Survey on aspherical manifolds

Abstract

Abstract. This is a survey on known results and open problems about closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. Many examples come from certain kinds of non-positive curvature conditions. The property aspherical, which is a purely homotopy theoretical condition, implies many striking results about the geometry and analysis of the manifold or its universal covering, and the ring theoretic properties and the K-and L-theory of the group ring associated to its fundamental group. The Borel Conjecture predicts that closed aspherical manifolds are topologically rigid. The article contains new results about product decompositions of closed aspherical manifolds and an announcement of a result joint with Arthur Bartels and Shmuel Weinberger about hyperbolic groups with spheres of dimension ≥ 6 as boundary. At the end we describe (winking) our universe of closed manifolds. IntroductionA space X is called aspherical if it is path connected and all its higher homotopy groups vanish, i.e., π n (X) is trivial for n ≥ 2. This survey article is devoted to aspherical closed manifolds. These are very interesting objects for many reasons. Often interesting geometric constructions or examples lead to aspherical closed manifolds. The study of the question which groups occur as fundamental groups of closed aspherical manifolds is intriguing. The condition aspherical is of purely homotopy theoretical nature. Nevertheless there are some interesting questions and conjectures about curvature properties of a closed aspherical Riemann manifold and about the spectrum of the Laplace operator on its universal covering. The Borel Conjecture predicts that aspherical closed topological manifolds are topologically rigid and that aspherical compact Poincaré complexes are homotopy equivalent to closed manifolds. We discuss the status of some of these questions and conjectures. Examples of exotic aspherical closed manifolds come from hyperbolization techniques and we list certain examples. At the end we describe (winking) our universe of closed manifolds.The results about product decompositions of closed aspherical manifolds in Section 6 are new and Section 8 contains an announcement of a result joint with Arthur Bartels and Shmuel Weinberger about hyperbolic groups with spheres of dimension ≥ 6 as boundary.

Fundamental classes not representable by products

“…In higher dimensions, Theorem 5.1 and its potential generalisations from bundles to other aspherical manifolds raise the following question: If M is a closed, oriented, connected, aspherical manifold whose fundamental group splits as a non-trivial direct product Γ 1 × Γ 2 , then can M be split up to homotopy or homeomorphism as a product of closed, oriented, connected manifolds with fundamental group Γ 1 and Γ 2 , respectively? Using the sophisticated machinery developed in the field of topological rigidity, this question can be answered affirmatively for a large class of such manifolds [28]. This solution relies on deep results concerning the Farrell-Jones conjecture, the Borel conjecture, the Novikov conjecture, and the resolution of homology manifolds.…”

On Proofs of the Farrell–Jones Conjecture

“…The conjecture is related to a number of other conjectures in geometric topology and K-theory, most prominently the Borel Conjecture. Detailed discussions of applications and the formulation of this conjecture (and related conjectures) can be found in [10,32,33,34,35].…”

Fundamental groups of aspherical manifolds and maps of non-zero degree

“…In particular, it is an open question whether aspherical manifolds with reducible fundamental groups (that is, virtual products of two infinite groups) are finitely covered -and therefore dominated -by products. Lück showed how to obtain an affirmative answer in dimensions higher than four, relying on very strong assumptions concerning the Farrell-Jones conjecture and the cohomological dimensions of the involved groups [24]. For non-positively curved manifolds an affirmative answer is given by Gromoll-Wolf's isometric splitting theorem [12].…”