Connected sums of simplicial complexes and equivariant cohomology

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.