Realisation of cycles by aspherical manifolds

Realisation of Cycles by Aspherical Manifolds

In the late 1940s N. Steenrod posed the following problem, which is now familiar as the problem on realisation of cycles. For a given homology class z ∈ Hn(X;Z), do there exist an oriented manifold N and a mapping f : N → X such that f∗[N ] = z? A famous theorem of R. Thom claims that each integral homology class is realisable in sense of Steenrod with some multiplicity. A classical problem is ...

Aspherical Manifolds*

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, has many striking implications about the geometry and analysis of the manifold or ...

Survey on Aspherical Manifolds

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 i...

Topological rigidity for non-aspherical manifolds by

The Borel Conjecture predicts that closed aspherical manifolds are topological rigid. We want to investigate when a non-aspherical oriented connected closed manifold M is topological rigid in the following sense. If f : N → M is an orientation preserving homotopy equivalence with a closed oriented manifold as target, then there is an orientation preserving homeomorphism h : N → M such that h an...

Exotic aspherical manifolds