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Novikov inequalities with symmetry

We establish an equivariant generalization of the Novikov inequalities which allow to estimate the topology of the set of critical points of a closed basic invariant 1-form by means of twisted equivariant cohomology of the manifold. 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 on M . 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 inequalities (corresponding to all the irreducible representations) put together give good enough estimate of the topology of C.
As a simple application we obtain Novikov type inequalities for manifolds with boundary.
1. Basic 1-forms. Consider a closed 1-form θ on a closed manifold M . Our problem is to find estimates on the topology of the set C of critical points of θ by using global topological invariants of M . Suppose that G is a compact Lie group acting on M . We will assume that θ is G invariant, i.e. g * θ = θ for any g ∈ G.
Moreover, we will assume that θ is basic; recall, that this means that θ vanishes on vectors tangent to the orbits of G.
Note that any exact invariant form θ = df is basic. Also, (cf., for example, [6,Lemma 3.4]), if M is connected and if the set of fixed points of the action of G on M is not empty, then any closed G-invariant 1-form on M is basic.

Equivariant flat vector bundles.
Let F → M be a flat vector bundle over M endowed with a smooth action of G such that the projection F → M is Gequivariant. We will assume that the action g : F x → F g·x is linear for any g ∈ G and x ∈ M .
The group G acts naturally on the space of differentials forms Ω * (M, F ) on M with values in F . For any element X ∈ g, (where g denotes the Lie algebra of G) we will denote by L F (X) : Ω * (M, F ) → Ω * (M, F ) the corresponding Lie algebra action.
A flat bundle F → M as above, is called G-equivariant flat vector bundle if the following two conditions are satisfied: for any g ∈ G, and for any X ∈ g. Here ∇ denotes the covariant derivative of F and for X ∈ g, X M denotes the vector field on M defined by the action of g. The second condition determines the covariant derivative in the directions tangent to the orbits. If the group G is connected, then first condition in (1) follows from the second. Conversely, if G is finite, the second condition in (1) carries no information. However, these conditions are independent in general.
As an example, consider a closed G-invariant 1-form θ on M with complex values. Consider the flat vector bundle determined by the form θ. Namely, let E θ = M × C with the G action coming from the factor M and with the flat connection ∇ = d + θ ∧ ·. This flat bundle always satisfy the first condition of (1) and it satisfies the second condition iff the form θ is basic.
3. The pushforward. Suppose (only in this section) that the action of G on M is free. Then the quotient B = M/G is a smooth manifold and the map q : M → B is a locally trivial fibration. We will discuss a construction which produces a flat vector bundle over B starting from an equivariant vector bundle over M .
Namely, let S(F ) denote the locally constant sheaf of flat sections of F . Then the direct image q * S(F ) of F is a locally constant sheaf over B. Let q * F denote the flat bundle corresponding to this sheaf. The group G acts naturally on q * F and this action is compatible with the flat structure. Moreover, the action of the connected component G 0 of the unit element e ∈ G is trivial. Thus, the bundle q * F representations of the finite group G/G 0 . The most important for us will be the subbundle corresponding to the trivial representation; we will denote it by q G * F . Note that if G is connected, then q G * F = q * F . We will say that the flat bundle q G * F is the pushforward of the bundle F . * p * F . We cannot literally apply the construction of the previous paragraph since our category is the category of smooth finite dimensional manifolds and the universal principal bundle EG → BG is usually infinite dimensional. The problem may be overcome by using finite dimensional approximations of EG. We refer to [6] (see also [2]) for details.  The formal power series will be called the equivariant Novikov series. If the group G is connected, then Z is a G-invariant submanifold of M . In general, we denote by G Z = {g ∈ G| g · Z ⊂ Z} the stabilizer of the component Z in G. Let |G : G Z | denote the index of G Z as a subgroup of G; it is always finite.
The compact Lie group G Z acts on the manifold Z and the flat vector bundles and define using it the following equivariant Morse counting series where the sum is taken over all connected components Z of C.

