A. K. Bousfield's $H\mathbb Z$-localization of groups inverts homologically two-connected homomorphisms of groups. J. P. Levine's algebraic closure of groups inverts homomorphisms between finitely generated and finitely presented groups which are homologically two-connected and for which the image normally generates. We resolve an old problem concerning Bousfield $H\mathbb Z$-localization of groups, and answer two questions of Levine regarding algebraic closure of groups. In particular, we show that the kernel of the natural homomorphism from a group $G$ to it's Bousfield $H\mathbb Z$-localization is not always a $G$-perfect subgroup. In the case of algebraic closure of groups, we prove the analogous result that this kernel is not always an invisible subgroup.
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…
Let M denote the mapping class group of S, a compact connected oriented surface with one boundary component. The action of M on the nilpotent quotients of the fundamental group of S allows to define the so-called Johnson filtration and the Johnson homomorphisms. J. Levine introduced a new filtration of M, called the Lagrangian filtration. He also introduced a version of the Johnson homomorphisms for this new filtration. The first term of the Lagrangian filtration is the Lagrangian mapping class group, whose definition involves a handlebody bounded by S, and which contains the Torelli group. Th…
Often a localization functor (in the category of groups) sends a finite simple group to another finite simple group. We study when such a localization also induces a localization between the automorphism groups and between the universal central extensions. As a consequence we exhibit many examples of localizations of finite simple groups which are not simple.
In characteristic 0 there are essentially two approaches to the conjectural theory of mixed motives, one due to Nori and the other one due to, independently, Hanamura, Levine, and Voevodsky. Although these approaches are apriori quite different it is expected that ultimately they can be reduced to one another. In this article we provide some evidence for this belief by proving that their associated motivic Galois groups are canonically isomorphic.
We give a Dehn-Nielsen type theorem for the homology cobordism group of homology cylinders by considering its action on the acyclic closure, which was defined by Levine, of a free group. Then we construct an additive invariant of those homology cylinders which act on the acyclic closure trivially. We also describe some tools to study the automorphism group of the acyclic closure of a free group generalizing those for the automorphism group of a free group or the homology cobordism group of homology cylinders.
A group homomorphism $i: H \to G$ is a localization of $H$ if for every homomorphism $φ: H\rightarrow G$ there exists a unique endomorphism $ψ: G\rightarrow G$, such that $i ψ=φ$ (maps are acting on the right). Göbel and Trlifaj asked in \cite[Problem 30.4(4), p. 831]{GT12} which abelian groups are centers of localizations of simple groups. Approaching this question we show that every countable abelian group is indeed the center of some localization of a quasi-simple group, i.e. a central extension of a simple group. The proof uses Obraztsov and Ol'shanskii's construction of infinite simple gr…
The purpose of this paper is to explore the concept of localization, which comes from homotopy theory, in the context of finite simple groups. We give an easy criterion for a finite simple group to be a localization of some simple subgroup and we apply it in various cases. Iterating this process allows us to connect many simple groups by a sequence of localizations. We prove that all sporadic simple groups (except possibly the Monster) and several groups of Lie type are connected to alternating groups. The question remains open whether or not there are several connected components within the f…
An n-dimensional μ-component boundary link is a codimension 2 embedding of spheres L=\bigsqcup_μS^n \subset S^{n+2} such that there exist μdisjoint oriented embedded (n+1)-manifolds which span the components of L. An F_μ-link is a boundary link together with a cobordism class of such spanning manifolds. The F_μ-link cobordism group C_n(F_μ) is known to be trivial when n is even but not finitely generated when n is odd. Our main result is an algorithm to decide whether two odd-dimensional F_μ-links represent the same cobordism class in C_{2q-1}(F_μ) assuming q>1. We proceed to compute the isomo…
Based on the algebraic cobordism theory of Levine and Morel, we develop a theory of algebraic cobordism modulo algebraic equivalence. We prove that this theory can reproduce Chow groups modulo algebraic equivalence and the semi-topological $K_0$-groups. We also show that with finite coefficients, this theory agrees with the algebraic cobordism theory. We compute our cobordism theory for some low dimensional varieties. The results on infinite generation of some Griffiths groups by Clemens and on smash-nilpotence by Voevodsky and Voisin are also lifted and reinterpreted in terms of this cobordis…
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 …
The purpose of this paper is to give a complete description of the Cohn localization of the augmentation map $Z[G]\rightarrow Z$ when $G$ is any finite group.
