Conway’s Question: The Chase for Completeness

We study various degrees of completeness for a Tychonoff space X. One of them plays a central role, namely X is called a Conway space if X is sequentially closed in its Stone–?ech compactification β X (a prominent example of Conway spaces is provided by Dieudonné complete spaces). The Conway spaces constitute a bireflective subcategory Conw of the category Tych of Tychonoff spaces. Replacing sequential closure by the general notion of a closure operator C, we introduce analogously the subcategory Conw _C of C-Conway spaces, that turns out to be again a bireflective subcategory of Tych. We show that every bireflective subcategory of Tych can be presented in this way by building a Galois connection between bireflective subcategories of Tych and closure operators of Top finer than the Kuratowski closure. Other levels of completeness are considered for the (underlying topological spaces of) topological groups. A topological group G is sequentially complete if it is sequentially closed in its Ra?kov completion ${ \ifmmode\expandafter\tilde\else\expandafter\~\fi{G}}$ . The sequential completeness for topological groups is stronger than Conway’s property, although they coincide in some classes of topological groups, for example: free (Abelian) topological groups, pseudocompact groups, etc.     

On the computational complexity of the theory of Abelian groups

We develop a series of Ehrenfeucht games and prove the following results:

• 1.(i) The first order theory of the divisible and indecomposable p-group, the first order theory of the group of rational numbers with denominators prime to p and the first order theory of a cyclic group of prime power order can be decided in 2^{2cn log n} Turing time.
• 2.(ii) The first order theory of the direct sum of countably many infinite cyclic groups, the first order theory of finite Abelian groups and the first order theory of all Abelian groups can be decided in 2^2dn Turing space.

     

Automatic continuity and representation of group homomorphisms defined between groups of continuous functions

Let C(X,G) be the group of continuous functions from a topological space X into a topological group G with pointwise multiplication as the composition law, endowed with the uniform convergence topology. To what extent does the group structure of C(X,G) determine the topology of X? More generally, when does the existence of a group homomorphism H between the groups C(X,G) and C(Y,G) implies that there is a continuous map h of Y into X such that H is canonically represented by h? We prove that, for any topological group G and compact spaces X and Y, every non-vanishing C-isomorphism (defined below) H of C(X,G) into C(Y,G) is automatically continuous and can be canonically represented by a continuous map h of Y into X. Some applications to specific groups and examples are given in the paper.     

The recognition of the class of indecomposable digraphs under low hemimorphy

Given a digraph G=(V,A), the subdigraph of G induced by a subset X of V is denoted by G[X]. With each digraph G=(V,A) is associated its dual G^?=(V,A^?) defined as follows: for any x,y∈V, (x,y)∈A^? if (y,x)∈A. Two digraphs G and H are hemimorphic if G is isomorphic to H or to H^?. Given k>0, the digraphs G=(V,A) and H=(V,B) are k-hemimorphic if for every X⊆V, with |X|≤k, G[X] and H[X] are hemimorphic. A class C of digraphs is k-recognizable if every digraph k-hemimorphic to a digraph of C belongs to C. In another vein, given a digraph G=(V,A), a subset X of V is an interval of G provided that for a,b∈X and x∈V−X, (a,x)∈A if and only if (b,x)∈A, and similarly for (x,a) and (x,b). For example, 0?, {x}, where x∈V, and V are intervals called trivial. A digraph is indecomposable if all its intervals are trivial. We characterize the indecomposable digraphs which are 3-hemimorphic to a non-indecomposable digraph. It follows that the class of indecomposable digraphs is 4-recognizable.     

Torsion and Abelianization in Equivariant Cohomology

Let X be a topological space upon which a compact connected Lie group G acts. It is well known that the equivariant cohomology H ^* _G (X; Q) is isomorphic to the subalgebra of Weyl group invariants of the equivariant cohomology H ^* _T (X; Q), where T is a maximal torus of G. This relationship breaks down for coefficient rings k other than Q. Instead, we prove that under a mild condition on k the algebra H ^* _G (X; k) is isomorphic to the subalgebra of H ^* _T (X; k) annihilated by the divided difference operators.     

