Barcelona Topology Workshop 2016, Spring Session

In classical algebraic topology, the genus set of a topological space X is the set of all homotopy types Y with the same rationalization and completion as X. In this talk I would like to emphasize that any localization has an associated genus set. The classical genus set is the genus set of rationalization. As a new example, we shall consider the genus set of the n^th Postnikov section and classify all spaces with the same n^th Postnikov approximation and the same n-connected cover as the n-sphere when n is odd. This is joint work with Jérôme Scherer.

In this talk I will present an extension theory for p-local finite groups which encompasses the known cases previously studied by Oliver-Ventura and by Broto-Castellana-Grodal-Levi-Oliver. The main tools, which I will mention in more or less detail depending on time, are partial groups and their simplicial structure, and the theory of (simplicial) fibre bundles.

In this talk, we will focus on Bouc's conjecture about B-groups.

Saturated fusion systems are categories generalizing important features of fusion in finite groups. Many concepts in finite group theory have analogues in the language of fusion systems. In particular, normal subsystems of fusion systems are defined. Broto, Levi and Oliver introduced centric linking systems to be able to study classifying spaces of fusion systems. It was proved by Andrew Chermak that there is a unique centric linking system associated to each saturated fusion system. For his proof he introduced the concept of a locality which can be thought of as a "partial group" with a multivariable product only defined on certain words. A centric linking system corresponds to a centric linking locality. Since localities are so group like, there is a very natural notion of a partial normal subgroup of a locality. I will report on a joint project with Andrew Chermak where we prove that there is a one to one correspondence between the normal subsystems of a fusion system and the partial normal subgroups of an associated linking locality.

Let $G$ be a Lie group with finitely many components, and let $G^\delta$ denote the group $G$ considered as a discrete group. Then the identity map on $G$ induces a map on classifying spaces $BG^\delta\to BG$. The Friedlander-Milnor Isomorphism Conjecture, which we will refer to as IC, is that this map induces an isomorphism on homology with any finite coefficients. The conjecture was generalized by Friedlander and Mislin, to include algebraic groups over arbitrary fields and proved it in certain cases. However, their treatment failed to handle IC. Milnor proved IC in the case where $G$ has a solvable identity component. He also showed that IC is true in general if it holds for all simple connected Lie group. Finally he showed that the map $BG^\delta\to BG$ is a split surjection on homology with finite (trivial) coefficients. In this talk we explain various aspects of IC and prove in particular that Milnor’s splitting in homology with $\mathbb{F}_p$ coefficients can be realised as a right homotopy inverse to the same map, after completion at a prime $p$. We discuss implications and statements that imply the isomorphism conjecture in general.

The following question arises when analyzing centralizers of involutions in simple fusion systems. Assume E ⊴ F are fusion systems over T ⊴ S, where E is “centric” in F in an appropriate sense, and E is realized by a finite simple group K. Under what conditions can one conclude that F is realized by an extension L of K such that C_L(K) = 1 (i.e., L ≤ Aut(K)).
We describe one answer to this question, as well as the related concept of tameness. We also list some recent results about tameness of fusion systems of finite simple groups.