Algebraic Topology and Transformation Groups

Geometric topology has seen significant advances in the understanding and application of infinite symmetries and of the principles behind them. On the one hand, for advances in (geometric) group theory, tools from algebraic topology are applied and extended; on the other hand, spectacular results in topology (e.g., the proofs of new cases of the Novikov conjecture or the Atiyah conjecture) were only possible through a combination of methods of homotopy theory and new insights in the geometry of groups. This workshop focused on the rich interplay between algebraic topology and geometric group theory.

1. In view of the recent work on topological groups it is natural to consider the situation which arises when such groups act as transformation groups on various types of spaces. Such a study is begun here from the point of view of almost periodic transformation groups, the definition of which is suggested by von Neumann's paper on almost periodic functions in a group.t Compact topological transformation groups are a special case of almost periodic transformation groups, at least for a rather wide class of spaces. The paper concerns itself chiefly with the nature of the minimal closed invariant sets of such groups. There are some results for general spaces but the main results are for Euclidean spaces and more particularly for threedimensional Euclidean spaces. One of the most interesting theorems states that if a compact one-dimensional group acts on three-space in such a way that its orbits are uniformly bounded in diameter, then every point of the space is fixed under the group, so that if such a group is to act in a nontrivial manner the diameters of its orbits must be unbounded. Under some restrictions a similar theorem is proved for one-parameter almost periodic groups. Furthermore it is shown that for this latter class of groups, many of the orbits must actually be simple closed curves if they have one-dimensional closures. 2. The group considered here will be denoted by G. It will be subjected to various conditions as the occasion demands but it will always be Abelian. In case it is the group of real numbers, it will be spoken of as a one-parameter group; in case it is the real numbers reduced modulo one, it will be spoken of as the circle group. The space on which the group acts will be denoted by R. It will be specialized in various ways, but in any case it will always be a locally compact metric space. If x and y are two points of R, the distance between them will be denoted by d(x, y).

Totally Lindelöf and totally ω-narrow paratopological groups

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

This paper is dedicated to the memory of Maurice Auslander, whose love of mathematics will always be an inspiration

Introduction. This article is based upon a principle which is so standard that it is almost a cliche: The first step to understanding a nonlinear phenomenon is to define and study a suitable linear approximation. To be more specific, we shall describe some applications of this to the symmetry questions of topological transformation groups. Given a topological space X, let Homeo(^) denote the set of self-homeomorphisms of X. This is a group under composition of mappings. If G is an arbitrary group, then a group action of G on I is a homomorphism Homeo( X). Frequently we wish to impose some weak assumptions on <p. For example, if G is a topological group, then we might want cp to have suitable continuity properties. The usual assumption is that the map

Contents: Introduction. §1: Computable rings and modules. §2: Ideal membership relation. §3: Effective structured domains. §4: Completion of a local ring as a domain. §5: The recursive completion. Epilogue. References.Introduction. Completion is an important general mathematical device. Often, but not always, a completion takes the following form. Let A be a topological algebraic structure whose topology is derived from a metric. For A, a topological algebra  and an embedding i: A →  are constructed such that  is a complete metric space in which A is densely embedded by i. The long list of structures for which such completions exist begins with Cantor's construction of the real number field and includes objects like the p-adic integers, Baire space, and Boolean algebras. In Bourbaki [6] a careful and thorough account of completions for arbitrary topological groups and fields is given, for which it is important to note that the topological structures need not be metrizable, but must possess a uniformity.The effectiveness of the completion process of a computable structure A cannot be readily studied using the tools of computable algebra, simply because the resulting structure  is almost invariably uncountable. However, in particular cases, it has been possible to define and study the substructure Ak of computable elements of Â; this has been done for the structures mentioned above, starting with the field of recursive real numbers.In this paper we analyse the effectivity of the completion of a local ring R. We do this using structured Scott-Ershov domains. Our study may be considered as a prototype containing methods applicable to a broad class of completions, including all the examples mentioned above, except for the real number field, which needs a generalisation of the domain concept.A Scott-Ershov domain D formalises how a set Dt of possibly “infinite” elements, called total elements, is constructed from a set Dc of “finite” elements, called compact elements. This is achieved by means of an approximation ordering which determines a topology on D and, in particular, on Dt. Our methodology is to associate to a given topological algebra A a structured domain D(A) such that the total elements D(A)t form a topological algebra topologically isomorphic to A. In such circumstances A is said to be domain definable by D(A). The theory of computability for domains is now applied to study the effectivity of the topological algebra A.

