CONNECTED SUMS OF SIMPLICIAL COMPLEXES AND EQUIVARIANT COHOMOLOGY

Abstract

In this paper, we introduce the notion of a connected sum K1 #(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 boolean AND K-2. The strong connected sum #(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 boolean AND 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.