Connected Sums of Simplicial Complexes and Equivariant Cohomology

Abstract In this paper, we introduce the notion of a connected sum K1 #Z K2 of simplicial complexes K1 and K2, 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 K1 #Z K2 is the connected sum of the Stanley–Reisner rings of K1 and K2 along the Stanley–Reisner ring of K1 \ K2. The strong connected sum K1 #Z K2 is defined in such a way that when K1, K2 are Gorenstein, and Z is a suitable subset of K1\ K2, then the Stanley–Reisner ring of K1 #Z K2 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.