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Let and be finite subgroups of . The double quotients of the form were introduced in materials science under the name misorientation spaces. In this paper we review several known results that allow one to study the topology of misorientation spaces. Neglecting the orbifold structure, one can say that all misorientation spaces are closed orientable topological -manifolds with finite fundamental groups. In the case when and are crystallographic groups, we compute the fundamental groups and apply the elliptization theorem to describe these spaces. Many misorientation spaces are homeomorphic to by Perelman's theorem. However, we explicitly describe the topological types of several misorientation spaces without appealing to Perelman's theorem. The classification of misorientation spaces yields new -valued group structures on the manifolds and . Finally, we outline the connection of the particular misorientation space with integrable dynamical systems and toric topology. [ABSTRACT FROM AUTHOR]

We study the actions of tori (standard compact tori as well as their quaternionic analogs) on products of spheres. We prove that the orbit space of a specific action of a torus on a product of spheres is homeomorphic to a sphere. A similar statement for the real torus was proved by the second author in 2019. We also extend this result to arbitrary compact topological groups, thus generalizing the results mentioned above as well as the results of the first author on the actions of a compact torus of complexity . [ABSTRACT FROM AUTHOR]

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Given a partial action of a topological group G on a space X we determine properties P which can be extended from X to its globalization. We treat the cases when P is any of the following: Hausdorff, regular, metrizable, second countable, and having invariant metric. Further, for a normal subgroup H, we introduce and study a partial action of G/H on the orbit space of X; applications to invariant metrics and inverse limits are presented.

In this paper, the dynamics (including shadowing property, expansiveness, topological stability and entropy) of several types of upper semi-continuous set-valued maps are mainly considered from differentiable dynamical systems points of view. It is shown that (1) if f is a hyperbolic endomor-phism then for each ε> 0 there exists a C1-neighborhood U of f such that the induced set-valued map F f , U has the ε-shadowing property, and moreover, if f is an expanding endomorphism then there exists a C1-neighborhood U of f such that the induced set-valued map F f , U has the Lipschitz shadowing property; (2) when a set-valued map F is generated by finite expanding endomorphisms, it has the shadowing property, and moreover, if the collection of the generators has no coincidence point then F is expansive and hence is topologically stable; (3) if f is an expanding endomorphism then for each ε> 0 there exists a C1-neighborhood U of f such that h (F f , U , ε) = h (f) (4) when F is generated by finite expanding endomorphisms with no coincidence point, the entropy formula of F is given. Furthermore, the dynamics of the set-valued maps based on discontinuous maps on the interval are also considered. [ABSTRACT FROM AUTHOR]

We discuss regularity statements for equidistant decompositions of Riemannian manifolds and for the corresponding quotient spaces. We show that any stratum of the quotient space has curvature locally bounded from both sides. [ABSTRACT FROM AUTHOR]

In this paper, we study the approximation of continuous functions on a subclass of singular space—the subcartesian space. As is well known, the orbit space of the proper action of a Lie group on a smooth manifold is a subcartesian space. We prove that continuous functions on the orbit space can be approximated by smooth functions. [ABSTRACT FROM AUTHOR]

Given a partial action of a topological group G on a topological space X we determine properties P which can be extended from X to its globalization. We treat the cases when P is any of the following: Hausdorff, regular, metrizable, second countable and having invariant metric. Further, for a normal subgroup H, we introduce and study a partial action of G/H on the orbit space X/~-H, applications to invariant metrics and inverse limits are presented. [ABSTRACT FROM AUTHOR]

A multidimensional projective space with a marked point (center) is considered. On the manifold of projective frames of the given space that are adapted to the center, the action of the stabilizer of the center of the group of projective transformations is introduced. We prove that linear frames, i.e., bases of the tangent vector space of the projective space at its center, can be identified with orbits of the adapted projective frames with respect to the action of the kernel of the epimorphism of Lie groups, which assigns to each transformation from the stabilizer its differential at the center. Using a multidimensional generalization of the Desargues theorem, we obtain a criterion for two adapted projective frames to belong to the same orbit. [ABSTRACT FROM AUTHOR]

