Aspherical manifolds

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Contents

1 Introduction

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 *.

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_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.

3 Invariants

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).

4 Classification

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.

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

5 Further discussion

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

6 References

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