Group-valued continuous functions with the topology of pointwise convergence

Let G be a topological group with the identity element e. Given a space X, we denote by Cp(X,G) the group of all continuous functions from X to G endowed with the topology of pointwise convergence, and we say that X is: (a) G-regular if, for each closed set F⊆X and every point x∈X?F, there exist f∈Cp(X,G) and g∈G?{e} such that f(x)=g and f(F)⊆{e}; (b) G^?-regular provided that there exists g∈G?{e} such that, for each closed set F⊆X and every point x∈X?F, one can find f∈Cp(X,G) with f(x)=g and f(F)⊆{e}. Spaces X and Y are G-equivalent provided that the topological groups Cp(X,G) and Cp(Y,G) are topologically isomorphic.We investigate which topological properties are preserved by G-equivalence, with a special emphasis being placed on characterizing topological properties of X in terms of those of Cp(X,G). Since R-equivalence coincides with l-equivalence, this line of research “includes” major topics of the classical Cp-theory of Arhangel'ski? as a particular case (when G=R).We introduce a new class of TAP groups that contains all groups having no small subgroups (NSS groups). We prove that: (i) for a given NSS group G, a G-regular space X is pseudocompact if and only if Cp(X,G) is TAP, and (ii) for a metrizable NSS group G, a G^?-regular space X is compact if and only if Cp(X,G) is a TAP group of countable tightness. In particular, a Tychonoff space X is pseudocompact (compact) if and only if Cp(X,R) is a TAP group (of countable tightness). Demonstrating the limits of the result in (i), we give an example of a precompact TAP group G and a G-regular countably compact space X such that Cp(X,G) is not TAP.We show that Tychonoff spaces X and Y are T-equivalent if and only if their free precompact Abelian groups are topologically isomorphic, where T stays for the quotient group R/Z. As a corollary, we obtain that T-equivalence implies G-equivalence for every Abelian precompact group G. We establish that T-equivalence preserves the following topological properties: compactness, pseudocompactness, σ-compactness, the property of being a Lindelöf Σ-space, the property of being a compact metrizable space, the (finite) number of connected components, connectedness, total disconnectedness. An example of R-equivalent (that is, l-equivalent) spaces that are not T-equivalent is constructed.     

On the continuity of left translations in the LUC-compactification

Consider the continuity of left translations in the LUC-compactification GLUC of a locally compact group G. For every X⊆G, let κ(X) be the minimal cardinality of a compact covering of X in G. Let U(G) be the points in GLUC that are not in the closure of any X⊆G with κ(X)<κ(G). We show that the points at which no left translation in U(G) is continuous are dense in U(G). This result is a generalization of a theorem by van Douwen concerning discrete groups. We obtain a new proof for the fact that the topological center of GLUC?G is empty.     

Distance-hereditary graphs and signpost systems

An ordered pair (U,R) is called a signpost system if U is a finite nonempty set, R⊆U×U×U, and the following axioms hold for all u,v,w∈U: (1) if (u,v,w)∈R, then (v,u,u)∈R; (2) if (u,v,w)∈R, then (v,u,w)∉R; (3) if u≠v, then there exists t∈U such that (u,t,v)∈R. (If F is a (finite) connected graph with vertex set U and distance function d, then U together with the set of all ordered triples (u,v,w) of vertices in F such that d(u,v)=1 and d(v,w)=d(u,w)−1 is an example of a signpost system). If (U,R) is a signpost system and G is a graph, then G is called the underlying graph of (U,R) if V(G)=U and xy∈E(G) if and only if (x,y,y)∈R (for all x,y∈U). It is possible to say that a signpost system shows a way how to travel in its underlying graph. The following result is proved: Let (U,R) be a signpost system and let G denote the underlying graph of (U,R). Then G is connected and every induced path in G is a geodesic in G if and only if (U,R) satisfies axioms (4)-(8) stated in this paper; note that axioms (4)-(8)-similarly as axioms (1)-(3)-can be formulated in the language of the first-order logic.     

Extending a function on a graph

Let G be a graph with vertex set V, and let h be a function mapping a subset U of V into the real numbers R. If ? is a function from V to R, we define δ (?) to be the sum of ∥?(b)? ?(a)∥ over all edges {a, b} of G. A best extension of h is such a function ? with ?(x) = h(x) for X ∈ U and minimum δ (?). We show that such a best extension exists and derive an algorithm for obtaining such an extension. We also show that if instead we minimise the sum of (?(b)??(a))², there is generally a unique best extension, obtainable by solving a system of linear equations.     

Joint spectrum and the infinite dihedral group

A subgroup H of a group G is called pronormal if, for any element g ∈ G, the subgroups H and H ^g are conjugate in the subgroup <H,H ^g >. We prove that, if a group G has a normal abelian subgroup V and a subgroup H such that G = HV, then H is pronormal in G if and only if U = N _U (H)[H,U] for any H-invariant subgroup U of V. Using this fact, we prove that the simple symplectic group PSp_6n (q) with q ≡ ±3 (mod 8) contains a nonpronormal subgroup of odd index. Hence, we disprove the conjecture on the pronormality of subgroups of odd indices in finite simple groups, which was formulated in 2012 by E.P. Vdovin and D.O. Revin and verified by the authors in 2015 for many families of simple finite groups.     

Every 2-group with all subgroups normal-by-finite is locally finite

A group G has all of its subgroups normal-by-finite if H/H _G is finite for all subgroups H of G. The Tarski-groups provide examples of p-groups (p a “large” prime) of nonlocally finite groups in which every subgroup is normal-by-finite. The aim of this paper is to prove that a 2-group with every subgroup normal-by-finite is locally finite. We also prove that if |H/H _G | 6 2 for every subgroup H of G, then G contains an Abelian subgroup of index at most 8.     

