Aspherical manifolds

Revision as of 12:03, 30 November 2009

1 Introduction

A path-connected space $<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == Introduction == <wikitex>; A path-connected space $X$ is called aspherical is its higher homotopy groups vanish: $\pi_i(X) = 0$ for all $i \geq 2$. This article is about closed, aspherical manifolds $M$ which are connected manifolds with contractible universal cover $\widetilde M \simeq *$. </wikitex> == Construction and examples == <wikitex>; $$K \backslash G/L$$ {{cite|Farrell&Jones1990}} </wikitex> == Invariants == <wikitex>; YOUR TEXT HERE ... </wikitex> == Classification/Characterization (if available) == <wikitex>; YOUR TEXT HERE ... </wikitex> == Further discussion == <wikitex>; YOUR TEXT HERE ... </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] {{Stub}}X$ is called aspherical is its higher homotopy groups vanish: $\pi_i(X) = 0$ for all $i \geq 2$. This article is about closed, aspherical manifolds $M$ which are connected manifolds with contractible universal cover $\widetilde M \simeq *$.

2 Construction and examples

• $S^1$ is aspherical.
• Any surface $F$, not homeomorphic to $S^2$ or $\Rr P^2$ is aspherical.
• 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 sectional curvature then $M$ is aspherical.
• If $L$ is a Lie group with $\pi_0L)$ finite, $K$ is a maximal compact subgroup of $L$ and $L$ is a discrete torsion free lattice in $L$ then

$$\displaystyle K \backslash G/L$$

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