We prove that for any isometric action of a group on a unit sphere of dimension larger than 1, the quotient space has diameter zero or larger than a universal dimension-independent positive constant. [ABSTRACT FROM AUTHOR]

The set of covariance matrices equipped with the Bures-Wasserstein distance is the orbit space of the smooth, proper, and isometric action of the orthogonal group on the Euclidean space of square matrices. This construction induces a natural orbit stratification on covariance matrices, which is exactly the stratification by the rank. Thus, the strata are the manifolds of symmetric positive semidefinite matrices of fixed rank endowed with the Bures-Wasserstein Riemannian metric. In this work, we study the geodesies of the Bures-Wasserstein distance. First, we complete the literature on geodesies in each stratum by clarifying the set of preimages of the exponential map and by specifying the injectivity domain. We also give explicit formulae of the horizontal lift, the exponential map, and the Riemannian logarithms that were kept implicit in previous works. Second, we give the expression of all the minimizing geodesic segments joining two covariance matrices of any rank. More precisely, we show that the set of all minimizing geodesies between two covariance matrices Σ and Δ is parametrized by the closed unit ball of ... for the spectral norm, where k, l,r are the respective ranks of Σ, Δ, Σ,Δ. In particular, the minimizing geodesic is unique if and only if r = min(k,l). Otherwise, there are infinitely many. As a secondary contribution, we provide a review of the definitions related to geodesies in metric spaces, affine connection manifolds, and Riemannian manifolds, which is helpful for the study of other spaces. [ABSTRACT FROM AUTHOR]

Let X be a finitistic space with the mod 2 cohomology of the product space of a projective space and a 4-sphere. Assume that X admits a free involution. In this paper we study the mod 2 cohomology algebra of the quotient of X by the action of the free involution and derive some consequences regarding the existence of Z2-equivariant maps between such X and an n-sphere. [ABSTRACT FROM AUTHOR]

Introduction Let G be a Lie group with a differentiable action on a differentiable manifold M. Consider the orbit space M/G with the quotient topology. Dimension of M/G is called the cohomogeneity of the action of G on M. Study of the orbit spaces has many important applications in invariant function theory and G-invariant variational problems associated to M. Many G-invariant objects associated to M can be related to similar objects associated to the orbit space. Therefore, if the dimension of the orbit space is small enough (the cohomgeneity of the action of G on M is small) we can effectively reduce many problems about G-invariant objects of M to generally easier problems on M/G . Because of this motivation, many mathematicians studied topological properties of the orbit spaces of Lie group actions on manifolds. A pioneer theorem in this area is a theorem proved by P. Mostert in 1957: If M is a differentiable manifold which is of cohomogeneity one under the action of G a compact connected Lie group, then the orbit space is homeomorphic to one of the spaces [0,1], (0,1], S¹ or R. This theorem has been generalized to noncompact Lie groups with proper actions on manifolds. Moreover, If M is endowed with a Riemannian metric, and G is a closed and connected subgroup of the isometries of M, which acts by cohomogeneity one on M, there are more interesting results about the orbit space and orbits. It is proved that if M is a Riemannian manifold of negative curvature and G is a connected and closed subgroup of the isometries of M, acting on M with cohomogeneity one, then the orbit space is not homeomorphic to [0,1], so by (generalized) Mostert's theorem, it would be homeomorphic to [0,1) or S¹ or R, and if in addition, M is simply connected then the orbit space is homeomorphic to [0,1) or R. This result, has been generalized to flat Riemannian manifolds, and recently it is proved for Riemannian manifolds of non-positive curvature. To extend Mostert's theorem, it is natural to ask, what may be the orbit space, when M is a cohomogeneity two G-manifold. There is no classification for orbit spaces of cohomogeneity two G-manifolds in general. Cohomogeneity two actions of compact Lie groups on Euclidean spaces are polar an all such actions are classified. It is clear in this case that the orbit space is homeomorphic to plane or half-plane. Also, It is proved that if G is a connected (compact or non-compact) group of the isometries which acts by cohomogeneity two on Rn, then the orbit space is homeomorphic to plane or half-plane. Classification of orbit spaces of cohomogeneity two actions on the standard sphere Sn has been described in a series of papers . Also, the orbit space of cohomogeneity two actions on simply connected spaces of constant curvature (hyperbolic spaces) has been classified before. To extend this classification to all Riemannian manifolds of constant positive curvature, we prove the following theorem: Theorem. If M is a coghomogeneity two Riemannian G-manifold of constant negative curvature, then the orbit space M/G is homeomorphic to one of the following spaces: R × S¹, R², R× [0, ∞). [ABSTRACT FROM AUTHOR]

