Osaka Journal of Mathematics

Connected sums of simplicial complexes and equivariant cohomology

Tomoo Matsumura and W. Frank Moore

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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

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1396966255

Mathematical Reviews number (MathSciNet)
MR3192548

Zentralblatt MATH identifier
1318.55007

Subjects
Primary: 55N91: Equivariant homology and cohomology [See also 19L47]
Secondary: 14M25: Toric varieties, Newton polyhedra [See also 52B20] 57R18: Topology and geometry of orbifolds 16S37: Quadratic and Koszul algebras 53D99: None of the above, but in this section

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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References

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