Roman Mikhailov

Derived functors of nonadditive functors and homotopy theory

The main purpose of this paper is to extend our knowledge of the derived functors of certain basic nonadditive functors. The discussion takes place over the integers, and includes a functorial description of the derived functors of certain Lie functors, as well as that of the main cubical functors. We also present a functorial approach to the study of the homotopy groups of spheres and of Moore spaces M.A;n/, based on the Curtis spectral sequence and the decomposition of Lie functors as iterates of simpler functors such as the symmetric or exterior algebra functors. As an illustration, we retrieve in a purely algebraic manner the 3‐torsion components of the homotopy groups of the 2‐sphere in low degrees, and give a unified presentation of the homotopy groups i.M.A;n// for small values of both i and n. 18G55, 18G10; 54E30, 55Q40

On Residual Nilpotence of Projective Crossed Modules

Abstract Let F be a free group, (M,∂, F) a non-aspherical projective F-crossed module. We prove that the action of Coker (∂) on Ker (∂) is faithful. Also we show that if (M,∂, F) is a residually nilpotent crossed module, then Coker (∂) is a residually nilpotent group. As a corollary, we get an alternative proof of Conduches translation of Whiteheads asphericity conjecture in terms of residual nilpotence of certain crossed modules.

On Certain Questions of the Free Group Automorphisms Theory

Certain subgroups of the groups Aut(Fn) of automorphisms of a free group Fn are considered. Comparing Alexander polynomials of two poly-free groups and P4 we prove that these groups are not isomorphic, despite the fact that they have a lot of common properties. This answers the question of Cohenet al. (preprint). The questions of linearity of subgroups of Aut(Fn) are considered. As an application of the properties of poison groups in the sense of Formanek and Procesi, we show that the groups of the type Aut(G * ℤ) for certain groups G and the subgroup of IA-automorphisms IA(Fn) ⊂ Aut(Fn) are not linear for n ≥ 3. This generalizes the recent result of Pettet that IA(Fn) are not linear for n ≥ 5 (Pettet, 2006).

FAITHFULNESS OF CERTAIN MODULES AND RESIDUAL NILPOTENCE OF GROUPS

Let F be a non-cyclic free group and R, S its normal subgroups. We study the abelian group , viewed as a module over F/RS, via conjugation in F, and residual nilpotence of the group F/[R, S]. An application to the asphericity of finite presentations is given.

On the Pure Virtual Braid Group PV3

We investigate various properties of the pure virtual braid group PV3. Out of its presentation, we get a free product decomposition of PV3. As a consequence, we show that PV3 is residually torsion free nilpotent, what implies that the set of the finite type invariants in the sense of Goussarov–Polyak–Viro is complete for virtual pure braids with three strands. Moreover, we prove that the presentation of PV3 is aspherical. We determine also the cohomology ring and the associated graded Lie algebra of PV3.

A higher limit approach to homology theories

A lot of well-known functors such as group homology, cyclic homology of algebras can be described as limits of certain simply defined functors over categories of presentations. In this paper, we develop a technique for the description of the higher limits over categories of presentations and show that certain homological functors can be described in this way. In particular, we give a description of Hochschild homology and the derived functors of tensor, symmetric and exterior powers in the sense of Dold and Puppe as higher limits.

HIGHER TRACES ON GROUP RINGS

ABSTRACT Motivated by the works of Cuntz and Quillen on cyclic homology and algebra extensions, we study higher traces on group rings.

On zero-divisors in group rings of groups with torsion

Nontrivial pairs of zero-divisors in group rings are introduced and discussed. A problem on the existence of nontrivial pairs of zero-divisors in group rings of free Burnside groups of odd exponent

INTERSECTION OF SUBGROUPS IN FREE GROUPS AND HOMOTOPY GROUPS

n \gg 1

A combinatorial approach to the exponents of Moore spaces

is solved in the affirmative. Nontrivial pairs of zero-divisors are also found in group rings of free products of groups with torsion.