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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.

The current paper continues consideration of geometry of projective frame orbits started in the author’s article in the previous issue. The ndimensional projective space with a distinguished point (the center) is considered. The action of matrix affine group of order n on the adapted projective frame manifold is given. It is shown that the linear frames, i. e., bases of the tangent space, can be identified with the orbits of adapted projective frames under the action of some normal subgroup of this group. Two adapted frames are said to be equivalent if they belong to the same orbit. The strict perspectivity relation between two adapted frames is introduced. The proofs of the theorem on the Desargues hyperplane and of the criterion of equivalence are simplified. According to this criterion, two adapted frames in strict perspective are equivalent if and only if the Desargues hyperplane generated by these frames is passing through the center.

Let G be a subgroup of the group Homeo(X) of homeomorphisms of a topological space X. Let G ¯ $\bar G$ be the closure of G in Homeo(X). The class of an orbit O of G is the union of all orbits having the same closure as O. We denote by X / G ˜ $X/\widetildeG$ the space of classes of orbits called the orbit class space. In this paper, we study the fundamental group of the spaces X/G, X / G ¯ $X/\bar G$ and X / G ˜ $X/\widetildeG$

We consider the orbit projection pE:E → E/G of a G-space E with only one orbit type. We show that pE admits a G-ANR-resolution consisting of G-fibrations if G is a compact metrizable group.

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.

Let G be a second-countable locally-compact Hausdorff groupoid with a Haar system, and let {x_n} be a sequence in the unit space of G. We show that the notions of strength of convergence of {x_n} in the orbit space and measure-theoretic accumulation along the orbits are equivalent ways of realising multiplicity numbers associated to a sequence of induced representation of the groupoid C*-algebra., Comment: 34 pages

Let ( G , X ) be a second-countable transformation group with G acting freely on X . It is shown that measure-theoretic accumulation of the action and topological strength of convergence in the orbit space X / G provide equivalent ways of quantifying the extent of nonproperness of the action. These notions are linked via the representation theory of the transformation-group C ∗ -algebra C 0 ( X ) ⋊ G .

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.

Given ap-blockbof a finite groupG, we show that theG-poset of Brauer pairs strictly containing (1,b) has contractibleG-orbit space. A similar result is proved for certainG-posets ofp-subgroups. Both results generalise P. Symonds' verification of a conjecture of P. Webb. © 1999 Academic Press.

Using the formalism defined by Lauterbach and Roberts (1992 J. Diff. Eqns. 100 22 - 48), we develop a geometric approach for the problem of forced symmetry breaking for periodic orbits in G-equivariant systems of ODEs. We show that this problem can be studied as the perturbation of the identity mapping on the double coset space LG/K where K is the maximal subgroup of G acting on the periodic orbit and L the symmetry of the perturbation. We exhibit some examples where this kind of symmetry breaking allows us to show the existence of heteroclinic cycles between periodic solutions.

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

We define an order structure on a nonseparatedn-manifold. Here, a nonseparated manifold denotes any topological space that is locally Euclidean and has a countable basis; the usual Hausdorff separation property is not required. Our result is that an ordered nonseparatedn-manifoldXcan be realized as an ordered orbit space of a completely unstable continuous flowϕon a Hausdorff(n+1)-manifoldE.