Let X be a uniform space, let T be a multiplicative topological group, and let T act as a transformation group on X. A subset A of T is said to be (left) syndetic provided that T= AK for some compact subset K of T. T he transformation group T is said to be almost periodic on X provided that if a is an index of X, then there exists a syndetic subset A of T such that xA Cxa for all xEX. If xGX, then the transformation group T is said to be locally almost periodic at x provided that if U is a neighborhood of x, then there exist a neighborhood V of x and a syndetic subset A of T such that VA CU. The transformation group T is said to be locally almost periodic on X in case T is locally almost periodic at every point of X. If xEX, then the transformation group T is said to be locally weakly almost periodic at x provided that if U is a neighborhood of x, then there exist a neighborhood V of x, a syndetic subset A of T, and a compact subset C of T such that y E V implies the existence of a subset B of T for which A CBC and yBC U. It is readily proved that if xcX, then T is locally weakly almost periodic at x if and only if for each neighborhood U of x there exist a neighborhood V of x and a compact subset K of T such that VTC UK. If xEX, then the transformation group T is said to be equicontinuous at x provided that if a is an index of X, then there exists an index e of X such that xftCxta for all tC T. The transformation group T is said to be equicontinuous on X in case T is equicontinuous at every point of X. The transformation group T is said to be uniformly equicontinuous on X provided that if a is an index of X, then there exists an index ,B of X such that xl3tCxta for all xEX and all tcT. It is readily verified that if X is compact, then T is uniformly equicontinuous if and only if T is equicontinuous. The transformation group T is said to be distal on X provided that if x, yEX with x5-zy, then there exists an index a of X such that (xt, yt) Ea for all t E T. We also consider T to be a transformation group acting on X X X in the following manner: if x, yEX and if tET, then (x, y)t is defined to be (xt, yt).

Introduction to topological groups

Introduction. In this paper, we investigate the proximal relation and related notions in coset transformation groups.In §1, we show that many of the interesting properties of the proximal relation can be characterized in terms of a group subset that arises naturally from the generating subgroup.A general criterion for distal in terms of subnormality of the generating subgroup is shown in §2.Proofs are given for [9, Remark (2.1) and Theorem (2.2)], establishing that coset transformation groups of nilpotent groups are distal.§3 is concerned with distal in relation to certain coset transformation groups that arise from solvable Lie groups.Finally, we establish in §4 some results concerning the existence of subgroups which yield either point-transitive or minimal distal actions on the phase space.These results are strengthened when applied to certain discrete flows and examined in Lie groups.Proofs for the rest of the results of [9] are given in this section.Let G be a topological group and H a subgroup of G. Consider the space H\G={Hg | ge G} of right cosets of G.The coset transformation group of G induced by H is defined to be the point-to-point transitive transformation group (H\G, G) with action (Hfi g) -> Hfg.The fact that (H\G, G) is indeed a transformation group under this action follows from [6, Definition 1.39].If H is syndetic in G, H\G is compact uniformizable, whence H\G has a unique compatible uniformity.Thus, the proximal relation of (H\G, G) is well defined and will be denoted by P(H\G).Throughout this paper, G will denote a topological group with identity e and 77 will denote a syndetic subgroup.If K and L are subgroups of G, we denote that K is normal in L by K <] L. If K is a subgroup of G, then K* will denote the closure of K.The additive groups of integers and reals will be denoted by Z and R respectively.The neighborhood filter of e will be denoted by r¡e.For other definitions, see [9].Finally, the author would like to thank Professor W. Gottschalk for his stimulating comments concerning the results in this paper.2. The proximal relation.When the phase space is not necessarily uniform, we can still introduce a notion of proximal.

