On the homotopy type of p -completions of infra-nilmanifolds

Abstract

Infra-nilmanifolds are compact K(G, 1)-manifolds with G a torsion-free, finitely generated, virtually nilpotent group. Motivated by previous results of various authors on p-completions of K(G, 1)-spaces with G a finite or a nilpotent group, we study the homotopy type of p-completions of infra-nilmanifolds, for any prime p. We prove that the p-completion of an infra-nilmanifold is a virtually nilpotent space which is either aspherical or has infinitely many nonzero homotopy groups. The same is true for plocalization. Moreover, we show by means of examples that rationalizations of infra-nilmanifolds may be elliptic or hyperbolic.

On the homotopy groups of p-completed classifying spaces

“…Later, Bastardas and Descheemaker discovered the same phenomenon holds for any virtually nilpotent group. This is done in [2] for torsion free groups and the general case is solved in [1]. We show that all these results can be deduced from Theorem 2.3 and we obtain the same statement for certain quasi p-perfect groups of finite virtual mod p cohomology and also for the new concept of p-local finite group, due to Broto, Levi, and Oliver [8].…”

“…In short if X is a virtually nilpotent space, its p-completion X ∧ p is an HF p -local space. Its second Postnikov section Y = X ∧ p [2] is a p-complete space with only two homotopy groups, which can be seen as the total space of a fibration of the form…”

Cellularization of classifying spaces and fusion properties of finite groups

“…An important role in the proof of this result is played by Levi [BD02]). All of these dichotomies share the common feature that they essentially concern objects whose homotopy, if infinite, is a p-torsion invariant.…”

The group of self homotopy equivalences of some localized aspherical complexes

“…F p = {1}). For instance, any group as in lemma above satisfies this for all p. In [2,9], the reader may find many examples of these groups which satisfy additional finiteness or nilpotency properties. Under these assumptions consider the free product G = * h∈H F and observe that BG = K(G, 1) = ∨ h∈H K(F, 1) = ∨ h∈H BF .…”

“…Therefore, we proceed by analyzing the group E m # (X p ) as X runs through different classes of aspherical spaces, i.e., Eilenberg MacLane spaces of the type K(G, 1). It is worth to mention that, out of the class of nilpotent spaces, p-localization functors do not, in general, preserve asphericity, even for virtually nilpotent K(G, 1)'s, i.e., when G contains a nilpotent normal subgroup of finite index [2,9]. Moreover, for many of these aspherical complexes, K(G, 1) p has an infinite number of non trivial homotopy groups.…”