We show that the product of infranilmanifolds with certain aspherical closed manifolds do not support Anosov diffeomorphisms. As a special case, we obtain that products of a nilmanifold and negatively curved manifolds of dimension at least 3 do not support Anosov diffeomorphisms. Mathematics Subject Classification. Primary 37D20; Secondary 55R10, 57R19, 37C25.

We prove an index theorem concerning the pushforward of flat B-vector bundles, where B is an appropriate algebra. We construct an associated analytic torsion form T . If Z is a smooth closed aspherical manifold, we show that T gives invariants of π∗(Diff(Z)).

Let X be a finite aspherical CW-complex whose fundamental group π1(X) possesses a subnormal series π1(X) ⊲ Gm ⊲ . . . ⊲ G0 with a non-trivial elementary amenable group G0. We investigate the L2-invariants of the universal covering of such a CW-complex X. We show that the NovikovShubin invariants αn(X̃) are positive. We further prove that the L2-torsion ρ(2)(X̃) vanishes if π1(X) has semi-integral...

We compute the Lagrangian Floer cohomology groups of certain tori in closed simply connected symplectic 4-manifolds arising from FintushelStern knot surgery. These manifolds are usually not symplectically aspherical. As a result of the computation we observe examples where HF (L0) = HF (L1) and L0 and L1 are smoothly isotopic but L0, L1 are not symplectically isotopic and are distinguished by H...

We give short new geometric proofs of theorems of Bogley and Pride and of Serre. Both follow quickly from a global fixed point theorem. The Bogley–Pride theorem concerns aspherical relative group presentations and was applied in [5] to the multivariable adjunction problem. Serre’s theorem is a basic result concerning group cohomology and finite subgroups.

We give examples of aspherical symplectic 4-manifolds, which are formal in the sense of rational homotopy theory but have degenerate Lefschetz pairings. (preliminary version) Compact Kähler manifolds have a number of topological properties which are not shared by general symplectic manifolds, see [1] for example. In particular, Kähler manifolds have the following cohomological properties: 1. th...

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 ...

We show that in all dimensions > 7 there are closed aspherical manifolds whose fundamental groups have nontrivial center but do not possess any topological circle actions. This disproves a conjectured converse (proposed by Conner and Raymond) to a classical theorem of Borel.

We present an overview of the results of our recent research in the field of adaptive optical components based on silicon microtechnologies, including membrane deformable mirrors, spatial light modulators, liquid-crystal correctors, wavefront sensors, and both spherical and aspherical micro-optical components. We aim at the realization of adaptive optical systems using standard-technology solut...