2010/08/05 by Pabiniak, Milena
#53D99 #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.1008.0900
We consider a Hamiltonian action of n-dimensional torus, Tn, on a compact symplectic manifold (M,ω) with d isolated fixed points. For every fixed point p there exists (though not unique) a class ap in H^*T(M; Q) such that the collection ap, over all fixed points, forms a basis for H^*T(M; Q) as an H^*(BT; Q) module. The map induced by the inclusion, ι^*:H^*T(M; Q) → H^*T(MT; Q)= ⊕j=1dQ[x1, ..., xn] is injective. We use such classes ap to give necessary and sufficient conditions for f=(f1, ...,fd) in ⊕j=1dQ[x1, ..., xn] to be in the image of ι^*, i.e. to represent an equiviariant cohomology class on M. In the case when T is a circle and present these conditions explicitly. We explain how to combine this 1-dimensional solution with Chang-Skjelbred Lemma in order to obtain the result for a torus T of any dimension. Moreover, for a GKM T-manifold M our techniques give combinatorial description of H^*K(M; Q), for a generic subgroup K \hookrightarrow T, even if M is not a GKM K-manifold.