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In this paper, we study differential forms and vector fields on the orbit space of a proper action of a Lie group on a smooth manifold, defining them as multilinear maps on the generators of infinitesimal diffeomorphisms, respectively. This yields an intrinsic view of vector fields and differential forms on the orbit space.

Let R be an open equivalence relation on a topological space E. We define on E a new equivalence relation ̃ℜ̅ by x̃ ̃ℜ̅y if the closure of the R-trajectory of x is equal to the closure of the R-trajectory of y. The quotient space E/̃ ̃ℜ̅ is called the trajectory class space. In this paper, we show that the space E/̃ ̃ℜ̅ is a simple model of the quotient space E/R. This model can provide a finite model. Some applications to orbit spaces of groups of homeomorphisms and leaf spaces are given.

For a given group $G$ and a collection of subgroups $\mathcal F$ of $G$, we show that there exist a left induced model structure on the category of right $G$-simplicial sets, in which the weak equivalences and cofibrations are the maps that induce weak equivalences and cofibrations on the $H$-orbits for all $H$ in $\mathcal F$. This gives a model categorical criterion for maps that induce weak equivalences on $H$-orbits to be weak equivalences in the $\mathcal F$-model structure., Comment: 9 pages

We show that any equivariant fibrant extension of a compact free G-space is also free. This result allows us to prove that the orbit space of any equivariant fibrant compact space E is also fibrant, provided that E has only one orbit type.

International audience; We give a definition of the Delaunay triangulation of a point set in a closed Euclidean d-manifold, i.e. a compact quotient space of the Euclidean space for a discrete group of isometries (a so-called Bieberbach group or crystallographic group). We describe a geometric criterion to check whether a partition of the manifold actually forms a triangulation (which subsumes that it is a simplicial complex). We provide an incremental algorithm to compute the Delaunay triangulation of the manifold defined by a given set of input points, if it exists. Otherwise, the algorithm returns the Delaunay triangulation of a finite-sheeted covering space of the manifold. The algorithm has optimal randomized worst-case time and space complexity. It extends to closed Euclidean orbifolds. An implementation for the special case of the 3D flat torus has been released in Cgal 3.5. To the best of our knowledge, this is the first general result on this topic.

Let G be a locally compact Hausdorff group. We study orbit spaces of equivariant absolute neighborhood extensors (G-ANE's) in the category of all proper G-spaces that are metrizable by a G-invariant metric. We prove that if a proper G-space X is a G-ANE (respectively, a G- ANE ( n ) , n ⩾ 0 ), and H a closed normal subgroup of G such that all the H-orbits in X are metrizable, then the H-orbit space X / H is a G / H -ANE (respectively, a G / H - ANE ( n ) ). Other related results are also established.

Let G be a locally compact Hausdorff group. We study orbit spaces and unions of equivariant absolute neighborhood extensors ( G - ANE s) in the category of all proper G -spaces that are metrizable by a G -invariant metric. We prove that if a proper G -space X is a G - ANE such that all the G -orbits in X are metrizable, then the G -orbit space X / G is an ANE . Equivariant versions of Hanner's theorem and Kodama's theorem about unions of absolute neighborhood extensors are established. We also introduce the notion of a G -polyhedron and prove that if G is any compact group, then every G - ANR is arbitrary closely dominated by a G -polyhedron. Each G -polyhedron is a G - ANE .

Extensorial properties of orbit spaces of locally compact proper group actions are investigated.

In classical covering space theory, a covering map induces an injection of fundamental groups. This paper reveals a dual property for certain quotient maps having connected fibers, with applications to orbit spaces of vector fields and leaf spaces in general., Comment: 12 pages, 4 figures; added references, keywords, and Remark 1.2; accepted at Topology and its Applications

Given an orbit space $M/\Gamma$ and an equivalence relation defined in it by means of the action of a group $G$, we obtain a miniversal deformation of an orbit through a miniversal deformation in $M$ with regard to a suitable group action of $G\times \Gamma$. We show some applications to the perturbations of $m$-tuples of subspaces and $(C,A)$-invariant subspaces.

Given a controllable system defined by a pair of matrices (A,B), we investigate the geometry of the set of controllability subspaces. This set is a subset of the set of (A,B)-invariant subspaces. We prove that, in fact, it is a stratified submanifold and we compute its dimension.

It is proved that: (1) every Lie group G can act properly (in sense of Palais) on each infinite-dimensional Hilbert space l2(τ) of a given weight τ such that (G,l2(τ)) becomes a universal G-space for all metrizable proper G-spaces admitting an invariant metric and having weight ⩽τ; (2) every Lie group G can act properly on Rτ⧹{0} such that (G,Rτ⧹{0}) becomes a universal G-space for all Tychonoff proper G-spaces of weight ⩽τ; (3) there is a dispersive dynamical system on l2, universal for all separable, metrizable, dispersive dynamical systems having a regular orbit space. Other universal proper G-spaces are constructed. As a corollary a shorter proof of Palais' invariant metric existence theorem is obtained. The metric cones con(G/H), with H⊂G a compact subgroup, are the main building blocs in our approach.

We find explicit miniversal deformations of flags in the generalized flag manifolds, with regard a natural equivalence relation defined by the group action that keeps fixed the reference flag.

In this paper, for G a locally compact group (or a Lie group), we study the relationship between the covering dimensions of a proper G-space X and its orbit space X / G . We prove also that dim X = Ind X for every proper G-space X with a metrizable orbit space provided that G is either pro-Lie or σ-compact or has a metrizable quotient group of connected components.

We prove that if G is a compact Lie group, Y a G -space equipped with a topological local convex structure compatible with the action of G , then Y is a G - ANE for metrizable G -spaces. If, in addition, Y has a G -fixed point and admits a global convex structure compatible with the action of G , then Y is a G - AE . This is applied to show that certain hyperspaces related to the Banach–Mazur compacta are equivariant absolute extensors.

For a compact Lie group G, we prove the existence of a universal G-space in the class of all paracompact (respectively, metrizable, and separable metrizable) free G-spaces. We show that such a universal free G-space cannot be compact. © 2009 Elsevier B.V.