Aspherical manifolds
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− | + | The primary invariant of an aspherical manifold $M$ is its fundamental group, $\pi_1(M)$. | |
+ | * $\pi_1(M)$ is finitely presented and torsion free. | ||
+ | * $\pi_i(M) = 0$ for $i > 1$ by definition. | ||
+ | Two aspherical manifolds $M$ and $N$ are homotopy equivalent if and only if there is an isomorphism $\pi_1(M) \cong \pi_1(N)$. | ||
+ | |||
+ | 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 $A$ | ||
+ | $$H^*(M; A) = H^*(\pi_1(M); A)~~~H_*(M;A) = H_*(\pi_1(M); A).$$ | ||
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Revision as of 12:08, 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
The primary invariant of an aspherical manifold is its fundamental group, .
- is finitely presented and torsion free.
- for by definition.
Two aspherical manifolds and are homotopy equivalent if and only if there is an isomorphism .
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/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. |