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**Number of pages:** ? **Last
modified:** October 2001

**Abstract:**

We study the behavior of localization functors on finite simple groups with respect to universal central extensions and automorphism groups. Often a localization between finite simple groups induces a localization between their automorphism groups and also between their universal central extensions. In particular we show that the localization of a finite simple group need not be simple.

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**Number of pages:** ? **Last
modified:** October 2001

**Abstract:**

In this paper we generalize the plus-construction given by M. Livernet
for algebras over rational differential graded Koszul operads to the framework
of ``nice'' operads (the category of algebras over such operads admits
a

closed model structure). We follow the modern approach of J. Berrick
and C. Casacuberta defining topological plus-construction as a nullification
with respect to a universal acyclic space. Similarly, we construct a universal
$H^Q_*$-acyclic algebra $\mathcal U$ and we define $F\longrightarrow F^+$
as the $\mathcal U$-nullification of $F$. This map induces an isomorphism
on Quillen homology and quotients out the maximal perfect ideal of $\pi_0(A)$.
As an application, we consider for any associative algebra $A$ the plus-constructions
of $sl(A)$ in the

categories of Lie and Leibniz algebras up to homotopy. This gives rise
to two new homology theories for associative algebras namely reduced cyclic
and Hochschild homologies up to homotopy. In particular, these theories
coincide with the classical reduced cyclic and Hochschild homologies over
the rational.

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**Number of pages:** 17 **Last
modified:** October 2000

**Abstract:**

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 family of finite simple groups. In some cases, we also consider automorphism groups and universal covering groups and we show that a localization of a finite simple group may not be simple.

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**Number of pages:** 17 **Last modified:**
February 2000

**Abstract:** (dvi
format)

This paper is devoted to Moore spaces of dimension 2. These are two-complexes $M$ with $H_2(M; \Z)=0$, hence $\widetilde H_*(M;R)=0$ for some appropriate ring $R$ which only depends on $H_1(M; \Z)$. We investigate which spaces that are also $HR$-acyclic admit an $M$-cellular decomposition, i.e. can be built as a pointed homotopy colimit of a diagram only involving~$M$.

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**Number of pages:** 11 **Last modified:**
November 1998

**Abstract:** (dvi
format)

Quillen's plus construction is a topological construction that kills the maximal perfect subgroup of the fundamental group of a space without changing the integral homology of the space. In this paper we show that there is a topological construction that, while leaving the integral homology of a space unaltered, kills even the intersection of the transfinite lower central series of its fundamental group. Moreover, we show that this is the maximal subgroup that can be factored out of the fundamental group without changing the integral homology of a space.

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**Number of pages:** 25 **Last modified:**
July 6, 1998

**Abstract:** (dvi
format)

Using recent techniques of unstable localization, we extend earlier results on homological localizations of Eilenberg--Mac Lane spaces, and show that several deep properties of such localizations can be explained by the preservation of certain algebraic structures under the effect of idempotent functors.

We study localizations $L_fK(G,n)$ of Eilenberg--Mac Lane spaces with respect to any map~$f$, where $n\ge 1$ and $G$ is abelian. We find that, if~$G$ is finitely generated, then the result is a $K(A,n)$, where $A$ can be computed using cohomological data derived from~$f$. If $G=\Z$, then $A$ is a commutative ring which is isomorphic to the ring $\End(A)$ of its own additive endomorphisms; such rings, which we call rigid, form a proper class which contains the set of solid rings. >From this fact it follows that there is a proper class of distinct homotopical localizations of the circle~$S^1$. Among other applications of our results, we show that, if~$X$ is a product of abelian Eilenberg--Mac Lane spaces and $f$ is any map, then the homotopy groups $\pi_m(L_f X)$ become modules over the ring $\pi_1(L_f S^1)$.

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**Number of pages:** 34 **Last modified:**
December 14, 1999

*Crelle's Journal, to appear.*

**Abstract:** (dvi
format)

A group homomorphism $\eta: H\to G$ is called a {\it localization} of~$H$ if every homomorphism $\varphi : H\to G$ can be `extended uniquely' to a homomorphism $\Phi :G\to G$ in the sense that $\Phi \eta = \varphi$.

Libman showed that a localization of a finite group need not be finite. This is exemplified by a well-known representation $A_n\to SO_{n-1}(\R)$ of the alternating group $A_n$, which turns out to be a localization for $n$ even and $n\geq 10$. Dror Farjoun asked if there is any upper bound in cardinality for localizations of $A_n$. In this paper we answer this question and prove, under the generalized continuum hypothesis, that every non abelian finite simple group $H$, has arbitrarily large localizations. This shows that there is a proper class of distinct homotopy types which are localizations of a given Eilenberg--Mac Lane space $K(H,1)$ for any non abelian finite simple group~$H$.

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**Number of pages:** 12 **Last modified:**
November 1999

**Abstract:** (dvi
format)

*Proceedings of the Arolla Conference on Algebraic Topology, 1999,
to appear.*

We use localizations with respect to group homomorphisms in order to study under which conditions homotopical localizations with respect to given maps commute. In this context we introduce the {\it iterating series\/} of two localizations, thus obtaining a description of the effect of a localization with respect to a wedge of maps. In particular this enables a better understanding of nullifications of nilpotent GEM spaces. Finally we consider nullifications with respect to torsion-free abelian groups of rank 1, and we show that even in this relative simple case, the length of an iterating series can be any integer.

