Jérôme Scherer

Cellular approximations using Moore spaces

For a two-dimensional Moore space M with fundamental group G,we identify the effect of the cellularization CW M and the fibre \({\overline P _M}\) 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 com-pletely M-cellular and M-acyclic spaces, in the case when M = M(Z/p κ , 1).

Topological Complexity Of H-Spaces

Let X be a (not-necessarily homotopy-associative) H-space. We show that TCn+1(X) = cat(X-n), for n >= 1, where TCn+1(-) denotes the so-called higher topological complexity introduced by Rudyak, and cat(-) denotes the Lusternik-Schnirelmann category. We also generalize this equality to an inequality, which gives an upper bound for TCn+1(X), in the setting of a space Y acting on X.

Representations of spaces

We explain how the notion of homotopy colimits gives rise to that of mapping spaces, even in categories which are not simplicial. We apply the technique of model approximations and use elementary properties of the category of spaces to be able to construct resolutions. We prove that the homotopy category of any monoidal model category is always a central algebra over the homotopy category of Spaces. 55U35, 18G55; 18G10

Nilpotent

We provide characterizations of

p\mspace{1mu}

p\mspace {1mu}

-local finite groups

-nilpotency for fusion systems and

Postnikov pieces and BZ/p-homotopy theory

p\mspace {1mu}

FINITE SIMPLE GROUPS AND LOCALIZATION

-local finite groups that are inspired by known result for finite groups. In particular, we generalize criteria by Atiyah, Brunetti, Frobenius, Quillen, Stammbach and Tate.

Non-simple localizations of finite simple groups

We present a constructive method to compute the cellularization with respect to B m Z/p for any integer m > 1 of a large class of H-spaces, namely all those which have a finite number of non-trivial B m Z/p-homotopy groups (the pointed mapping space map * (B m Z/p, X) is a Postnikov piece). We prove in particular that the B m Z/p-cellularization of an H-space having a finite number of B m Z /p-homotopy groups is ap-torsion Postnikov piece. Along the way, we characterize the BZ/p r -cellular classifying spaces of nilpotent groups.

Goodwillie calculus and Whitehead products

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.