Dirk Scevenels

Universal epimorphic equivalences for group localizations

Abstract Recent work by Bousfield shows the existence, for any map φ , of a universal space that is killed by homotopical φ -localization. Nullification with respect to this so-called universal φ -acyclic space is related to φ -localization in the same way as Quillens 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 homomorphism, we show that localization with respect to E (f) has in general nicer properties than f-localization itself.

Homology equivalences inducing an epimorphism on the fundamental group and Quillen's plus construction

Quillens 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.

Non-cancellation and Mislin genus of certain groups and H0-spaces

Abstract The non-cancellation set of a group G measures the extent to which the infinite cyclic group cannot be cancelled as a direct factor of G× Z . If G is a finitely generated group with finite commutator subgroup, then there is a group structure on its non-cancellation set, which coincides with the Hilton–Mislin genus group when G is nilpotent. Using a notion closely related to Nielsen equivalence classes of presentations of a finite abelian group, we give an alternative description of the group structure on the non-cancellation set of groups of a certain kind, and we include some computations. Analogously, we consider non-cancellation, up to homotopy, of the circle as a direct factor of a topological space. In particular, we show how the Mislin genera of certain H 0 -spaces with two non-vanishing homotopy groups can be identified with the genera of certain nilpotent groups.

Localizations associated to semidirect products

Abstract For homotopical localization with respect to any continuous map, there are results describing the relations among the localization functors associated to the 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 explicitly the localization functors associated to a semidirect product of finitely generated Abelian groups.

The genus of a direct product of nilpotent groups of a certain type and non-cancellation phenomena

We study the relation between the genus of a direct product of nilpotent groups of a certain type and the direct product of their genera. We show that the natural map φ : G(N 1) × … × G(N k) − g{N 1 × … × N k) is surjective if the groups N i are of a certain type. We also deduce some non-cancellation phenomena.

Closure Operations and Localization of Groups

Abstract Any localization functor on the category Grp of groups gives rise to a radical and an idempotent radical (possibly coinciding) which determine an epireflection, respectively a reduction. We describe how the classes of local groups for this epireflection and reduction can be obtained by application of standard closure operations on the class of local groups for the original localization. We furthermore characterize the class of acyclic groups for a given localization as a certain closed complement of its local groups. This approach through closure operations allows us to associate with any orthogonal pair on Grp, not necessarily reflective, an epireflection and a reduction which are in some sense maximal.

Strong genus of nilpotent groups of Hirsch length six

We give a characterization in terms of finitely many arithmetical invariants for two finitely generated, torsion-free, nilpotent groups of nilpotency class two and Hirsch length six to have isomorphic localizations at every finite set of primes. This result is obtained through adequate reduction of an arbitrary binary integral quadratic form. We derive some consequences and, in particular, we characterize completely those groups N in the above class of groups, for which the Mislin genus coincides with the strong genus (i.e. those groups N for which, whenever a nilpotent group M satisfies M p ≅ N p for every prime p , then M P ≅ N P for every finite set of primes P ).

From Mislin Genus to Strong Genus

If two finitely generated, torsion-free, nilpotent groups of class two satisfy the two-arrow property (that is, they embed into each other with finite, relatively prime indices), then they necessarily belong to the same Mislin genus (that is, they have isomorphic localizations at every prime). Here we show that the other implication is false in general. We even provide counterexamples in the case where both groups have isomorphic localizations at every finite set of primes of bounded cardinality. The latter equivalence relation leads us to introduce the notion of n -genus for every positive integer n , which we show to be meaningful in various contexts. In particular, the two-arrow property is related to the n -genus in the context of topological spaces.

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

SummaryWe study embeddings between torsion-free nilpotent groups having isomorphic localizations. Firstly, we show that for finitely generated torsion-free nilpotent groups of nilpotency class 2, the property of having isomorphicP-localizations (whereP denotes any set of primes) is equivalent to the existence of mutual embeddings of finite index not divisible by any prime inP. We then focus on a certain family Γ of nilpotent groups whose Mislin genera can be identified with quotient sets of ideal class groups in quadratic fields. We show that the multiplication of equivalence classes of groups in Γ induced by the ideal class group structure can be described by means of certain pull-back diagrams reflecting the existence of enough embeddings between members of each Mislin genus. In this sense, the family Γ resembles the family N0 of infinite, finitely generated nilpotent groups with finite commutator subgroup. We also show that, in further analogy with N0, two groups in Γ with isomorphic localizations at every prime have isomorphic localizations at every finite set of primes. We supply counterexamples showing that this is not true in general, neither for finitely generated torsion-free nilpotent groups of class 2 nor for torsion-free abelian groups of finite rank.