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 $\pi_0(L)$ finite, $K$ is a maximal compact subgroup of $L$ and $G$ is a discrete torsion free lattice in $L$ then | + | * 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. |
− | $G \backslash L/K$ is aspherical. | + | |
* A product of aspherical manifolds is again aspherical: | * A product of aspherical manifolds is again aspherical: | ||
** $T^n$, the $n$-torus is aspherical. | ** $T^n$, the $n$-torus is aspherical. |
Revision as of 12:29, 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. |
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Contents |
1 Introduction
Tex syntax errorwhich 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
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is aspherical if and only if it is irreducible and is torsion free. - In any dimension, if
Tex syntax error
admits a metric of non-positive sectional curvature thenTex syntax error
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.
- A product of aspherical manifolds is again aspherical:
- , the -torus is aspherical.
3 Invariants
Tex syntax erroris its fundamental group, .
- is finitely presented and torsion free.
- for by definition.
Tex syntax erroris a , the homology and cohomology of
Tex syntax errorare by definition the homology and cohomology of . For any coefficient module :
- ,
- .
4 Classification
Tex syntax errorand are homotopy equivalent if and only if there is an isomorphism .
The main conjecture organising the classification of aspherical manifolds is the Borel Conjecture.
Conjecture 4.1. Let be a homotopy equivalence between aspherical manifolds. Then is homotopic to a homeomorphism.
5 Further discussion
For further information see [Farrell&Jones1990] and [Lück2008].
6 References
- [Farrell&Jones1990] F. T. Farrell and L. E. Jones, Classical aspherical manifolds, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1990. MR1056079 (91k:57001) Zbl 0729.57001
- [Lück2008] W. Lück, Survey on aspherical manifolds, to appear in the proceedings of the 5-th ECM in Amsterdam (2008).