Orbit configuration spaces of small covers and quasi-toric manifolds

Details Orbit configuration spaces of small covers and quasi-toric manifolds Orbit configuration spaces of small covers and quasi-toric manifolds

2011-11-29

arxiv.org/abs/1111.6699v1

Mathematics

Algebraic Topology

30 pages with 6 figures

Scientific paper

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 are all non-free and noncompact, but still built via simple convex polytopes. We obtain an explicit formula of 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 Euler characteristic for the classical configuration space, which generalizes the F\'elix-Thomas formula. In addition, we also study the homotopy type of such orbit configuration spaces. In particular, we determine an equivariant strong deformation retract of the orbit configuration space of 2 distinct orbit-points in a small cover or a quasi-toric manifold, which turns out that we are able to further study the Betti numbers and (equivariant) cohomology of such an orbit configuration space.

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