Novikov-Betti numbers and the fundamental group

Novikov-betti Numbers and the Fundamental Group

This result may appear striking as the Novikov-Betti numbers carry “abelian” information about X. We refer to [4], [3] for the definition of the Novikov-Betti numbers; an explicit definition will also be given below in the proof of Theorem 1. An alternative description of bi(ξ) uses homology of complex flat line bundles. Consider the variety Vξ of all complex flat line bundles L over X having t...

Von Neumann Betti Numbers and Novikov Type Inequalities

In this paper we show that Novikov type inequalities for closed 1-forms hold with the von Neumann Betti numbers replacing the Novikov numbers. As a consequence we obtain a vanishing theorem for L cohomology. We also prove that von Neumann Betti numbers coincide with the Novikov numbers for free abelian coverings. §0. Introduction S. Novikov and M. Shubin [NS] proved that Morse inequalities for ...

Self Coincidence Numbers and the Fundamental Group

For M and N closed oriented connected smooth manifolds of the same dimension, we consider the mapping space Map(M, N ; f) of continuous maps homotopic to f : M → N . We will show that the evaluation map from the space of maps to the manifold N induces a nontrivial homomorphism on the fundamental group only if the self coincidence number of f , denoted Λf,f , equals zero. Since Λf,f is equal to ...

-betti Numbers

The Atiyah conjecture predicts that the L-Betti numbers of a finite CW -complex with torsion-free fundamental group are integers. We show that the Atiyah conjecture holds (with an additional technical condition) for direct and inverse limits of groups for which it is true. As a corollary it holds for residually torsion-free solvable groups, e.g. for pure braid groups or for positive 1-relator g...

Coverings and Betti Numbers