The complexity of biological systems, and the increasingly large amount of associated experimental data, necessitates that we develop mathematical models to further our understanding of these systems. Because biological systems are generally not well understood, most mathematical models of these systems are based on experimental data, resulting in a seemingly heterogeneous collection of models that ostensibly represent the same system. To understand the system we therefore need to understand how the different models are related to each other, with a view to obtaining a unified mathematical description. This goal is complicated by the fact that a number of distinct mathematical formalisms may be employed to represent the same system, making direct comparison of the models very difficult. A methodology for comparing mathematical models based on their underlying conceptual structure is therefore required. In previous work we developed an appropriate framework for model comparison where we represent models, specifically the conceptual structure of the models, as labelled simplicial complexes and compare them with the two general methodologies of comparison by distance and comparison by equivalence. In this article we continue the development of our model comparison methodology in two directions. First, we present a rigorous and automatable methodology for the core process of comparison by equivalence, namely determining the vertices in a simplicial representation, corresponding to model components, that are conceptually related and the identification of these vertices via simplicial operations. Our methodology is based on considerations of vertex symmetry in the simplicial representation, for which we develop the required mathematical theory of group actions on simplicial complexes. This methodology greatly simplifies and expedites the process of determining model equivalence. Second, we provide an alternative mathematical framework for our model-comparison methodology by representing models as groups, which allows for the direct application of group-theoretic techniques within our model-comparison methodology. [ABSTRACT FROM AUTHOR]

Let G be a group acting properly and by isometries on a metric space X; it follows that the quotient or orbit space X/G is also a metric space. We study the Vietoris–Rips and Čech complexes of X/G. Whereas (co) homology theories for metric spaces let the scale parameter of a Vietoris–Rips or Čech complex go to zero, and whereas geometric group theory requires the scale parameter to be sufficiently large, we instead consider intermediate scale parameters (neither tending to zero nor to infinity). As a particular case, we study the Vietoris–Rips and Čech thickenings of projective spaces at the first scale parameter where the homotopy type changes. [ABSTRACT FROM AUTHOR]

The orbit space of a distributive binary -space is studied. A number of its properties in the case of a compact binarily acting group are established. [ABSTRACT FROM AUTHOR]

We study the orbit space of the standard action of the compact torus on the complex Grassmann manifold . We describe the structure of the set of critical points of the generalized moment map whose image is a hypersimplex . The canonical projection maps the set to the set , which by definition consists of the orbits with nontrivial stabilizer subgroup in , where is the diagonal one-dimensional torus. Introducing the notion of a singular point in terms of the parameter spaces of the orbits, we prove that the set is an open manifold and is dense in . We show that for , but . Our central result is the construction of a projection , , where is a universal parameter space. Earlier, we have proved that is a closed smooth manifold diffeomorphic to a known manifold . We show that the map is a diffeomorphism, and describe the structure of the sets for . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of free involutions on some Wall manifolds and we compute the mod 2 cohomology algebra of the correspondent orbit space. [ABSTRACT FROM AUTHOR]

The current paper continues consideration of geometry of projective frame orbits started in the author’s article in the previous issue. The n-dimensional 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.

