Connected sums of simplicial complexes and equivariant cohomology

Abstract

In this paper, we discuss the connected sum K_1#^Z K_2 of simplicial complexes K_1 and K_2, as well as define the notion of a strong connected sum. Geometrically, the connected sum is motivated by Lerman's symplectic cut applied to a toric orbifold, and algebraically, it is motivated by the connected sum of rings introduced by Ananthnarayan-Avramov-Moore. We show that the Stanley-Reisner ring of a connected sum K_1#^Z K_2 is the connected sum of the Stanley-Reisner rings of K_1 and K_2 along the Stanley-Reisner ring of the intersection of K_1 and K_2. The strong connected sum K_1 #^Z K_2 is defined in such a way that when K_1 and K_2 are Gorenstein, and Z is a suitable subset of the intersection of K_1 and K_2, then the Stanley-Reisner ring of the connected sum is Gorenstein, by the work of Ananthnarayan-Avramov-Moore. These algebraic computations can be interpreted in terms of the equivariant cohomology of moment angle complexes and we also describe the symplectic cut of a toric orbifold in terms of moment angle complexes.