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 smooth functions remain true with the usual Betti numbers being replaced by the von Neumann Betti numbers. Novikov [N] initiated an analog of the Morse theory for closed 1-forms. He showed that the Morse inequalities for functions can be generalized to closed 1-forms if instead of the Betti numbers one uses the Novikov numbers (they will be briefly reviewed in §3.1 below). In this paper we show that the inequalities of Novikov and Shubin [NS] hold for closed 1-forms as well. This result gives new Novikov type inequalities for closed 1-forms. Viewed differently, this gives a vanishing theorem for L cohomology, generalizing a theorem of W. Lück [L3]. The proof of our Theorem 1 uses an idea of Gromov and Eliashberg [EG], which consist in counting additional critical points which appear when transforming the given closed 1-form into a function; the proof also uses the multiplicativity of the von Neumann Betti numbers under finitely sheeted coverings. In papers [BF2] and [MS] different Novikov type inequalities using the von Neumann Betti numbers were put forward. These inequalities involve cohomology of flat bundles of Hilbertian modules twisted by a generic line bundle determined by the closed 1-form (similarly to the finite dimensional case). Our approach in this paper does not require the twisting, and therefore it is simpler. On the other hand V. Mathai and M. Shubin [MS] allow more general Hilbertian flat bundles. It is clear that one may easily generalize Theorem 1 below for closed 1-forms with Bott type singularities, in a fashion similar to [BF1], [BF2]. I would like to thank Andrew Ranicki for stimulating discussions. Partially supported by US Israel Binational Science Foundation, by the Herman Minkowski Center for Geometry, and by EPSRC grant GR/M20563 Typeset by AMS-TEX 1