Certain generalized higher derived functors associated to quasitoric manifolds

Abstract

In this paper, we show that the higher derived functors of the primitive element functor of $$E_*(M)$$ where M is a quasitoric manifold and E is a complex-orientable homology theory are independent of the torus action, implying they depend on the orbit space. This extends the results of an earlier work by both authors. By applying cosimplicial methods to coalgebras that are free modules over a commutative ring, we are able to generalize the aforementioned functors that appeared in Bousfield’s work. In addition, certain results that appeared in earlier work of Larry Smith have been generalized and applied to faithful systems to expose how and to what extent ESP sequences show up in the arguments. As an application, we are able to prove that a necessary condition for a simplicial complex dual to a simple convex polytope being rigid is an isomorphism of these derived functors in sufficiently large dimensions; this further exposes the relation between the homotopy type of the manifold, the torus action and the combinatorics of the orbit.