In the paper to some known standard constructions in algebraic topology connected with the cohomological structure of a continuous mapping the author's standard construction is added. The last one is served to study perfect zero-dimensional mappings and it is proved that in this case all these constructions lead to equivalent resolvents for Abelian group sheaves, thus to the same spectral sequences. The results of the paper intend to study actions of p-adic group action on topological manifolds, the case that motivated the analysis of spectral sequences connecting the cohomology of a space with p-adic group of transformations and the cohomology of the orbit-space.The paper is presented to the special issue dedicated to the memory of professor Yuri Michailovich Smirnov. It is a pleasure to the author to recall that Yu.M. Smirnov was the supervisor of his dissertation and that one of his first articles (Zarelua and Smirnov, 1963 [25]) was written in collaboration with Yu.M. Smirnov. (C) 2013 Elsevier B.V. All rights reserved.

The interest to p-Adic groups actions on topological spaces, especially on manifolds, is arisen from the well-known Hilbert–Smith problem: is any action of such groups on compact manifolds non-effective? that is, does there exist an element of the group other than unit which fixes all elements of the space? In the present paper we develop new techniques for study the structure of the ring of continuous functions on spaces with p-Adic group action, based on integration along the orbits, and give some its applications. The text is a widened version of a part of the talk presented by the author on the 2010 International Conference on Topology and its Applications (Nafpactos, Greece, June 26–30, 2010) [14].

In a series of recent papers, V.I. Arnold studied many questions concerning the statistics and dynamics of powers of elements in algebraic systems. In particular, on the basis of experimental data, he proposed an Euler-type congruence for the traces of powers of integer matrices as a conjecture. The proof of this conjecture was deduced from the author's theorem (obtained at the end of 2004) on congruences for the traces of powers of elements in number fields. Recently, it turned out that there also exist other approaches to congruences for the traces of powers of integer matrices. In the present paper, the author's results of 2004 are strengthened and a survey of their relations to number theory, theory of dynamical systems, combinatorics, and p-adic analysis is given. The main conclusion of this survey is that all approaches considered here ultimately reflect different points of view on a certain simple but important phenomenon in mathematics.

In several papers author applied methods of algebraic topology to some topics in general topology; main applications are in dimension theory. An important tool of these investigations were two new types of spectral sequences. The first among them coincides with Cartan–Grothendieck sequence and thus involves the cohomology of a finite group. A description of second spectral sequence needs a new type of homology (cohomology) of a finite group which based on skew-symmetric invariant functions on the group ring of a finite group. The purpose of this article is to give a few principal approaches to study the structure of skew-symmetric invariant functions and, generally, of skew-symmetrical invariant tensors on the group ring of a finite group, bearing in mind further applications to topology.

The theorem proved in this paper gives a congruence for the traces of powers of an algebraic integer for the case in which the exponent of the power is a prime power. The theorem implies a congruence in Gauss’ form for the traces of the sums of powers of algebraic integers, generalizing many familiar versions of Fermat’s little theorem. Applied to the traces of integer matrices, this gives a proof of Arnold’s conjecture about the congruence of the traces of powers of such matrices for the case in which the exponent of the power is a prime power.

CONTENTSIntroduction ??1. Locally homotopy contractible resolutions of sheaves and DG-sheaves ??2. Elements of the homotopy theory of DG-sheaves ??3. Obstruction to a homotopy equivalence of a cochain map of locally homotopy contractible DG-sheavesReferences

A spectral sequence is defined for a closed map of finite multiplicity which coincides with the Cartan-Grothendieck spectral sequence in the case of a map onto a quotient space by a finite group acting freely [1, 2]. It is proved that the resolution by means of which the spectral sequence is defined can be described within the framework of the so-called theory of triples. A definition of this sequence is given for an arbitrary continuous map. It is shown that the spectral sequences of coverings are the spectral sequences of special continuous maps.