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Sergei O. Ivanov

Publications and source records attributed to Sergei O. Ivanov.

At least 37 records · Page 2Linked to original sources
arXiv2021

An overview of rationalization theories of non-simply connected spaces and non-nilpotent groups

We give an overview of five rationalization theories for spaces (Bousfield-Kan's $\mathbb Q$-completion; Sullivan's rationalization; Bousfield's homology rationalization; Casacuberta-Peschke's $Ω$-rationalization; Gómez-Tato-Halperin-Tanré's $π_1$-fiberwise rationalization) that extend the classical rationalization of simply connected spaces. We also give an overview of the corresponding rationalization theories for groups ($\mathbb Q$-completion; $H\mathbb Q$-localization; Baumslag rationalization) that extend the classical Malcev completion.

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arXiv2021

On commutator length in free groups

Let $F$ be a free group. We present for arbitrary $g\in\mathbb{N}$ a LogSpace (and thus polynomial time) algorithm that determines whether a given $w\in F$ is a product of at most $g$ commutators; and more generally an algorithm that determines, given $w\in F$, the minimal $g$ such that $w$ may be written as a product of $g$ commutators (and returns $\infty$ if no such $g$ exists). The algorithm also returns words $x_1,y_1,\dots,x_g,y_g$ such that $w=[x_1,y_1]\cdots[x_g,y_g]$. The algorithms we present are also efficient in practice. Using them, we produce the first example of a word in the free group whose commutator length decreases under taking a square. This disproves in a very strong sense a conjecture by Bardakov.

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arXiv2021

Baumslag rationalization of spaces

Using the functor of Baumslag rationalization of groups we construct a functor on the category of all (non necessarily simply connected) spaces that extends the classical rationalization of simply connected spaces. We study this functor and compare it with other extensions of the classical rationalization: Bousfield-Kan $\mathbb Q$-completion; Bousfield's homology rationalization; Gómez-Tato--Halperin--Tantré's $π_1$-fiberwise rationalization; and the localization with respect to the maps $n:S^1\to S^1$ that we call $Ω$-rationalization.

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arXiv2021

On homology of Lie algebras over commutative rings

We study five different types of the homology of a Lie algebra over a commutative ring which are naturally isomorphic over fields. We show that they are not isomorphic over commutative rings, even over $\mathbb Z,$ and study connections between them. In particular, we show that they are naturally isomorphic in the case of a Lie algebra which is flat as a module. As an auxiliary result we prove that the Koszul complex of a module $M$ over a principal ideal domain that connects the exterior and the symmetric powers $0\to Λ^n M\to M \otimes Λ^{n-1} M \to \dots \to S^{n-1}M \otimes M \to S^nM\to 0 $ is purely acyclic.

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arXiv2021

On the cokernel of the Baumslag rationalization

We prove that for the free group of rank two $F$ the cokernel of the homomorphism to its Baumslag rationalization $F\to {\sf Bau}(F)$ is not abelian. Moreover, this cokernel contains a free subgroup of countable rank. This answers a question of Emmanuel Farjoun.

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arXiv2021

Right exact localizations of groups

We introduce several classes of localizations (idempotent monads) on the category of groups and study their properties and relations. The most interesting class for us is the class of localizations which coincide with their zero derived functors. We call them right exact (in the sense of Keune). We prove that a right exact localization $L$ preserves the class of nilpotent groups and that for a finite $p$-group $G$ the map $G\to LG$ is an epimorphism. We also prove that some examples of localizations (Baumslag's $P$-localization with respect to a set of primes $P,$ Bousfield's $HR$-localization, Levine's localization, Levine-Cha's $\mathbb Z$-localization) are right exact. At the end of the paper we discuss a conjecture of Farjoun about Nikolov-Segal maps and prove a very special case of this conjecture.

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arXiv2021

Right exact group completion as a transfinite invariant of the homology equivalence

We consider a functor from the category of groups to itself $G\mapsto \mathbb Z_\infty G$ that we call right exact $\mathbb Z$-completion of a group. It is connected with the pronilpotent completion $\hat G$ by the short exact sequence $1\to {\varprojlim}^1\: M_n G \to \mathbb Z_\infty G \to \hat G \to 1,$ where $M_n G$ is $n$-th Baer invariant of $G.$ We prove that $\mathbb Z_\infty π_1(X)$ is an invariant of homological equivalence of a space $X$. Moreover, we prove an analogue of Stallings' theorem: if $G\to G'$ is a 2-connected group homomorphism, then $\mathbb Z_\infty G\cong \mathbb Z_\infty G'.$ We give examples of $3$-manifolds $X,Y$ such that $ \hat{π_1(X)}\cong \hat{π_1( Y)}$ but $\mathbb Z_\infty π_1(X)\not \cong \mathbb Z_\infty π_1(Y).$ We prove that for a finitely generated group $G$ we have $(\mathbb Z_\infty G)/ γ_ω= \hat G.$ So the difference between $\hat G$ and $\mathbb Z_\infty G$ lies in $γ_ω.$ This allows us to treat $\mathbb Z_\infty π_1(X)$ as a transfinite invariant of $X.$ The advantage of our approach is that it can be used not only for $3$-manifolds but for arbitrary spaces.

