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Cohomology of the graphical real toric manifold
[The 5th Korea Toric Topology Winter Workshop]
Date: 2019-01-22
Speaker : Park, Hanchul (Jeju National University)
Abstract : The topology of the real toric manifold MR has been less known than that of its complex counterpart. In 1985, Jurkiewicz gave the formula for the Z2-cohomology ring of MR. Its Q-Betti numbers were calculated by Suciu and Trevisan in their unpublished paper, and the result was strengthened for coecient ring R in which 2 is a unit by Suyoung Choi and the speaker. Recently, the cup product for H(MR;R) is computed by Choi and the speaker. For any simple graph G, the graph associahedron 4G and the graph cubeahedron G are simple polytopes which support natural projective toric manifolds XG and YG respectively. For connected G, they are obtained by cutting faces of the simplex and the cube respectively. In this talk, we describe the structure of H(XR G; F) and H(Y R G ; F) in terms of the graph G, where F is a eld with characteristic other than 2. Note that their Q-Betti numbers can be calculated using the graph invariants known as the a-number and b-number of G respectively. A weird thing about XR G and Y R G is that it seems that a slight generalization of them would make the computation of the cohomology too dicult. For example, nestohedra or simple generalized permutohedra de ne real toric manifolds, but in general their cohomology rings are very dicult to compute. It is suspected that H(XR G; F) determines G, and in particular, the family of XR G is Q-cohomologically rigid.
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