# Homotopy Theory and Toric Spaces

# Homotopy Theory and Toric Spaces

Suyoung Choi of Ajou University, Korea - Toric rigidity of simple polytopes and moment-angle manifoldsA simple convex polytope $P$ is \emph{cohomologically rigid} if its combinatorial structure is determined by the cohomology ring of a quasitoric manifold over $P$, and is \emph{combinatorially rigid} if its combinatorial structure is determined by its graded Betti numbers, which are important invariants coming from combinatorial commutative algebra. Not every $P$ has these properties, but some important polytopes such as simplices or cubes are known to be cohomologically and combinatorially rigid. In general, it is known that if $P$ is combinatorially rigid and it supports a quasitoric manifold, then $P$ is cohomologically rigid. In this talk, we survey results on toric rigidity of polytopes, and we provide two simple polytopes of dimension 3 having the discuss about the identical bigraded Betti numbers but non-isomorphic Tor-algebras. Furthermore, they turn out to be the first examples which are cohomologically rigid and not combinatorially. Moreover, one can see that the moment-angle manifolds arising from these two polytopes are homotopically different. Before this example, as far as I know, in all known examples of combinatorially different polytopes with same bigraded Betti numbers (such as vertex truncations of simplices), the moment-angle manifolds are diffeomorphic.

Fred Cohen of University of Rochester - Polyhedral products: Decompositions, monodromy, and cohomology

The purpose of this talk is to develop new, further decompositions of the polyhedral product functor, to apply these constructions to address natural monodromy representations, and to present one (possibly curious) example in cohomology. These new decompositions are joint work of T. Bahri, M. Bendersky, S. Gitler, and the speaker. The possibly curious related computation arises in joint work with F. Callegaro, and M. Salvetti.