In the setting of homotopy type theory, each type can be interpreted as a space. Moreover, given an element of a type, i.e. a point in the corresponding space, one can define another type which encodes the space of loops based at this point. In particular, when the type we started with is a groupoid, this loop space is always a group. Conversely, to every group we can associate a type (more precisely, a pointed connected groupoid) whose loop space is this group: this operation is called delooping. The generic procedures for constructing such deloopings of groups (based on torsors, or on descriptions of Eilenberg-MacLane spaces as higher inductive types) are unfortunately equipped with elimination principles which do not directly allow eliminating to untruncated types, and are thus difficult to work with in practice. Here, we construct deloopings of the cyclic groups Zm which are cellular, and thus do not suffer from this shortcoming. In order to do so, we provide type-theoretic implementations of lens spaces, which constitute an important family of spaces in algebraic topology. Our definition is based on the computation of an iterative join of suitable maps from the circle to an arbitrary delooping of Zm. In some sense, this work generalizes the construction of real projective spaces by Buchholtz and Rijke, which handles the case m = 2, although the general setting requires more involved tools. Finally, we use this construction to also provide cellular descriptions of dihedral groups, and explain how we can hope to use those to compute the cohomology and higher actions of such groups.

In ?1 of this paper it is shown that a variety of conditions implying nice behavior for topological transformation groups are, in the presence of separability, equivalent. In ?2 the continuity properties of the stability subgroups are studied. The conditions of ?1 exclude the line acting on the torus in such a way that each orbit is dense. They exclude the integers acting on the circle by rotation through multiples of an irrational angle and they exclude the group of those sequences of zeros and ones which have all but a finite number of their terms equal to zero when this group acts on the space of all sequences of zeros and ones by coordinatewise addition (mod 2). As we shall see in the proof of Theorem 1, the latter transformation group is a prototype for all excluded transformation groups. This is analogous to the following fact in the theory of Rings of Operators: Every factor of type II, contains a hyperfinite factor of type II,. The conditions were suggested by [3, Theorem 1] and the proof of their equivalence is somewhat analogous to the proof of [3, Theorem 1]. However, the proof does not depend upon [3] nor upon the theory of C*-algebras.

Let K[?] denote the group algebra of an infinite locally finite group?. In recent years, the lattice of ideals of K[?] has been extensively studied under the assumption that? is simple. From these many results, it appears that such group algebras tend to have very few ideals. While some work still remains to be done in the simple group case, we nevertheless move on to the next stage of this program by considering certain abelian-by-(quasi-simple) groups. Standard arguments reduce this problem to that of characterizing the ideals of an abelian group algebra K[V] stable under the action of an appropriate automorphism group of V. Specifically, in this paper, we let B be a quasi-simple group of Lie type defined over an infinite locally finite field F, and we let V be a finite-dimensional vector space over a field E of the same characteristic p. If B acts nontrivially on V by way of the homomorphism Φ: G → GL(V), and if V has no proper G-stable subgroups, then we show that the augmentation ideal ωK[V] is the unique proper G-stable ideal of K[V] when char K ¬= p. The proof of this result requires, among other things, that we study characteristic p division rings D, certain multiplicative subgroups G of D*, and the action of G on the group algebra K[A], where A is the additive group D + . In particular, properties of the quasi-simple group G come into play only in the final section of this paper.

The cohomology of groups

This article delves into algebraic topology, specifically (co)homology theory, which is essential in various mathematical fields. It explores different types of (co)homology groups such as Hochschild, cyclic, reflexive, and dihedral, focusing on dihedral cohomology generated by the dihedral group operating on simplicial complexes. The text discusses when dihedral and reflexive cohomology vanishes and provides examples, along with proving a homomorphism between dihedral cohomology groups. Mathematical concepts like Banach spaces, C*-algebras, and tensor products are elucidated to understand the algebraic structures involved. Theorems and proofs establish the relationships and properties of these cohomology groups, culminating in a canonical isomorphism theorem for the product of two algebras. Overall, the article aims to provide a comprehensive exploration of dihedral cohomology and its interplay with other algebraic structures, offering insights into its applications and theoretical foundations. Additionally, it studies the (co)homology theory of C*-algebras, focusing on the triviality of cohomology groups of operator algebras and discovering the canonical isomorphism between any two unital K -algebras A and A′: HDn(A×A′)≅HDn(A)⊕HDn(A′).