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Complexity one spaces are an important class of examples in symplecticgeometry. Karshon and Tolman classify them in terms of combinatorial andtopological data. In this paper, we compute the equivariant cohomology for anycomplexity one space $T^{n-1}$ acting on $M^{2n}$. The key step is to computethe equivariant cohomology for any Hamiltonian $S^1$ action on $M^4$.

The equivariant cohomology of complexity one spaces