The equivariant cohomology of complexity one spaces

TL;DR

This paper computes the equivariant cohomology of complexity one symplectic spaces, extending the understanding of their topological invariants by analyzing Hamiltonian circle actions, with a key step involving four-dimensional cases.

Contribution

It provides a general computation method for equivariant cohomology of complexity one spaces, building on the classification by Karshon and Tolman.

Findings

Explicit equivariant cohomology formulas for complexity one spaces

Reduction of the problem to Hamiltonian $S^1$ actions on 4-manifolds

Enhanced understanding of symplectic invariants in complexity one cases

Abstract

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