We extend the standard localization theory for function and section spaces due to Hilton-Mislin-Roitberg and Moller outside the CW category to the case of compact metric domain in the presence of a grouplike structure. We study applications in two cases directly generalizing the gauge group of a principal bundle. We prove an identity for the monoid of fibre-homotopy self-equivalences of a Hurewicz fibration -- due to Gottlieb and Booth-Heath-Morgan-Piccinini in the CW category -- in the compact case. This leads to an extended localization result for this monoid. We also obtain an extended loca…
We introduce an equivariant algebraic cobordism theory Ω^G for algebraic varieties with G-action, where G is a split diagonalizable group scheme over a field k. It is done by combining the construction of the algebraic cobordism theory Ωby F. Morel and M. Levine, with the notion of (G, F)-formal group law with respect to a complete G-universe and complete G-flag F as introduced by M. Cole, J. P. C. Greenlees and I. Kriz. In particular, we use their corresponding representing ring L_G(F) in place of the Lazard ring L. We show that localization property and homotopy invariance property hold in Ω…
We study the relationship between the transchromatic localizations of Morava $E$-theory, $L_{K(n-1)}E_n$, and formal groups. In particular, we show that the coefficient ring $π_0L_{K(n-1)}E_n$ has a modular interpretation, representing deformations of formal groups with certain extra structure, and derive similar descriptions of the cooperations algebra and $E_{n-1}$-homology of this spectrum. As an application, we show that $L_{K(1)}E_2$ has exotic $\mathcal{E}_\infty$ structures not obtained by $K(1)$-localizing the $\mathcal{E}_\infty$ ring $E_2$.
The aim of this article is to introduce Vogel's localization theorem for classes of D-complexes: this generalization of Waldhausen's localization theorem is especially useful and powerful in that it gives an explicit and computable description of the local objects. Next we present an excision theorem for transverse classes. Finally, we apply these methods to deduce partial results on Vogel's [Conjecture] on a regular ring R (cf [Bih01]).
We discuss the Bousfield localization $L_E X$ for any spectrum $E$ and any $HR$-module $X$, where $R$ is a ring with unit. Due to the splitting property of $HR$-modules, it is enough to study the localization of Eilenberg-Mac Lane spectra. Using general results about stable $f$-localizations, we give a method to compute the localization of an Eilenberg-Mac Lane spectrum $L_E HG$ for any spectrum $E$ and any abelian group $G$. We describe $L_E HG$ explicitly when $G$ is one of the following: finitely generated abelian groups, $p$-adic integers, Prüfer groups, and subrings of the rationals. The …
Let $\widetilde{JO}(X)=\widetilde{KO}(X)/TO(X)$ be the J-group of a connected finite CW complex X. We Obtain two computable formulas of $TO(X)_{(p)}$, the localization of $TO(X)$ at a prime p. Then we show how to use these two formulas of $TO(X)_{(p)}$ to find the J-orders of elements of $\widetilde{KO}(CP^m)$, at least the 2 and 3 primary factors of the canonical generators of $\widetilde{JO}(CP^m).$ Here $CP^m$ is the complex projective space.
We prove that localization functors of crossed modules of groups do not always admit fiberwise (or relative) versions. To do so we characterize the existence of a fiberwise localization by a certain normality condition and compute explicit examples and counter-examples. In fact, some nullification functors do not behave well and we also prove that the fiber of certain nullification functors, known as acyclization functors in other settings such as groups or spaces, is not acyclic.
We analyze the stable isomorphism type of polynomial rings on degree 1 generators as modules over the sub-algebra A(1) = <Sq^1, Sq^2> of the mod 2 Steenrod algebra. Since their augmentation ideals are Q_1-local, we do this by studying the Q_i-local subcategories and the associated Margolis localizations. The periodicity exhibited by such modules reduces the calculation to one that is finite. We show that these are the only localizations which preserve tensor products, by first computing the Picard groups of these subcategories and using them to determine all idempotents in the stable category …
In this paper we investigate the structure of algebraic cobordism of Levine-Morel as a module over the Lazard ring with the action of Landweber-Novikov and symmetric operations on it. We show that the associated graded groups of algebraic cobordism with respect to the topological filtration $Ω^*_{(r)}(X)$ are unions of finitely presented $\mathbb{L}$-modules of very specific structure. Namely, these submodules possess a filtration such that the corresponding factors are either free or isomorphic to cyclic modules $\mathbb{L}/I(p,n)x$ where $\mathrm{deg\ } x\ge \frac{p^n-1}{p-1}$. As a corollar…
Motivated by calculations of motivic homotopy groups, we give widely attained conditions under which operadic algebras and modules thereof are preserved under (co)localization functors
