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 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 Félix-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 algebraic topology of such an orbit configuration space by using the Mayer-Vietoris spectral sequence.

Almost-quasifibrations and Fundamental Groups of Orbit Configuration Spaces

“…Remark 1.5. The first part of Corollary 1.4 supports the Asphericity conjecture of [11], which says that 2 , and G is acting on S as a group of deck transformations (hence free and properly discontinuous action), so that S is the universal cover of S/G , then π 1 (O n (S, G), * n ) has an iterated semi-direct product of free groups structure. This was achieved using the fact that the fibration p has a section, in this particular case.…”

“…See [16] and [8] for some works and relevant literatures on orbit configuration spaces. Also see [2] for some non-free action case. However, most of these studies are related to (co)homological computations or to know the homotopy type of the orbit configuration space.…”

Almost-quasifibrations and fundamental groups of orbit configuration spaces

“…See [13] and [6] for some works and relevant literatures. Also see [1] for some non-free action case. However, most of these studies are related to (co)homological computations or to know the homotopy type of the configuration spaces.…”

Connecting Homomorphism and Separating Cycles

“…More abstract point of view on the generalization of the Mayer-Vietoris sequence for homology of the union relates to consideration of two spectral sequences of the double complex C = (C p,q ; δ, ∂) (see [2,3]). This double complex is a first quarter complex (C p,q = 0 for p < 0 or q < 0).…”

“…All the necessary information about the homology of the union of open subspaces (the homology of the topological space with open cover U) is discussed in Section 1. The case of the union of more than two subspaces leads to the study of the double complex of a cover and the Mayer-Vietoris spectral sequence (see [3]).…”