







The 5th Korea Toric Topology Winter Workshop 








Title 


Date 
20190122 


Host 



Place 
Gyeongju, K Hotel 




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
Z2cohomology ring of MR. Its QBetti 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and Trevisan in their unpublished paper, and the result was strengthened for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 QBetti
numbers can be calculated using the graph invariants known as the anumber
and bnumber 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 dene 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 Qcohomologically rigid. 





