## Osaka Journal of Mathematics

### 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.

#### Article information

Source
Osaka J. Math., Volume 51, Number 2 (2014), 405-425.

Dates
First available in Project Euclid: 8 April 2014

https://projecteuclid.org/euclid.ojm/1396966255

Mathematical Reviews number (MathSciNet)
MR3192548

Zentralblatt MATH identifier
1318.55007

#### Citation

Matsumura, Tomoo; Moore, W. Frank. Connected sums of simplicial complexes and equivariant cohomology. Osaka J. Math. 51 (2014), no. 2, 405--425. https://projecteuclid.org/euclid.ojm/1396966255

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• \begingroup S. Luo, T. Matsumura and W.F. Moore: Moment angle complexes and big Cohen–Macaulayness, arXiv:1205.1566. \endgroup