Aspherical manifolds
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* $\pi_1(M)$ is finitely presented and torsion free. | * $\pi_1(M)$ is finitely presented and torsion free. | ||
* $\pi_i(M) = 0$ for $i > 1$ by definition. | * $\pi_i(M) = 0$ for $i > 1$ by definition. | ||
− | + | As each aspherical manifold $M$ is a $K(\pi_1(M), 1)$, the homology and cohomology of $M$ are by definition the homology and cohomology of $\pi_1(M)$. For any coefficient module $A$: | |
− | + | *$H^*(M; A) = H^*(\pi_1(M); A)$, | |
− | As each aspherical manifold $M$ is a $K(\pi_1(M), 1)$, the homology and cohomology of $M$ are by definition the homology and cohomology of $\pi_1(M)$. For any coefficient module $A$ | + | *$H_*(M;A) = H_*(\pi_1(M); A)$. |
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Revision as of 12:25, 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
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
The primary invariant of an aspherical manifold is its fundamental group, .
- is finitely presented and torsion free.
- for by definition.
As each aspherical manifold is a , the homology and cohomology of are by definition the homology and cohomology of . For any coefficient module :
- ,
- .
4 Classification
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).