Title   name
The 5th Korea Toric Topology Winter Workshop
 
  Title
  Speaker   Park, Hanchul    
  Date 2019-01-22
  Host
  Place Gyeongju, K- Hotel
  File  의 1 번째 Real Media 동영상입니다.
 
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and Trevisan in their unpublished paper, and the result was strengthened for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.