Homology, Homotopy and Applications

Volume 21 (2019)

Number 1

Local face rings and diffeomorphisms of quasitoric manifolds

Pages: 303 – 322

DOI: https://dx.doi.org/10.4310/HHA.2019.v21.n1.a14


David Allen (Department of Mathematics, City University of New York, N.Y., U.S.A.)

José La Luz (Departmento de Matemáticas, Universidad de Puerto Rico, Bayamón, Puerto Rico, U.S.A.)


In this paper we apply presheaves to develop an invariant that can distinguish diffeomorphism classes of quasitoric manifolds in the category of quasitoric pairs $\mathcal{Q}$. 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 $\mathcal{Q}$ and then we develop local versions of the Stanley–Reisner ring and the left higher derived functors of the indecomposable functor. We prove that diffeomorphisms between certain objects do not lift to equivalence in the category $\mathcal{Q}$. The main application is geared toward the quasitoric manifolds $\#_4 \mathbb{C}P^3$ with orbit spaces that come from double vertex truncations of the prism that has appeared in the work of Masuda, Panov, and their collaborators.


quasitoric manifold, toric topology, higher homotopy group, homotopy type, moment angle complex, Davis–Januszkiewicz space, diffeomorphism, rigidity

2010 Mathematics Subject Classification

Primary 14M25. Secondary 57N65.

Received 10 May 2018

Received revised 11 July 2018

Accepted 2 September 2018

Published 31 October 2018