Carles Casacuberta

A UNIVERSAL SPACE FOR PLUS-CONSTRUCTIONS

Abstract We exhibit a two-dimensional, acyclic, Eilenberg–Mac Lane space W such that, for every space X, the plus-construction X+ with respect to the largest perfect subgroup of π1(X) coincides, up to homotopy, with the W-nullification of X; that is, the natural map X→X+ is homotopy initial among maps X→Y where the based mapping space map ∗ (W, Y) is weakly contractible. Furthermore, we describe the effect of W-nullification for any acyclic W, and show that some of its properties imply, in their turn, the acyclicity of W.

Extending localization functors

We prove that, under suitable restrictions, an idempotent monad t defined on a full subcategory A of a category C can be extended to an idempotent monad T on C in a universal (terminal) way. Our result applies in particular to the case when t is P-localization of nilpotent groups (where P denotes a set of primes) and C is the category of all groups. The corresponding monad T on C is, in a certain precise sense, the best idempotent approximation to the usual Zp-completion of groups; it turns out to be a strict epimorphic image of Bousfields HZp-localization.

On Towers Approximating Homological Localizations

Our object of study is the natural tower which, for any given map f : A → 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 Z/p or Z[1/p]. Our main tool is a comparison theorem for nullification functors (that is, localizations with respect to maps of the form f : A → pt), 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(Z[1/p], 1).

Models for torsion homotopy types

Given an integern>1 and any setP of positive integers, one can assign to each topological spaceX a homotopy universal mapX(P,n)→X whereX(P,n) is an (n−1)-connected CW-complex whose homotopy groups areP-torsion. We analyze this construction and its properties by means of a suitable closed model category structure on the pointed category of topological spaces.

Localizations as idempotent approximations to completions

Given a monad (also called a triple) T on an arbitrary category, an idempotent approximation to T is dened as an idempotent monad ^ T rendering invertible precisely the same class of morphisms which are rendered invertible by T. One basic example is homological localization with coecients in a ring R; which is an idempotent approximation to R-completion in the homotopy category of CW-complexes. We give general properties of idempotent approximations to monads using the machinery of orthogonal pairs, aiming to a better understanding of the relationship between localizations and completions. c 1999 Elsevier Science B.V. All rights reserved.

On weak homotopy equivalences between mapping spaces

Abstract Let S+n denote the n-sphere with a disjoint basepoint. We give conditions ensuring that a map h: X → Y that induces bijections of homotopy classes of maps [S+n, X] ≅ [S+n, Y] for all n ⩾ 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: map∗(B, W) → 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 → B. From the latter fact we infer that, for each map ƒ, the class of ƒ-local spaces is precisely the class of spaces orthogonal to ƒ and ƒ λ S + n for n ⩾ 1 i homotopy category. This has useful implications in the theory of homotopical localization.

A homotopy idempotent construction by means of simplicial groups

AbstractWe obtain a simplicial group model for localization of (not necessarily nilpotent) spaces at sets of primes by applying a suitable functor dimensionwise, as in earlier work of Quillen and Bousfield-Kan. For a set of primesP and any groupG, letG→LPG be a universal homomorphism fromG into a group which is uniquely divisible by primes not inP, and denote also byLP the prolongation of this functor to simplicial groups. We prove that, ifX is any connected simplicial set andJ is any free simplicial group which is a model for the loop space ΩX, then the classifying spacen

On genus and embeddings of torsion-free nilpotent groups of class two

overline W LpJ