be incomplete or provisional. |