Some inverse problems involving conditional expectations

Let (Ω, F, P) be a probability space, let H be a sub-σ-algebra of F, and let Y be positive and H-measurable with E[Y] = 1. We discuss the structure of the convex set CE(Y; H) = {X ∈ pF: Y = E[X|H]} of random variables whose conditional expectation given H is the prescribed Y. Several characterizations of extreme points of CE(Y; H) are obtained. A necessary and sufficient condition is given in order that CE(Y; H) be the closed, convex hull of its extreme points. For the case of finite F we explicitly calculate the extreme points of CE(Y; H), identify pairs of adjacent extreme points, and characterize extreme points of CE(Y; H) ? CE(Z; G), where G is a second sub-σ-algebra of F and Z ∈ pG. When H = σ(Y) and appropriate topological hypotheses hold, extreme points of CE(Y; H) are shown to be in explicit one-to-one correspondence with certain left inverses of Y. Finally, it is shown how the same approach can be applied to the problem of extremal random measures on $\text{R}$₊ with a prescribed compensator, to deduce that the number of extreme points is zero or one.     

Homomorphisms and composition operators on algebras of analytic functions of bounded type

Let U and V be convex and balanced open subsets of the Banach spaces X and Y, respectively. In this paper we study the following question: given two Fréchet algebras of holomorphic functions of bounded type on U and V, respectively, that are algebra isomorphic, can we deduce that X and Y (or X^* and Y^*) are isomorphic? We prove that if X^* or Y^* has the approximation property and H_wu(U) and H_wu(V) are topologically algebra isomorphic, then X^* and Y^* are isomorphic (the converse being true when U and V are the whole space). We get analogous results for H_b(U) and H_b(V), giving conditions under which an algebra isomorphism between H_b(X) and H_b(Y) is equivalent to an isomorphism between X^* and Y^*. We also obtain characterizations of different algebra homomorphisms as composition operators, study the structure of the spectrum of the algebras under consideration and show the existence of homomorphisms on H_b(X) with pathological behaviors.     

On the reconstruction problem for factorizable homeomorphism groups and foliated manifolds

For a group G of homeomorphisms of a regular topological space X and a subset U⊆X, set . We say that G is a factorizable group of homeomorphisms, if for every open cover U of X, _?U∈UG generates G.

Theorem I. Let G, H be factorizable groups of homeomorphisms of X and Y respectively, and suppose that G, H do not have fixed points. Let φ be an isomorphism between G and H. Then there is a homeomorphism τ between X and Y such thatφ(g)=τ○g○τ−1for everyg∈G.     

Bounded sets in topological groups and embeddings

We show that the existence of a non-metrizable compact subspace of a topological group G often implies that G contains an uncountable supersequence (a copy of the one-point compactification of an uncountable discrete space). The existence of uncountable supersequences in a topological group has a strong impact on bounded subsets of the group. For example, if a topological group G contains an uncountable supersequence and K is a closed bounded subset of G which does not contain uncountable supersequences, then any subset A of K is bounded in G?(K?A). We also show that every precompact Abelian topological group H can be embedded as a closed subgroup into a precompact Abelian topological group G such that H is bounded in G and all bounded subsets of the quotient group G/H are finite. This complements Ursul's result on closed embeddings of precompact groups to pseudocompact groups.     

Essential subgroups of topological groups

We study the class Wof Hausdorff topological groups Gfor which the following two cardinal invariants coincide

ES(G)=min{|H|:H≤ Gdense and essential}

TD(G)=min{|H|:H≤ Gtotally dense}

We prove that W contains the following classes:locally compact abelian groups, compact connected groups, countably compact totally discon¬nected abelian groups, topologically simple groups, locally compact Abelian groups when endowed with their Bohr topology, totally minimal abelian groups and free Abelian topological groups. For all these classes we are also able to giv ean explicit computation of the common value of ESand TD.     

The Fibre of a Pinch Map in a Model Category

In the category of pointed topological spaces, let F be the homotopy fibre of the pinching map X?∪?CA?→?X?∪?CA?/?X from the mapping cone on a cofibration A?→?X onto the suspension of A. Gray (Proc Lond Math Soc (3) 26:497–520, 1973) proved that F is weakly homotopy equivalent to the reduced product (X, A)_∞. In this paper we prove an analogue of this phenomenon in a model category, under suitable conditions including a cube axiom.     

Group distance magic and antimagic graphs

Given a graph G with n vertices and an Abelian group A of order n, an A-distance antimagic labelling of G is a bijection from V (G) to A such that the vertices of G have pairwise distinct weights, where the weight of a vertex is the sum (under the operation of A) of the labels assigned to its neighbours. An A-distance magic labelling of G is a bijection from V (G) to A such that the weights of all vertices of G are equal to the same element of A. In this paper we study these new labellings under a general setting with a focus on product graphs. We prove among other things several general results on group antimagic or magic labellings for Cartesian, direct and strong products of graphs. As applications we obtain several families of graphs admitting group distance antimagic or magic labellings with respect to elementary Abelian groups, cyclic groups or direct products of such groups.