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arXiv2020

Homological properties of parafree Lie algebras

In this paper, an explicit construction of a countable parafree Lie algebra over $\mathbb Z/2$ with nonzero second homology is given. It is also shown that the cohomological dimension of the pronilpotent completion of a free noncyclic finitely generated Lie algebra over $\mathbb Z$ is greater than two. Moreover, it is proven that there exists a countable parafree group with nontrivial $H_2$.

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arXiv2020

Parafree augmented algebras and Gröbner-Shirshov bases for complete augmented algebras

We develop a theory of parafree augmented algebras similar to the theory of parafree groups and explore some questions related to the Parafree Conjecture. We provide an example of finitely generated parafree augmented algebra of infinite cohomological dimension. Motivated by this example, we prove a version of the Composition-Diamond lemma for complete augmented algebras and provide a sufficient condition for augmented algebra to be residually nilpotent on the language of its relations.

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arXiv2019

A Simple proof of Curtis' connectivity theorem for Lie powers

We give a simple proof of the Curtis' theorem: if $A_\bullet$ is $k$-connected free simplicial abelian group, then $L^n(A_\bullet)$ is an $k+ \lceil \log_2 n \rceil$-connected simplicial abelian group, where $L^n$ is the functor of $n$-th Lie power. In the proof we do not use Curtis' decomposition of Lie powers. Instead of this we use the Chevalley-Eilenberg complex for the free Lie algebra.

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arXiv2019

A finite Q-bad space

We prove that for a free noncyclic group $F$, $H_2(\hat F_\mathbb Q, \mathbb Q)$ is an uncountable $\mathbb Q$-vector space. Here $\hat F_\mathbb Q$ is the $\mathbb Q$-completion of $F$. This answers a problem of A.K. Bousfield for the case of rational coefficients. As a direct consequence of this result it follows that, a wedge of circles is $\mathbb Q$-bad in the sense of Bousfield-Kan. The same methods as used in the proof of the above results allow to show that, the homology $H_2(\hat F_\mathbb Z,\mathbb Z)$ is not divisible group, where $\hat F_\mathbb Z$ is the integral pronilpotent completion of $F$.

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arXiv2019

Mod-2 (co)homology of an abelian group

It is known that for a prime $p\ne 2$ there is the following natural description of the homology algebra of an abelian group $H_*(A,\mathbb F_p)\cong Λ(A/p)\otimes Γ({}_pA)$ and for finitely generated abelian groups there is the following description of the cohomology algebra of $H^*(A,\mathbb F_p)\cong Λ((A/p)^\vee)\otimes {\sf Sym}(({}_pA)^\vee).$ We prove that there are no such descriptions for $p=2$ that `depend' only on $A/2$ and ${}_2A$ but we provide natural descriptions of $H_*(A,\mathbb F_2)$ and $H^*(A,\mathbb F_2)$ that `depend' on $A/2,$ ${}_2A$ and a linear map $\tilde β:{}_2A\to A/2.$ Moreover, we prove that there is a filtration by subfunctors on $H_n(A,\mathbb F_2)$ whose quotients are $Λ^{n-2i}(A/2)\otimes Γ^i({}_2A)$ and that for finitely generated abelian groups there is a natural filtration on $H^n(A,\mathbb F_2)$ whose quotients are $ Λ^{n-2i}((A/2)^\vee)\otimes {\sf Sym}^i(({}_2A)^\vee).$

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arXiv2019

Limits, standard complexes and fr-codes

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.

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arXiv2019

Higher colimits, derived functors and homology

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.

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arXiv2018

On two letter identities in Lie rings

Let $L=L(a,b)$ be a free Lie ring on two letters $a,b.$ We investigate the kernel $I$ of the map $L\oplus L \to L $ given by $(A,B)\mapsto [A,a]+[B,b].$ Any homogeneous element of $L$ of degree $\geq 2$ can be presented as $[A,a]+[B,b].$ Then $I$ measures how far such a presentation from being unique. Elements of $I$ can be interpreted as identities $[A(a,b),a]=[B(a,b),b]$ in Lie rings. The kernel $I$ can be decomposed into a direct sum $I=\bigoplus_{n,m} I_{n,m},$ where elements of $I_{n,m}$ correspond to identities on commutators of weight $n+m,$ where the letter $a$ occurs $n$ times and the letter $b$ occurs $m$ times. We give a full description of $I_{2,m};$ describe the rank of $I_{3,m};$ and present a concrete non-trivial element in $I_{3,3n}$ for $n\geq 1.$

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arXiv2017

On discrete homology of a free pro-$p$-group

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.

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