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**Graph equivariant cohomological rigidity for GKM graphs.**
*(English)*
Zbl 1442.55006

Goresky-Kottwitz-MacPherson (GKM) manifolds are torus manifolds with only finitely many fixed points and satisfying some other technical conditions. The class of GKM manifolds includes toric manifolds. Every GKM manifold \(X\) determines some combinatorial data. In particular, one can associate to \(X\) a graph \({\mathcal G}_X\), called the GKM graph of \(X\), that encodes the structure of the equivariant \(1\)-skeleton of \(X\) and the weights of the tangential real representations.

In [Asian J. Math. 3, No. 1, 49–76 (1999; Zbl 0971.58001)], V. Guillemin and C. Zara introduced the concept of an abstract GKM graph \({\mathcal{G}}\) and its graph equivariant cohomology \(H^\ast_T({\mathcal{G}})\), which is a graded algebra over the integral cohomology \(H^\ast(BT)\). Here \(BT\) denotes the classifying space of the torus \(T\). Let \({\mathcal{G}}\) and \({\mathcal{G}}'\) be two abstract GKM graphs defined for the same torus \(T\). In this paper, the authors introduce the notion of an isomorphism \({\mathcal{G}}'\to {\mathcal{G}}\). The main theorem states that \(H^\ast_T({\mathcal{G}})\) and \(H^\ast_T({\mathcal{G}}')\) are isomorphic as \(H^\ast(BT)\)-algebras if and only if \({\mathcal{G}}\) and \({\mathcal{G}}'\) are isomorphic as GKM graphs.

In [Asian J. Math. 3, No. 1, 49–76 (1999; Zbl 0971.58001)], V. Guillemin and C. Zara introduced the concept of an abstract GKM graph \({\mathcal{G}}\) and its graph equivariant cohomology \(H^\ast_T({\mathcal{G}})\), which is a graded algebra over the integral cohomology \(H^\ast(BT)\). Here \(BT\) denotes the classifying space of the torus \(T\). Let \({\mathcal{G}}\) and \({\mathcal{G}}'\) be two abstract GKM graphs defined for the same torus \(T\). In this paper, the authors introduce the notion of an isomorphism \({\mathcal{G}}'\to {\mathcal{G}}\). The main theorem states that \(H^\ast_T({\mathcal{G}})\) and \(H^\ast_T({\mathcal{G}}')\) are isomorphic as \(H^\ast(BT)\)-algebras if and only if \({\mathcal{G}}\) and \({\mathcal{G}}'\) are isomorphic as GKM graphs.

Reviewer: Marja Kankaanrinta (Helsinki)

### MSC:

55N91 | Equivariant homology and cohomology in algebraic topology |

57S15 | Compact Lie groups of differentiable transformations |

### Citations:

Zbl 0971.58001
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\textit{M. Franz} and \textit{H. Yamanaka}, Proc. Japan Acad., Ser. A 95, No. 10, 107--110 (2019; Zbl 1442.55006)

### References:

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[4] | V. Guillemin and C. Zara, Equivariant de Rham theory and graphs, Asian J. Math. 3 (1999), no. 1, 49-76. · Zbl 0971.58001 |

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