Borsuk–Ulam theorem for filtered spaces

Let X and Y be pathwise connected and paracompact Hausdorff spaces equipped with free involutions T : X → X {T:X\to X} and S : Y → Y {S:Y\to Y} , respectively. Suppose that there exists a sequence ( X i , T i ) ⁢ ⟶ h i ⁢ ( X i + 1 , T i + 1 ) for ⁢ 1 ≤ i ≤ k , (X_{i},T_{i})\overset{h_{i}}{\longrightarrow}(X_{i+1},T_{i+1})\quad\text{for }% 1\leq i\leq k, where, for each i, X i {X_{i}} is a pathwise connected and paracompact Hausdorff space equipped with a free involution T i {T_{i}} , such that X k + 1 = X {X_{k+1}=X} , and h i : X i → X i + 1 {h_{i}:X_{i}\to X_{i+1}} is an equivariant map, for all 1 ≤ i ≤ k {1\leq i\leq k} . To achieve Borsuk–Ulam-type theorems, in several results that appear in the literature, the involved spaces X in the statements are assumed to be cohomological n-acyclic spaces. In this paper, by considering a more wide class of topological spaces X (which are not necessarily cohomological n-acyclic spaces), we prove that there is no equivariant map f : ( X , T ) → ( Y , S ) {f:(X,T)\to(Y,S)} and we present some interesting examples to illustrate our results.

Some Chaotic Properties of G- Average Shadowing Property

Let be a metric space and be a continuous map. The notion of the -average shadowing property ( ASP ) for a continuous map on –space is introduced and the relation between the ASP and average shadowing property(ASP)is investigated. We show that if has ASP, then has ASP for every . We prove that if a map be pseudo-equivariant with dense set of periodic points and has the ASP, then is weakly mixing. We also show that if is a expansive pseudo-equivariant homeomorphism that has the ASP and is topologically mixing, then has a -specification. We obtained that the identity map on has the ASP if and only if the orbit space of is totally disconnected. Finally, we show that if is a pseudo-equivariant map, and the trajectory map is a covering map, then has the ASP if and only if the induced map has ASP.

General Properties of Equivariant Cohomology

This chapter assesses the general properties of equivariant cohomology. Both the homotopy quotient and equivariant cohomology are functorial constructions. Equivariant cohomology is particularly simple when the action is free. Throughout the chapter, by a G-space, it means a left G-space. Let G be a topological group and consider the category of G-spaces and G-maps. A morphism of left G-spaces is a G-equivariant map (or G-map). Such a morphism induces a map of homotopy quotients. The map in turn induces a ring homomorphism in cohomology. The chapter then looks at the coefficient ring of equivariant cohomology, as well as the equivariant cohomology of a disjoint union.

Augmented, free and tensor generalized digroups

The concept of generalized digroup was proposed by Salazar-Díaz, Velásquez and Wills-Toro in their paper “Generalized digroups” as a non trivial extension of groups. In this way, many concepts and results given in the category of groups can be extended in a natural form to the category of generalized digroups. The aim of this paper is to present the construction of the free generalized digroup and study its properties. Although this construction is vastly different from the one given for the case of groups, we will use this concept, the classical construction for groups and the semidirect product to construct the tensor generalized digroup as well as the semidirect product of generalized digroups. Additionally, we give a new structural result for generalized digroups using compatible actions of groups and an equivariant map from a group set to the group corresponding to notions of associative dialgebras and augmented racks.

Garden of Eden and specification

We establish a Garden of Eden theorem for expansive algebraic actions of amenable groups with the weak specification property, i.e. for any continuous equivariant map $T$ from the underlying space to itself, $T$ is pre-injective if and only if it is surjective. In particular, this applies to all expansive principal algebraic actions of amenable groups and expansive algebraic actions of $\mathbb{Z}^{d}$ with completely positive entropy.

Equivariant Map Queer Lie Superalgebras

An equivariant map queer Lie superalgebra is the Lie superalgebra of regular maps from an algebraic variety (or scheme) X to a queer Lie superalgebra q that are equivariant with respect to the action of a finite group Γ acting on X and q. In this paper, we classify all irreducible finite-dimensional representations of the equivariant map queer Lie superalgebras under the assumption that Γ is abelian and acts freely on X. We show that such representations are parameterized by a certain set of Γ-equivariant finitely supported maps from X to the set of isomorphism classes of irreducible finite-dimensional representations of q. In the special case where X is the torus, we obtain a classification of the irreducible finite-dimensional representations of the twisted loop queer superalgebra.

Rigidity of maximal holomorphic representations of Kähler groups

We investigate representations of Kähler groups [Formula: see text] to a semisimple non-compact Hermitian Lie group [Formula: see text] that are deformable to a representation admitting an (anti)-holomorphic equivariant map. Such representations obey a Milnor–Wood inequality similar to those found by Burger–Iozzi and Koziarz–Maubon. Thanks to the study of the case of equality in Royden’s version of the Ahlfors–Schwarz lemma, we can completely describe the case of maximal holomorphic representations. If [Formula: see text], these appear if and only if [Formula: see text] is a ball quotient, and essentially reduce to the diagonal embedding [Formula: see text]. If [Formula: see text] is a Riemann surface, most representations are deformable to a holomorphic one. In that case, we give a complete classification of the maximal holomorphic representations, which thus appear as preferred elements of the respective maximal connected components.