Aspherical manifolds

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1 Introduction

A path-connected space $X$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == Introduction == <wikitex>; A path-connected space $X$ is called aspherical is its higher homotopy groups vanish: $\pi_i(X) = 0$ for all $i \geq 2$. This article is about closed, aspherical manifolds $M$ which are connected manifolds with contractible universal cover $\widetilde M \simeq *$. </wikitex> == Construction and examples == <wikitex>; * $S^1$ is aspherical. * Any [[Surface|surface]] $F$, not homeomorphic to $S^2$ or $\Rr P^2$ is aspherical. * A closed, oriented $-manifold $M$ is aspherical if and only if it is [[irreducible]] and $\pi_1(M)$ is torsion free. * In any dimension, if $M$ admits a metric of non-positive [[Wikipedia:Sectional_curvature|sectional curvature]] then $M$ is aspherical. * If $L$ is a Lie group with $\pi_0(L)$ finite, $K$ is a maximal compact subgroup of $L$ and $G$ is a discrete torsion free lattice in $L$ then $$G \backslash L/K$$ is aspherical. </wikitex> == Invariants == <wikitex>; The primary invariant of an aspherical manifold $M$ is its fundamental group, $\pi_1(M)$. * $\pi_1(M)$ is finitely presented and torsion free. * $\pi_i(M) = 0$ for $i > 1$ by definition. Two aspherical manifolds $M$ and $N$ are homotopy equivalent if and only if there is an isomorphism $\pi_1(M) \cong \pi_1(N)$. As each aspherical manifold $M$ is a $K(\pi_1(M), 1)$, the homology and cohomology of $M$ are by definition the homology and cohomology of $\pi_1(M)$. For any coefficient module $A$ $$H^*(M; A) = H^*(\pi_1(M); A)~~~H_*(M;A) = H_*(\pi_1(M); A).$$ </wikitex> == Classification/Characterization (if available) == <wikitex>; YOUR TEXT HERE ... </wikitex> == Further discussion == <wikitex>; YOUR TEXT HERE ... </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] {{Stub}}X$ is called aspherical is its higher homotopy groups vanish: $\pi_i(X) = 0$$\pi_i(X) = 0$ for all $i \geq 2$$i \geq 2$. This article is about closed, aspherical manifolds $M$$M$ which are connected manifolds with contractible universal cover $\widetilde M \simeq *$$\widetilde M \simeq *$.

2 Construction and examples

• $S^1$$S^1$ is aspherical.
• Any surface $F$$F$, not homeomorphic to $S^2$$S^2$ or $\Rr P^2$$\Rr P^2$ is aspherical.
• A closed, oriented $3$$3$-manifold $M$$M$ is aspherical if and only if it is irreducible and $\pi_1(M)$$\pi_1(M)$ is torsion free.
• In any dimension, if $M$$M$ admits a metric of non-positive sectional curvature then $M$$M$ is aspherical.
• If $L$$L$ is a Lie group with $\pi_0(L)$$\pi_0(L)$ finite, $K$$K$ is a maximal compact subgroup of $L$$L$ and $G$$G$ is a discrete torsion free lattice in $L$$L$ then

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

is aspherical.

3 Invariants

The primary invariant of an aspherical manifold $M$$M$ is its fundamental group, $\pi_1(M)$$\pi_1(M)$.

• $\pi_1(M)$$\pi_1(M)$ is finitely presented and torsion free.
• $\pi_i(M) = 0$$\pi_i(M) = 0$ for $i > 1$$i > 1$ by definition.

Two aspherical manifolds $M$$M$ and $N$$N$ are homotopy equivalent if and only if there is an isomorphism $\pi_1(M) \cong \pi_1(N)$$\pi_1(M) \cong \pi_1(N)$.

As each aspherical manifold $M$$M$ is a $K(\pi_1(M), 1)$$K(\pi_1(M), 1)$, the homology and cohomology of $M$$M$ are by definition the homology and cohomology of $\pi_1(M)$$\pi_1(M)$. For any coefficient module $A$$A$

$\displaystyle H^*(M; A) = H^*(\pi_1(M); A)~~~H_*(M;A) = H_*(\pi_1(M); A).$ $$\displaystyle H^*(M; A) = H^*(\pi_1(M); A)~~~H_*(M;A) = H_*(\pi_1(M); A).$$

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.