Volume 22, Issue 4

In this paper we investigate the basic algorithmic problems for class-2 nilpotent MR-groups. It is proved that, under an additional assumption of finite definiteness, all these problems have a positive solution and, in the general case, they have a negative solution for finitely generating groups.

This note provides a theorem on good groups in the sense of Hopkins–Kuhn–Ravenel [J. Amer. Math. Soc. 13 (2000), no. 3, 553–594] and some relevant examples.

In the category of soft topological spaces, a singular homology group is defined and the homotopic invariance of this group is proved [Internat. J. Engrg. Innovative Tech. (IJEIT) 3 (2013), no. 2, 292–299]. The first aim of this study is to define relative homology groups in the category of pairs of soft topological spaces. For these groups it is proved that the axioms of dimensional and exactness homological sequences hold true. The axiom of excision for singular homology groups is also proved.

The modal logic S4.3 defines the class of hereditarily extremally disconnected spaces (HED-spaces). We construct a countable HED-subspace X of the Gleason cover of the real closed unit interval [0,1] such that S4.3 is the logic of X .

For a bitopological space (X,τ1<τ1(A))${(X,\tau _1&lt;\tau _1(A))}$, where A∉τ1${A\notin \tau _1}$ and τ1(A)${\tau _1(A)}$ is a simple extension of the initial topology τ1${\tau _1}$, we show under which relations between the topologies τ1${\tau _1}$ and τ1(A)${\tau _1(A)}$ such properties as in particular the Blumberg property, pseudocompleteness, connectedness, submaximality, to be a Baire space, to be an H -closed space, to be an H D -space, to be an e D -space and to be a nodec space are passed from τ1${\tau _1}$ over to τ1(A)${\tau _1(A)}$. Moreover, we give the necessary and sufficient conditions both for the connectedness of a simple extension and for the maximal connectedness of some special topology. We obtain a solution for one of Everett and Ulam's problems dealing with the case of coincidence of two classes of homeomorphisms of a space onto itself.

In this paper, the notion of a truncating twisting function from a cubical set to a permutahedral set and the corresponding notion of a twisted Cartesian product of these sets are introduced. The latter becomes a permutocubical set that models, in particular, the path fibration on a loop space. The chain complex of this twisted Cartesian product is in fact a comultiplicative twisted tensor product of cubical chains of base and permutahedral chains of fibre. This construction is formalized as a theory of twisted tensor products for Hirsch algebras.

In this paper we give a direct generalization of Carleson–Ullrich's theorem in Besov spaces and geometric characterization of removable sets for harmonic functions in those spaces in terms of Hausdorff measure. In particular, for a compact set K⊂ℝd${K\subset \mathbb {R}^d}$ and the Besov space Λα, loc p,q(1≤p≤∞,1≤q≤∞,0<α<2)${\Lambda _{\alpha ,\mathrm {loc}}^{p,q}~(1\le p\le \infty ,~1\le q\le \infty ,0&lt;\alpha &lt;2)}$, we formulate the exact criteria to decide whether K is Λα, loc p,q${\Lambda _{\alpha ,\mathrm {loc}}^{p,q}}$-removable or not.

A Hopf algebra object in Loday and Pirashvili's category of linear maps entails an ordinary Hopf algebra and a Yetter–Drinfel'd module. We equip the latter with a structure of a braided Leibniz algebra. This provides a unified framework for examples of racks in the category of coalgebras discussed recently by Carter, Crans, Elhamdadi and Saito.

The notion of a differential module with simplicial F ∞ -faces is considered. It is shown that this notion is the homotopy analogue of the notion of a differential module with simplicial faces. The homotopical invariance of the structure of a differential module with simplicial F ∞ -faces is proved, and basic homotopical and categorial properties of differential modules with simplicial F ∞ -faces are considered. The connections between the concept of a differential module with simplicial F ∞ -faces and the theory of differential A ∞ -algebras and the theory of D ∞ -differential modules are established. The applications of differential modules with F ∞ -faces to description of homologies of realizations of simplicial topological spaces are given.

In this paper we prove that for good topological spaces the assignment U↦Π1(U)${U\mapsto \Pi _1(U)}$ is the terminal object in the 2-category of costacks over X . Here Π1(U)${\Pi _1(U)}$ denotes the fundamental groupoid of U .

The concept of relative topological finiteness is introduced. A Hausdorff compact space is constructed, which is homeomorphic to none of its proper G δ -subspaces, but is homeomorphic to its proper (not G δ )-subspace. This example shows that the class of all normal spaces which are topologically finite with respect to the class of all their proper G δ -subspaces does not coincide with the class of all normal topologically finite spaces.