Princeton-Rider Workshop on Homotopy Theory and Toric Spaces (sponsored by Algebraic Topology Seminar)

2/24/2012

Suyoung Choi
Ajou University, Korea

Toric rigidity of simple polytopes and moment-angle manifolds

A 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.