Novikov Inequalities with Symmetry

We suggest Novikov type inequalities in the situation of a compact Lie groups action assuming that the given closed 1-form is invariant and basic. Our inequalities use equivariant cohomology and an appropriate equivariant generalization of the Novikov numbers. We test and apply our inequalities in the case of a finite group. As an application we obtain Novikov type inequalities for a manifold with boundary. In 1981 S.Novikov found a generalization of the classical Morse inequalities to closed 1-forms. In this paper we suggest an equivariant version of the Novikov inequalities. We consider compact G manifold M , where G is a compact Lie group, and an invariant closed 1-form θ onM . We assume that the form θ is non-degenerate in the sense of Bott, and our problem is to find estimates on the topology of the set C of critical points of θ using global topological invariants of M . We construct the equivariant Morse counting series, which combines information about the equivariant cohomology of all the connected components of C. Assuming that the form θ is basic (cf. below) we define an equivariant generalization of the Novikov numbers and, using these numbers, we construct the Novikov counting series. Our main theorem (Theorem 7) states that the equivariant Morse series is greater (in an appropriate sense) than the Novikov counting series. This statement contains an infinite number of inequalities involving the dimensions of the equivariant cohomology of connected components of C and the equivariant Novikov numbers. We use in this paper equivariant cohomology twisted by equivariant flat vector bundles, which is crucial for our approach. On one hand, any closed invariant basic 1-form determines a one-parameter family of equivariant flat vector bundles, which we use to define the equivariant generalizations of the Novikov numbers. On the other hand we observe, that using this cohomology allows to strengthen the inequalities considerably (cf. [BF1,§1.7]. Simple examples show, that applying the well-known equivariant Morse inequalities of Atiyah and Bott [AB] to the case when the group G is finite, one obtains the estimates, which are sometimes worse than the standard Morse inequalities (ignoring the group action!). The situation may be improved, however, by using the twisted equivariant cohomology. If G is a finite group, then any representation of G gives rise to an equivariant flat vector bundle and then (applying our general construction) to a family of inequalities. Examples show that only all these The research was supported by grant No. 449/94-1 from the Israel Academy of Sciences and Humanities