Theorem.
Suppose that G is a compact Lie group, acting on a closed manifold M and let F be an equivariant flat vector bundle over M . Then for any closed non-degenerate (in the sense of Bott) basic 1-form θ on M , there exists a formal power series Q(λ) with non-negative integer coefficients, such that The proof (cf. [6]) is based in its main part on the Novikov type inequalities for forms with non-isolated zeros, obtained in [4,5].
Consider the case when G acts freely on M . Then the basic form θ defines a closed 1-form θ ′ on M/G. It is straightforward to see, that in this case the inequalities of Theorem 7 (with F being the trivial line bundle) reduce to the usual Novikov inequalities with respect to the form θ ′ on the quotient manifold M/G. In particular, we see that in this situation the equivariant Novikov numbers β G i (ξ, F ) vanish for large i.
Note that in the case of a (non free) circle action G = S 1 , the equivariant Novikov numbers β G i (ξ, F ) for large i are two-periodic and coincide with the sum of even or odd (depending on the parity of i) usual (i.e. non-equivariant) Novikov numbers of the fixed point set; to see this one uses the localization theorem.
8. The finite group case. The rest of this paper is devoted to illustrations of Theorem 7 in the case when the group G is finite. In this situation, one can explicitly calculate the equivariant cohomology in terms of the action of G on the usual cohomology.
Let ρ : G → End V ρ be a representation of G on a finite dimensional complex vector space V ρ . Consider the bundle F ρ = M × V ρ over M with the trivial connection and with the diagonal G action. It is an equivariant flat bundle over M , which is trivial as the flat bundle but it may be not trivial as an equivariant flat bundle. We will apply Theorem 7 with the bundle F = F ρ .
The twisted equivariant cohomology H * G (M, F ρ ⊗ E tθ ) can be calculated as Here V * ρ denotes the representation dual to ρ; the cohomology H * (M, E tθ ) is considered with its induced G-action.
For an irreducible representation ρ the equivariant Novikov number β G i (ξ, F ρ ) = β G i (ξ, ρ) equals to the background multiplicity (i.e. generic with respect to t) of V * ρ in the decomposition of H i (M, E tθ ). The equivariant Poincaré series (4) can be calculated using a formula similar to (7) in terms of the action of G on the usual cohomology. We will denote M G θ,F ρ (λ) = M G θ (λ; ρ) and N G ξ,F ρ (λ) = N G ξ (λ; ρ) and view them as functions of two variables: λ (which will be formal) and ρ (which will run over the set of irreducible representations of G). Applying Theorem 7, we obtain where ρ is an irreducible representations of G and Q(λ; ρ) denotes a polynomial in λ with non-negative integral coefficients for any ρ. We may view the last statement as establishing one family of inequalities of Novikov type for any irreducible representation ρ. The inequalities of M.Atiyah and R.Bott [1] correspond (in the case ξ = 0) to the inequalities (8) with ρ the trivial representation. The usual (non-equivariant) Novikov inequalities [7,8] correspond to the regular representation in (8). Using these remarks, one may construct very simple examples (one and two dimensional!) such that the non-equivariant inequalities are better than the inequalities of [1].
9. Novikov inequalities for manifolds with boundary. Our strategy will be to reduce the problem on a manifold with boundary to a problem on the double with its natural Z 2 action. From section 8 we know that we should expect two families of inequalities -since there are two irreducible representations of Z 2 .
Let M be a compact manifold with boundary Γ = ∂M and let θ be a closed 1-form on M . We will denote by C the set of all critical points of θ. We will suppose that θ| int(M ) and θ| Γ are both non-degenerate in the sense of Bott (cf. §6) and that any critical point of θ| Γ is a critical point of θ as well. Additionally, we will suppose that for any connected component Z ⊂ Γ of the critical point set of θ| Γ holds either (1) Z is nondegenerate as a critical manifold of θ, or (2) Z is the boundary of a connected component Z ′ ⊂ C such that Z = Z ′ ∩ Γ and the intersection Z ′ ∩ Γ is transversal. If the first possibility holds, then the Hessian h θ (·, ·) of θ is a nondegenerate unique nonvanishing vector field X on Z normal to Γ such that h θ splits as a direct sum: h θ (X, Y ) = 0 for any Y ∈ T Γ. The function h θ (X, X) is everywhere positive or negative on Z; we will call the corresponding component Z ⊂ Γ ∩ C positive or negative respectively.
Represent C as the union of 4 disjoint submanifolds C in ∪ C + ∪ C − ∪ C bd , where C in denotes the union of the connected components of C which do not intersect Γ, C bd denotes the union of the components which are manifolds with nonempty boundary, and C ± denotes the union of the positive (negative) components in Γ (as defined in the previous paragraph).
For simplicity we will assume that M and all submanifolds Z ⊂ C are orientable -without this assumption the notations will be more complicated.
For any connected component Z ⊂ C denote by ind + (Z) and ind − (Z) the dimensions of the positive and the negative subbundles ν + (Z) and ν − (Z) of the normal bundle ν(Z) in M correspondingly, compare §6. Now we will define two Morse counting polynomials where P Z (λ) = λ i dim H i (Z, o(Z)) is the Poincaré polynomial of Z; in this formula Z runs over the connected components contained in C in ∪ C bd ∪ C + in the case of + and contained in C in ∪ C bd ∪ C − in the case of −.
The Novikov numbers β i (ξ) will be defined as the background values of the dimension of H i (M, E tθ ), cf. §5. Here ξ ∈ H 1 (M, R) denotes the class of θ.
Note that in the case of a closed manifold, these two statements (for ± equal to + and − correspondingly) are equivalent (as follows from Poincaré duality), but for the case of manifolds with boundary they are independent.
The proof of Theorem 10 follows by applying Theorem 7 to the double D(M ), two copies of M glued along the boundary Γ. The double D(M ) has the natural Z 2 action. Using our assumptions on θ we may construct an appropriate invariant 1-formθ on D(M ) and then use Theorem 7. The computation of the twisted equivariant cohomology of the double is based on the following simple Lemma: