On the non-existence of torus actions (Q2630465)

Quasitoric manifolds \(M(\lambda)\) are certain manifolds with an action of a half-dimensional torus \(T\) which generalize smooth projective toric varieties. The orbit space of a quasitoric manifold is a simple convex polytope. Let \(E\) be a complex orientable theory. A quasitoric manifold \(M(\lambda)\) is called nice if the higher right derived functors \(R^iP(\cdot)\) of the primitive element functor vanish on \(E_*(M(\lambda))\) for all \(i>1\). The main result of the paper under review is as follows: A quasitoric manifold \(M(\lambda)\) is nice if and only if \(r_i\cap r_j=\empty\) for all relations \(r_i,r_j\) with \(i\neq j\) in the \(E\)-face ring, that is in the ring \[ E^*(\mathrm{pt})[v_1,\dots,v_m]/(v_{i_1}\cdot\dots\cdot v_{i_k}; F_{i_1}\cap\dots\cap F_{i_k}=\emptyset). \] Here the \(v_i\) have degree two and \(F_1,\dots,F_m\) are the facets of the orbit polytope of \(M(\lambda)\). This answers a question of M. Bendersky.