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 the following property: L has trivial monodromy along any loop γ in X assuming that 〈ξ, [γ]〉 = 0. It is easy to see that (a) Vξ is an algebraic variety isomorphic to (C∗)r for some integer r and (b) the dimension dim Hi(X;L) is independent of L assuming that L ∈ Vξ is generic, see [3], Theorem 1.50. The number dim Hi(X; L) for a generic L ∈ Vξ coincides with the Novikov-Betti number bi(ξ). The proof of Theorem 1 given below is based on the results of R. Bieri, W. Neumann, R. Strebel [1] and J.-Cl. Sikorav [6]. Theorem 1 implies the following vanishing result: