arXiv subjects
Publications and source records attributed to Roman Mikhailov.
For a strongly connected category $\mathcal C$ with pair-wise coproducts, we introduce a cosimplicial object, which serves as a sort of resolution for computing higher derived functors of ${\sf lim} : \mathrm{Ab}^{\mathcal C}\to \mathrm{Ab}$. Applications involve Künneth theorem for higher limits and ${\sf lim}$-finiteness of ${\bf fr}$-codes. A dictionary for the ${\bf fr}$-codes with words of length $\leq 3$ is given.
A theory of higher colimits over categories of free presentations is developed. It is shown that different homology functors such as Hoshcshild and cyclic homology of algebras over a field of characteristic zero, simplicial derived functors, and group homology can be obtained as higher colimits of simply defined functors. Connes' exact sequence linking Hochschild and cyclic homology was obtained using this approach as a corollary of a simple short exact sequence. As an application of the developed theory it is shown that the third reduced $K$-functor can be defined as the colimit of the second reduced $K$-functor applied to the fibre square of a free presentation of an algebra. A Hopf-type formula for odd dimensional cyclic homology of an algebra over a field of characteristic zero is also proved.
By exploring simplicial structure of pure virtual braid groups, we give new connections between the homotopy groups of the 3-sphere and the virtual braid groups that are related to the theory of Brunnian virtual braids. The group structure of VP_n with n > 4 is determined by VP_3, VP_4 and virtual cablings given by iterated degeneracy operations on the generators and defining relations. The complete proofs will be published in the two forthcoming papers.
A finitely generated solvable group with unbounded iterated identity is constructed.
A description of the derived functors of Lie functors for free abelian groups is given.
For a prime $p$, let $\hat F_p$ be a finitely generated free pro-$p$-group of rank $\geq 2$. We show that the second discrete homology group $H_2(\hat F_p,\mathbb Z/p)$ is an uncountable $\mathbb Z/p$-vector space. This answers a problem of A.K. Bousfield.
K. Orr defined a Milnor-type invariant of links that lies in the third homotopy group of a certain space $K_ω.$ The problem of non-triviality of this third homotopy group has been open. We show that it is an infinitely generated group. The question of realization of its elements as links remains open.
This paper presents a description of the fourth dimension quotient, using the theory of limits of functors from the category of free presentations of a given group to the category of abelian groups. A functorial description of a quotient of the third Fox subgroup is given and, as a consequence, an identification (not involving an isolator) of the third Fox subgroup is obtained. It is shown that the limit over the category of free representations of the third Fox quotient represents the composite of two derived quadratic functors.
This paper is about application of various homological methods to classical problems in the theory of group rings. It is shown that the third homology of groups plays a key role in Narain Gupta's three normal subgroup problem. For a free group $F$ and its normal subgroups $R,\,S,\,T,$ and the corresponding ideals in the integral group ring $\mathbb Z[F]$, ${\bf r}=(R-1)\mathbb Z[F],\ {\bf s}=(S-1)\mathbb Z[F],\ {\bf t}=(T-1)\mathbb Z[F],$ a complete description of the normal subgroup $F\cap (1+{\bf rst})$ is given, provided $R\subseteq T$ and the third and the fourth homology groups of $R/R\cap S$ are torsion groups.
In this paper, the $H\mathbb Z$-length of different groups is studied. By definition, this is the length of $H\mathbb Z$-localization tower or the length of transfinite lower central series of $H\mathbb Z$-localization. It is proved that, for a free noncyclic group, its $H\mathbb Z$-length is $\geq ω+2$. For a large class of $\mathbb Z[C]$-modules $M,$ where $C$ is an infinite cyclic group, it is proved that the $H\mathbb Z$-length of the semi-direct product $M\rtimes C$ is $\leq ω+1$ and its $H\mathbb Z$-localization can be described as a central extension of its pro-nilpotent completion. In particular, this class covers modules $M$, such that $M\rtimes C$ is finitely presented and $H_2(M\rtimes C)$ is finite.
An approach to identify the normal subgroups determined by ideals in free group rings with the help of the derived functors of non-additive functors is explored. A similar approach, i.e., via derived functors, for computing limits of functors from the category of free presentations to the category of abelian groups, arising from commutator structure of free groups, is also discussed.
This text is based on lectures given by authors in summer 2015. It contains an introduction to the theory of limits over the category of presentations, with examples of different well-known functors like homology or derived functors of non-additive functors in a form of derived limits. The theory of so-called ${\bf fr}$-codes also is developed. This is a method how different functors from the category of groups to the category of abelian groups, such as group homology, tensor products of abelianization, can be coded as sentences in the alphabet with two symbols ${\bf f}$ and ${\bf r}$.
For normal subgoups $R$ and $S$ of a free group $F$, an identification of the subgroup $F\cap (1+\mathfrak r\mathfrak f\mathfrak s)$ is derived, and it is shown that the the quotient $\frac{F\cap (1+\mathfrak r\mathfrak f\mathfrak s)}{[R'\cap S',\, R\cap S][R'\cap S,\,R'\cap S][R\cap S',\, R\cap S']}$ is, in general, non-trivial.
Let $G$ be a group and $R,S,T$ its normal subgroups. There is a natural extension of the concept of commutator subgroup for the case of three subgroups $\|R,S,T\|$ as well as the natural extension of the symmetric product $\|\bf r,\bf s,\bf t\|$ for corresponding ideals $\bf r,\bf s, \bf t$ in the integral group ring $\mathbb Z[G]$. In this paper, it is shown that the generalized dimension subgroup $G\cap (1+\|\bf r,\bf s,\bf t\|)$ has exponent 2 modulo $\|R,S,T\|.$ The proof essentially uses homotopy theory. The considered generalized dimension quotient of exponent 2 is identified with a subgroup of the kernel of the Hurewicz homomorphism for the loop space over a homotopy colimit of classifying spaces.
Every two-sided ideal $\mathfrak a$ in the integral group ring $\mathbb Z[G] $ of a group $G$ determines a normal subgroup $G \cap (1 + \mathfrak a)$ of $G$. In this paper certain problems related to the identification of such subgroups, and their relationship with derived functors in the sense of Dold-Puppe, are discussed.
A natural composition $\odot$ on all pages of the lower central series spectral sequence for spheres is defined. Moreover, it is defined for $p$-lower central series spectral sequence of a simplicial group. It is proved that $r$th differential satisfies a "Leibiz rule with suspension": $d^r(a\odot σb)=\pm d^ra\odot b+a\odot d^rσb,$ where $σ$ is the suspension homomorphism.
In this article, we give a combinatorial approach to the exponents of the Moore spaces. Our result states that the projection of the $p^{r+1}$-th power map of the loop space of the $(2n+1)$-dimensional mod $p^r$ Moore space to its atomic piece containing the bottom cell $T^{2n+1}\{p^r\}$ is null homotopic for $n>1$, $p>3$ and $r>1$. This result strengthens the classical result that $ΩT^{2n+1}\{p^r\}$ has an exponent $p^{r+1}$.
In this article, we study the exponents of metastable homotopy of mod $2$ Moore spaces. Our result gives that the double loop space of $4n$-dimensional mod $2$ Moore spaces has a multiplicative exponent $4$ below the range of $4$ times the connectivity. As a consequence, the homotopy groups of $4n$-dimensional mod $2$ Moore spaces have an exponent $4$ below the range of $4$ times the connectivity.