Connected sums of simplicial complexes and equivariant cohomology

TL;DR

This paper explores the algebraic and geometric properties of connected sums of simplicial complexes, linking them to equivariant cohomology and symplectic cuts in toric orbifolds, and introduces a strong connected sum concept.

Contribution

It defines the strong connected sum of simplicial complexes and shows how it preserves Gorenstein properties in Stanley-Reisner rings, connecting algebraic and geometric perspectives.

Findings

Stanley-Reisner ring of connected sum is a sum along the intersection ring.

Strong connected sum preserves Gorenstein property under certain conditions.

Algebraic constructions correspond to geometric symplectic cuts in toric orbifolds.

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.