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**Number of pages:** 21 **Last modified:**
December 7, 1998

*Proceedings of the Barcelona Conference on Algebraic Topology, 1998,
to appear.*

**Abstract:** (dvi
format)

For a two-dimensional Moore space $M$ with fundamental group $G$, we identify the effect of the cellularization $CW_M$ and the fiber $\ov P_M$ of the nullification on an Eilenberg--Mac Lane space $K(N,1)$, where $N$ is any group: both induce on the fundamental group a group theoretical analogue, which can also be described in terms of certain universal extensions. We characterize completely $M$-cellular and $M$-acyclic spaces, in the case when $M=M(\Z/p^k,1)$.

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**Number of pages:** 11 **Last modified:**
August 5, 1998

*Topology and its Applications,* 108 (2) (2000), 169-177.

**Abstract:** (dvi
format)

For homotopical localization with respect to any continuous map, there are results describing the relations among the localization functors associated to the spaces and maps of a given fibration. Here we treat an analogous question in a group-theoretical context: we study localization functors associated to a short exact sequence of groups. We further specialize to a split short exact sequence of groups. In particular, we describe explicitely the functors associated to a semidirect product of finitely generated abelian groups.

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**Final version:** November 30, 1998

*J. Pure Appl. Algebra* (148) 3 (2000), 309-316.

**Abstract:** (dvi
format)

Recent work by Bousfield shows the existence, for any map $\phi$, of a universal space that is killed by $\phi$-localization. Nullification with respect to this so called universal $\phi$-acyclic space is related to $\phi$-localization in the same way as Quillen's plus construction is related to homological localization. Here we construct a universal $f$-acyclic group for any group homomorphism $f$. Moreover, we prove that there is a universal epimorphism $\E(f)$ that is inverted by $f$-localization. Although the kernel of the $\E(f)$-localization homomorphism coincides with that of the $f$-localization, we show that localization with respect to $\E(f)$ has in general nicer properties than $f$-localization itself.

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**Final version:** May 25, 1998

*Proceedings of Groups St Andrews 1997 in Bath I, London Math. Soc.,
Lecture Note Series 260, Cambridge University Press (1999).*

**Abstract:** (dvi
format)

To every variety of groups $W$ one can associate an idempotent radical
$P_W$ by iterating the verbal subgroup. The basic example is the perfect
radical, which is the intersection of the transfinite derived series. We
prove that each such radical $P_W$ is generated by a single locally free
group $F$, in the sense that, for every group $G$, the subgroup $P_W G$
is

generated by images of homomorphisms $F \to G$.

Our motivation comes from algebraic topology. In fact we show that every variety of groups $\calW$ determines a localization functor in the homotopy category, which kills the radical $\calPW$ of the fundamental group while preserving homology with certain coefficients.

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**Final version:** July 28, 1997

*Topology* 37 (1998), 709-717.

**Abstract:** (dvi
format)

Let $S^n_+$ denote the $n$-sphere with a disjoint basepoint. We give conditions ensuring that a map $h\colon X\to Y$ that induces bijections of homotopy classes of maps $[S^n_+,X]\cong [S^n_+,Y]$ for all $n\ge 0$ is a weak homotopy equivalence. For this to hold, it is sufficient that the fundamental groups of all path-connected components of $X$ and $Y$ be inverse limits of nilpotent groups. This condition is fulfilled by any map between based mapping spaces $h\colon{\rm map}_*(B,W)\to{\rm map}_*(A,V)$ if $A$ and $B$ are connected CW-complexes. The assumption that $A$ and $B$ be connected can be dropped if $W=V$ and the map $h$ is induced by a map $A\to B$. From the latter fact we infer that, for each map $f$, the class of $f$-local spaces is precisely the class of spaces orthogonal to $f$ and $f\wedge S^n_+$ for $n\ge 1$ in the based homotopy category. This has useful implications in the theory of homotopical localization.

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**Final version:** July 16, 1997

*Israel Journal of Mathematics* 107 (1998), 301-318.

**Abstract:** (dvi
format)

Given an integer $n>1$ and any set $P$ of positive integers, one can assign to each topological space $X$ a homotopy universal map $X^{(P,n)}\to X$ where $X^{(P,n)}$ is an $(n-1)$-connected CW-complex whose homotopy groups are $P$-torsion. We analyze this construction and its properties by means of a suitable closed model category structure on the pointed category of topological spaces.

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**Final version:** September 22, 1995

*J. London Math. Soc.* (2) 56 (1997), 645-656.

**Abstract:** (dvi
format)

Our object of study is the natural tower which, for any given map $f\colon A\to B$ and each space $X$, starts with the localization of $X$ with respect to $f$ and converges to $X$ itself. These towers can be used to produce approximations to localization with respect to any generalized homology theory $E_*$, yielding e.g.~an analogue of Quillen's plus-construction for each $E_*$. We discuss in detail the case of ordinary homology with coefficients in ${\bf Z}/p$ or ${\bf Z}[1/p]$. Our main tool is a comparison theorem for nullification functors (that is, localizations with respect to maps of the form $f\colon A\to *$), which allows us, among other things, to generalize Neisendorfer's observation that $p$-completion of simply-connected spaces coincides with nullification with respect to a Moore space $M({\bf Z}[1/p],1)$.

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