LOCAL FACE RINGS AND DIFFEOMORPHISIMS OF QUASITORIC MANIFOLDS

In this paper we apply presheaves to develop an invariant that can distinguish diffeomorphism classes of quasitoric manifolds in the category of quasitoric pairs Q. The objects in this category are pointed topological spaces (M, p) where M is a quasitoric manifold and p is a fixed point under the torus action. Maps between pairs are continuous, base-point preserving with respect to a certain topology that depends on the submanifolds of M. It is shown that the category of quasitoric manifolds is a subcategory of Q and then we develop local versions of the Stanley-Reisner ring and the left higher derived functors of the indecomposable functor. We prove that diffeoukorphisms between certain objects do not lift to equivalence in the category Q. The main application is geared toward the quasitoric manifolds #4CP3 with orbit spaces that come from double vertex truncations of the prism that has appeared in the work of Masuda, Panov and their collaborators.