Aspherical manifolds

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.

This page has not been refereed. The information given here might be incomplete or provisional.

1 Introduction

A path-connected space $X$$== 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: {{cite|???}}. * 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. * A product of aspherical manifolds is again aspherical: ** $T^n$, the $n$-torus 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. 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 == <wikitex>; Two aspherical manifolds $M$ and $N$ are homotopy equivalent if and only if there is an isomorphism $\pi_1(M) \cong \pi_1(N)$. The main conjecture organising the classification of aspherical manifolds is the Borel Conjecture. {{beginthm|Conjecture}} Let $f : N \simeq M$ be a homotopy equivalence between aspherical manifolds. Then $f$ is homotopic to a homeomorphism. {{endthm}} </wikitex> == Further discussion == <wikitex>; For further information see {{cite|Farrell&Jones1990}} and {{cite|Lück2008}}. </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Theory]] {{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$$L$ then $G \backslash L/K$$G \backslash L/K$ is aspherical.
• A product of aspherical manifolds is again aspherical:
  • $T^n$$T^n$, the $n$$n$-torus 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.

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$:

• $H^*(M; A) = H^*(\pi_1(M); A)$$H^*(M; A) = H^*(\pi_1(M); A)$,
• $H_*(M;A) = H_*(\pi_1(M); A)$$H_*(M;A) = H_*(\pi_1(M); A)$.

4 Classification

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)$.

The main conjecture organising the classification of aspherical manifolds is the Borel Conjecture.

Conjecture 4.1. Let $f : N \simeq M$$f : N \simeq M$ be a homotopy equivalence between aspherical manifolds. Then $f$$f$ is homotopic to a homeomorphism.

5 Further discussion

For further information see [Farrell&Jones1990] and [Lück2008].

6 References

• [???] Template:???
• [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).