Aspherical manifolds
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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.
- A product of aspherical manifolds is again aspherical:
- , the -torus 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
Two aspherical manifolds and 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).