A homotopical version of the cohomological rigidity problem

Xin Fu, Beijing Institute of Mathematical Sciences and Applications

Please note the start time for this online talk.

A toric manifold is a compact, smooth toric variety that arises from a complete and regular fan. Its cohomology is the Stanley-Reisner ring of the fan quotient by the ideal generated by certain linear terms. The cohomological rigidity problem, a longstanding challenge in toric topology, asks whether two toric manifolds are homeomorphic/diffeomorphic if their integral cohomology rings are isomorphic. It is known that cohomological rigidity holds for a few special cases of toric manifolds, such as Bott manifolds and 6-dimensional toric manifolds associated with Pogorelov polytopes.

 On the other hand, toric orbifolds are the orbifold version of toric manifolds, and they behave differently from toric manifolds. Their cohomology rings remain largely unknown, and counterexamples disproving their cohomology rigidity have been found.

 In this talk, I will report on recent progress in investigating the integral cohomology rings of certain toric orbifolds and the cohomological rigidity with respect to homotopy equivalence based on joint work with Tseleung So, Jongbaek Song and Stephen Theriault.