m at h . SG ] 5 O ct 1 99 8 SIGNATURE VIA NOVIKOV NUMBERS
نویسندگان
چکیده
It is shown that the signature of a manifold with a symplectic circle action, having only isolated fixed points, equals the alternating sum of the Novikov numbers corresponding to the cohomology class of the generalized moment map. The same is true for more general fixed point sets 1. Theorem. Let M be a symplectic manifold with a symplectic circle action having only isolated fixed points. Then the signature of M is given by σ(M) = b0(ξ)− b2(ξ) + b4(ξ)− b6(ξ) + · · ·+ (−1)b2n(ξ), (1) where ξ ∈ H(M ;R) denotes the cohomology class of the generalized moment map and bi(ξ) denotes the corresponding Novikov number. This generalizes a theorem proven in [JR], which concerns Hamiltonian circle actions. In the Hamiltonian case ξ = 0 and the Novikov numbers become the usual Betti numbers bi(ξ) = bi(M). 2. Let us explain the terms used in the statement of Theorem 1. First note that any symplectic manifold has a canonical orientation, and the signature σ(M) is understood with respect to this orientation. Let ω denote the symplectic form of M . The S action is assumed to be symplectic, which means that for any g ∈ S holds gω = ω. Let X denote the vector field generating the S-action. Then θ = ι(X)ω (2) is a closed 1-form on M , which is called the generalized moment map. We consider the De Rham cohomology class ξ = [θ] ∈ H(M ;R) of θ. For the definition of the Novikov numbers bi(ξ) we refer to [BF], [F], [N]. For n odd both sides of formula (1) vanish. Indeed, we follow the convention that the signature of any 4k + 2-dimensional manifold is zero. The RHS of (1) vanishes because of the relation bi(ξ) = b2n−i(ξ), which follows directly from definition 1.2 in [BF] of the Novikov numbers and the classical Poincaré duality. The research was supported by EPSRC Visiting Fellowship.
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