In this paper, we show that the statement in the study of Srivastava (1987) holds also for the Hewitt realcompactification. The mentioned statement showed that when the action of a finite topological group on a Tychonoff space is given, the Stone-Čech compactification of the orbit space of the action is the orbit space of the Stone-Čech compactification of the space. As an application, we show that Srivastava's result can be obtained using the main theorem of the present study. [ABSTRACT FROM AUTHOR]

We study isometric actions of compact Lie groups on complete orientable positively curved n -manifolds whose orbit spaces have non-empty boundary in the sense of Alexandrov geometry. In particular, we classify quotients of the unit sphere by actions of compact simple Lie groups with non-empty boundary. We deduce from this the list of representations of compact simple Lie groups that admit non-trivial reductions. As a tool of special interest, we introduce a new geometric invariant of a compact symmetric space, namely, the minimal number of points in a "spanning set" of the space. [ABSTRACT FROM AUTHOR]

Let be a differentiable action of a Lie group on a differentiable manifold and consider the orbit space with the quotient topology. Dimension of is called the cohomogeneity of the action of on . If is a differentiable manifold of cohomogeneity one under the action of a compact and connected Lie group, then the orbit space is homeomorphic to one of the spaces , , or . In this paper we suppose that the hyperbolic space is of cohomogeneity two under the action of , a connected and closed subgroup of Then we prove that its orbit space is homeomorphic to or Also we prove that either all orbits are diffeomorphic to or there are nonnegative integers such that some orbits are diffeomorphic to , and the other orbits are diffeomorphic to , where may be a sphere, a homogeneous hypersurface of sphere or a helix in some Euclidean space. ../files/site1/files/0Abstract6.pdf

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.

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 G be discrete group and F be a collection of subgroups of G. We show that there exists 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 H-orbits for all H in F . This gives a model categorical criterion for maps that induce weak equivalences on H-orbits to be weak equivalences in the F -model structure. [ABSTRACT FROM AUTHOR]

We consider a group G acting on a local dendrite X (in particular on a graph). We give a full characterization of minimal sets of G by showing that any minimal set M of G (whenever X is different from a dendrite) is either a finite orbit, or a Cantor set, or a circle. This result extends that of the authors for group actions on dendrites. On the other hand, we show that, for any group G acting on a local dendrite X different from a circle, the following properties are equivalent: (1) (G, X) is pointwise almost periodic. (2) The orbit closure relation R = { (x , y) ∈ X × X : y ∈ G (x) ¯ } is closed. (3) Every non-endpoint of X is periodic. In addition, if G is countable and X is a local dendrite, then (G, X) is pointwise periodic if and only if the orbit space X / G is Hausdorff. [ABSTRACT FROM AUTHOR]

For a Lie group G , we prove the existence of a universal G -space in the class of all paracompact (respectively, metrizable, and separable metrizable) free proper G -spaces which have a paracompact orbit space. Our approach is based on Milnor's construction E G. We show that E G and the universal free G -spaces in question are G -AE's. Besides, we show how to extend these results to the case of arbitrary metrizable, locally compact, almost connected group actions. [ABSTRACT FROM AUTHOR]

Let X be a locally compact and noncompact G--space with a compact group G. In this paper, we give some useful description of a compactification of the orbit space X/G when it is an orbit space of a G--compactification of X. As an application, we show that the closed bounded interval [a; b] is homeomorphic to the space of maximal ideals with Stone topology of uniformly continuous even functions subring of C*(R) . [ABSTRACT FROM AUTHOR]

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 give a topological classification of the orbit space of cohomogeneity two isometric actions on at Riemannian manifolds. [ABSTRACT FROM AUTHOR]

Let G be a group acting freely, properly discontinuously and cellularly on some finite dimensional C W-complex Σ(2 n) which has the homotopy type of the 2 n -sphere 𝕊2 n . Then, that action induces a homomorphism G → Aut(H 2 n (Σ(2 n))). We classify all pairs (G , φ), where G is a virtually cyclic group and φ: G → Aut(ℤ) is a homomorphism, which are realizable in the way above and the homotopy types of all possible orbit spaces as well. Next, we consider the family of all groups which have virtual cohomological dimension one and which act on some Σ(2 n). Those groups consist of free groups and semi-direct products F ⋊ ℤ2 with F a free group. For a group G from the family above and a homomorphism φ: G → Aut(ℤ), we present an algebraic criterion equivalent to the realizability of the pair (G , φ). It turns out that any realizable pair can be realized on some Σ(2 n) with dim Σ(2 n) ≤ 2 n + 1. [ABSTRACT FROM AUTHOR]