Volume 16, Issue 2

The aim of this paper is to study the focal points of 4-transnormal tori in R 4 . For 4-transnormal embeddings of tori which are not usual product embeddings, different behavior is deduced.

We consider dimensional type functions defined on the class of π -bicompact spaces. Using these functions, the dimensional properties of generalized manifolds are studied. The class of weakly net-like spaces is introduced, which is a natural generalization of net-like spaces in the sense of V. V. Proizvolov and the behavior of this class is studied in the case of (monotone, open) mappings of special type. Generalizations of V. V. Proizvolov's theorems are obtained for weakly net-like spaces.

The concepts of coshapes of topological spaces and continuous maps are given. Applications of the obtained results include constructions of long exact sequences of homology and homotopy inj-groups and cohomology pro-groups of spaces and maps.

Let Y be an integral projective curve. For any torsion free sheaf F on Y set I ( F ) ≔ { L ∈ Pic 0 ( Y ) : F ⊗ L ≅ F }. Here we study pairs ( Y , F ) such that the abelian group I ( F ) is very large.

In this paper we define the concepts of g -Λ δ -continuous maps, g -Λ δ -irresolute maps and g -V δ -closed maps by using generalized Λ δ -sets and generalized V δ -sets. Also we introduce a new class of topological spaces called T V δ -spaces.

We describe algorithms for computing the induced nilpotent orbits in semisimple Lie algebras. We use them to obtain the induction tables for the Lie algebras of exceptional type. This also yields the classification of the rigid nilpotent orbits in those Lie algebras.

The sum theorem is proved and its corollaries are given for a countable family of zero-dimensional (in the sense of small and large inductive bidimensions) p -closed sets, using a new bitopological notion of relative normality whose topological correspondence is also new. The notion of almost n -dimensionality is considered from the bitopological point of view.

We study the conformal curvature tensor and the contact conformal curvature tensor in Sasakian and/or K -contact manifolds. We find a necessary and sufficient condition for a Sasakian manifold to be φ -conformally flat. We also find some necessary conditions for a K -contact manifold to be φ -contact conformally flat. Then we give a structure theorem for φ -contact conformally flat Sasakian manifolds. It is also proved that a Sasakian manifold cannot be ξ -contact conformally flat.

We consider a particular finite dimensional example of an L ∞ algebra in which a 2-dimensional Lie algebra acts on a 1-dimensional vector space in a non-trivial non-Lie manner. In order to understand the nature of this action, we show that this algebra is in fact an example of an open-closed homotopy algebra.

It is proved, using inverse systems, that if X is a non-metric confluently tree-like continuum, then it is confluently σ -tree-representable and has the fixed point property.

As different from the ordinary singular cohomology, where a group of coefficients is abelian, in this paper a continuous singular cohomology with coefficients in a topological abelian group is defined. It is proved that for arbitrary topological spaces and any topological abelian group the continuous singular cohomology satisfies all Eilenberg–Steenrod axioms and the additivity axiom, except the excision axiom. However, if a group of coefficients is an absolute retract, then for metric spaces the continuous singular cohomology satisfies the excision axiom and also there is Mayer–Vietoris sequence and the suspension axiom is satisfied.

For the leaf space of the null foliation on a presymplectic manifold, we study group actions, moment maps and reductions.

Recently Dupont proved that the categories of discrete and codiscrete (or connected) objects in an abelian 2-category are equivalent abelian categories. In this paper for any abelian category 𝔸 with enough projective or injective objects we construct an abelian 2-category 𝔗 such that the categories of discrete, resp. codiscrete objects of 𝔗 are equivalent to 𝔸.

We use the multiplicative structure of the Koszul resolution to give short and simple proofs of some known estimates for the total dimension of the cohomology of spaces which admit free torus actions and some analogous results for filtered differential modules over polynomial rings. We also point out the possibility of improving these results in the presence of a multiplicative structure on the so-called minimal Hirsch–Brown model for the equivariant cohomology of the space.

It is shown that for any integer-valued function d defined on the class T SM of all separable and metrizable spaces and satisfying a minimal (in the natural sense) independent system of axioms (consisting of axioms of topological invariance, normability in dimensions –1 and 1 , monotonity with respect to F σ -subsets, subadditivity, decomposability and G δ -enlargement) the inequality d ( X ) ≤ dim X holds for any X ∈ T SM . Note that the proposed system of axioms does not contain the countable sum axiom.

In this paper, the concept of a σ -locally countable mapping is introduced, by which the relations between metric spaces and spaces with σ -locally countable networks, or spaces with σ -locally countable k-networks, or spaces with σ -locally countable cs-networks, or spaces with σ -locally countable bases are established.