Emmanuel Dror Farjoun

Fundamental group of homotopy colimits

We give a general version of theorems due to Seifert–van Kampen and Brown about the fundamental group of topological spaces. We consider here the fundamental group of a general homotopy colimit of spaces. This includes unions, direct limits and quotient spaces as special cases. The fundamental group of the homotopy colimit is determined by the induced diagram of fundamental groupoids via a simple commutation formula. We use this framework to discuss homotopy (co-)limits of groups and groupoids as well as the useful Classification Lemma 6.4. Immediate consequences include the fundamental group of a quotient spaces by a group action π1(K/G) and of more general colimits. The Bass–Serre and Haefligers decompositions of groups acting on simplicial complexes is shown to follow effortlessly. An algebraic notion of the homotopy colimit of a diagram of groups is treated in some detail.

Cellular properties of nilpotent spaces

We show that cellular approximations of nilpotent Postnikov stages are always nilpotent Postnikov stages, in particular classifying spaces of nilpotent groups are turned into classifying spaces of nilpotent groups. We use a modified Bousfield-Kan homology completion tower zkX whose terms we prove are all X-cellular for any X. As straightforward consequences, we show that if X is K-acyclic and nilpotent for a given homology theory K, then so are all its Postnikov sections PnX, and that any nilpotent space for which the space of pointed self-maps map ∗ (X,X) is “canonically” discrete must be aspherical.

Bousfield–Kan completion of homotopy limits

Abstract We consider the commutation of R∞, the Bousfield–Kan R-completion functor, with homotopy (inverse) limits over categories I with compact classifying spaces. We get a generalization of the usual fibre lemma regarding preservation of a fibration sequence by R∞. The basic result is that for such I-diagrams N of nilpotent spaces the canonical commutation map R ∞ holim I N → c holim I R ∞ N is always a covering projection. This has clear implications for Sullivan–Quillen localization and completion theory and for rational models. On the way we are lead to a sufficient condition for the homotopy limit over a finite diagram to be non-empty or in fact r-connected for a given r⩾−1.

Homology of jet groups

In this paper we compute the second homology of the discrete jet groups. Let R be the additive group of real numbers and R the multiplicative group of positive reals. The n jet group Jn = {rx+a2x + · · ·+anx | r ∈ R, ai ∈ R} is the group, under composition followed by truncation, of invertible, orientationpreserving real n-jets at 0. Consider the homomorphism D : Jn → R obtained by projecting onto the first coefficient, i.e. Df = first derivative of f at 0. Every jet with slope not equal to 1 is conjugate to its linear part. It follows there is a split exact sequence

On the third homotopy group of Orr's space

K. Orr defined a Milnor-type invariant of links that lies in the third homotopy group of a certain space

Crossed modules as maps between connected components of topological groups

K_\omega.

Idempotent Functors and Nilpotent Spaces

The problem of non-triviality of this third homotopy group has been open. We show that it is an infinitely generated group. The question of realization of its elements as links remains open.

Bousfield localizations of classifying spaces of nilpotent groups

Abstract The purpose of this paper is to observe that a homomorphism of discrete groups f : Γ → G {f:\Gamma\to G} arises as the induced map π 0 ⁢ ( 𝔐 ) → π 0 ⁢ ( 𝔛 ) {\pi_{0}(\mathfrak{M})\to\pi_{0}(\mathfrak{X})} on path components of some closed normal inclusion of topological groups 𝔐 ⊆ 𝔛 {\mathfrak{M}\subseteq\mathfrak{X}} if and only if the map f can be equipped with a crossed module structure. In that case an essentially unique realization 𝔐 ⊆ 𝔛 {\mathfrak{M}\subseteq\mathfrak{X}} exists by homotopically discrete topological groups. The results here are topological elaboration and exposition using existing simplicial techniques.

Cellular covers of groups

We show that cellular approximations of nilpotent Postnikov stages are always nilpotent Postnikov stages and, in particular, classifying spaces of nilpotent groups are turned into classifying spaces of nilpotent groups. We use a modified Bousfield–Kan homology completion tower \(z_k X\) whose terms are all X-cellular for any X.

Homotopy localization and V1-periodic spaces

Let G be a finitely generated nilpotent group. The object of this paper is to identify the Bousfield localization LhBG of the classifying space BG with respect to a multiplicative complex oriented homology theory h*. We show that LhBG is the same as the localization of BG with respect to the ordinary homology theory determined by the ring ho.