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 Félix-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.
orbit configuration space small cover quasi-toric manifold (real) moment-angle manifold Euler characteristic homotopy type
55R80 57S25 52B20 55P91 55N91 14M25
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The first author and the second author were supported by National Natural Science Foundation of China (Grant Nos. 11371093, 11431009 and 11661131004). The third author was supported by National Natural Science Foundation of China (Grant No. 11028104).
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