Connected sums of simplicial complexes and equivariant cohomology

Abstract

In this paper, we introduce the notion of a connected sum $K_{1} \#^{Z} K_{2}$ of simplicial complexes $K_{1}$ and $K_{2}$, as well as define 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 [1]. 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 $K_{1} \cap K_{2}$. The strong connected sum $K_{1} \#^{Z} K_{2}$ is defined in such a way that when $K_{1}$, $K_{2}$ are Gorenstein, and $Z$ is a suitable subset of $K_{1} \cap K_{2}$, then the Stanley--Reisner ring of $K_{1} \#^{Z} K_{2}$ is Gorenstein, by work appearing in [1]. We also show that cutting a simple polytope by a generic hyperplane produces strong connected sums. These algebraic computations can be interpreted in terms of the equivariant cohomology of moment angle complexes and toric orbifolds.