Orbit configuration spaces of small covers and quasi-toric manifolds

Abstract

In this article, we investigate the orbit configuration spaces of some equivariant closed manifolds over simple convex polytopes in toric topology, such as small covers, quasi-toric manifolds and (real) moment-angle manifolds; especially for the cases of small covers and quasi-toric manifolds. These kinds of orbit configuration spaces have non-free group actions, and they are all noncompact, but still built via simple convex polytopes. We obtain an explicit formula of the Euler characteristic for orbit configuration spaces of small covers and quasi-toric manifolds in terms of the h-vector of a simple convex polytope. As a by-product of our method, we also obtain a formula of the Euler characteristic for the classical configuration space, which generalizes the Felix-Thomas formula. In addition, we also study the homotopy type of such orbit configuration spaces. In particular, we determine an equivariant strong deformation retraction of the orbit configuration space of 2 distinct orbit-points in a small cover or a quasi-toric manifold, which allows to further study the algebraic topology of such an orbit configuration space by using the Mayer-Vietoris spectral sequence.