# Schedule of the Workshop "Fusion systems and equivariant algebraic topology"

## Monday, November 21

10:30 - 11:00 |
Registration & Welcome coffee |

11:00 - 12:00 |
Radu Stancu: Fusion systems: survival kit |

12:00 - 13:50 |
Lunch break |

13:50 - 14:50 |
Antonio Díaz Ramos: Mackey functors for fusion systems |

15:00 - 16:00 |
Sejong Park: Double Burnside rings and Mackey functors with applications to fusion systems |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:30 |
Oihana Garaialde: Cohomology of the J_{2} group over F_{3} using a spectral sequence in fusion systems |

afterwards |
Reception |

## Tuesday, November 22

09:30 - 10:30 |
Jesper Grodal: Burnside rings in algebra and topology (part 1) |

10:30 - 11:00 |
Group photo and coffee break |

11:00 - 12:00 |
Radu Stancu: Saturation and the double Burnside ring |

12:00 - 15:00 |
Lunch break and free time |

15:00 - 16:00 |
Matthew Gelvin: Minimal characteristic bisets of fusion systems |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:30 |
Nathaniel Stapleton: Transchromatic character theory for fusion systems |

## Wednesday, November 23

09:30 - 10:30 |
Benjamin Böhme: The Dress splitting and equivariant commutative multiplications |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Bob Oliver: Local structure of finite groups and of their p-completed classifying spaces |

12:00 - 15:00 |
Lunch break and free time |

15:00 - 16:00 |
Justin Lynd: Control of fixed points and centric linking systems |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:30 |
Isabelle Laude: Maps between (uncompleted) classifying spaces of p-local finite groups |

## Thursday, November 24

09:30 - 10:30 |
Jesper Grodal: Burnside rings in algebra and topology (part 2) |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Rémi Molinier: Cohomology with twisted coefficients of linking systems and stable elements |

12:00 - 14:00 |
Lunch break |

14:00 - 15:00 |
Ergün Yalcin: Representation rings for fusion systems and dimension functions |

16:00 - 16:30 |
Tea and cake |

# Abstracts

## Benjamin Böhme: The Dress splitting and equivariant commutative multiplications

Let G be a finite group. The p-local Burnside ring of G splits into a product of rings which can be described in terms of Dress' classification of idempotent elements. The "first" factor is the Grothendieck ring of G-sets with isotropy a p-group and coincides with the Burnside ring of the p-fusion system of G upon p-localization. It plays an important role in Grodal's work on the uncompleted Segal conjecture.

On the level of G-spectra, the Dress splitting induces a wedge decomposition of the p-local G-equivariant sphere spectrum, but only little is known about the multiplicative structure of the factors. Grodal showed that the first summand is a G-commutative ring spectrum in the strongest possible sense, but this is not true for the other summands, which in fact become contractible upon restriction to any p-subgroup. In light of recent work of Blumberg, Hill and Hopkins, it is clear that the existence of genuinely equivariant commutative multiplications on the wedge summands (so-called N_{∞} ring structures) is obstructed by the behaviour of co-induction of finite G-sets.

## Antonio Díaz Ramos: Mackey functors for fusion systems

Mackey functors naturally appear in the context of (stable) equivariant cohomology. In this talk, we will introduce Mackey functors for fusion system and comment on some of their applications. We will treat in more detail how to use Mackey functors to construct spectral sequences. In particular, we will explain how to build a "Lyndon-Hochschild-Serre"-type spectral sequence from a strongly closed subgroup.

## Oihana Garaialde: Cohomology of the J_{2} group over F_{3} using a spectral sequence in fusion systems

Let p be a prime number, let F_{p} denote the finite field of p elements and let G be a p-group. Our aim is to compute the cohomology algebra H*(G; F_{p}) using spectral sequences. When G contains a non-trivial normal subgroup N, the Lyndon-Hochschild-Serre spectral sequence allows us computing H*(G;F_{p}) from H*(N; F_{p}) and H*(G/N; F_{p}). However, if G is a simple group, no such a spectral sequence can be used anymore. Recently, in [1], the author constructs a new spectral sequence in fusion systems that can be used for certain simple groups.

In this talk, we shall compute the cohomology algebra of the sporadic Janko two group J_{2} over F_{3} [2] using the aforementioned spectral sequence in fusion systems.

References:

[1] Antonio Díaz Ramos: A spectral sequence for fusion systems, Algebraic and Geometric Topology 14 (2014), 349-378.

[2] Antonio Díaz Ramos, Oihana Garaialde Ocaña: The cohomology of the sporadic group J_{2} over F_{3}, Forum Mathematicum, Volume 28, Issue 1 (2014), 77-87.

## Matthew Gelvin: Minimal characteristic bisets of fusion systems

If G is a finite group with Sylow p-subgroup S, the left and right multiplications of S on G give it a biset structure that is closely connected to the p-fusion system of G. The key properties of this biset were axiomatized by Linckelmann and Webb, resulting in the notion of a characteristic biset for an arbitrary saturated fusion system.

In this talk I will outline joint work with Sune Reeh, which begins by parameterizing the characteristic bisets of a fusion system. An important consequence of this parameterization is the existence of a unique minimal characteristic biset. I will describe how in several respects, the MCB is the smallest group-like structure that induces the fusion system. Of particular note are the cases of constrained fusion systems — where the model actually is the MCB — and the centric linking system associated to a fusion system, which can be viewed as the centric part of the MCB. I will also describe how the construction of normalizers and centralizers can be realized in the context of MCBs, which leads to a pleasing coherence of definition in the case of centric subgroups.

## Jesper Grodal: Burnside rings in algebra and topology

The Burnside ring of a finite group is the group completion of the semi-ring of finite G-sets under direct sum and cartesian product. This ring made its debut into algebraic topology via Segal's equivariant Hopf theorem, identifying the zeroth equivariant stable homotopy group as this ring — it has been a central object in equivariant algebraic topology ever since.

In my two talks I'll survey some of the ways this ring, and its variants, show up in equivariant stable and unstable homotopy theory. In particular I'll look at the difference between "genuine" and "derived" equivariant homotopy theory. Stably this is dictated by the classical Segal conjecture proved by Carlsson in the 80's, whose modern formulation involves fusion systems. I'll also explain a more refined "uncompleted" version of this result, lying between stable and unstable, that I recently obtained.

## Isabelle Laude: Maps between (uncompleted) classifying spaces of p-local finite groups

In the literature there are many results concerning the space of maps between p-completed classifying spaces of p-local finite groups, most notable work of Dwyer-Zabrodsky, Mislin and Broto-Levi-Oliver, but very little is known in the uncompleted case. In this talk I will present some of the first complete calculations in the uncompleted case and relate them to previously known results.

## Justin Lynd: Control of fixed points and centric linking systems

The centric linking system of a saturated fusion system is an extension category that provides the bridge to the classifying space of the fusion system. The unique existence of linking systems was shown by Chermak, and Oliver subsequently showed how to interpret Chermak's proof within the homological obstruction theory for existence/uniqueness of centric linking systems that was outlined early by Broto, Levi, and Oliver. I will discuss some group/representation theoretic aspects of joint work with G. Glauberman that, once plugged in to the Chermak-Oliver framework, help to give a proof of Chermak's theorem that does not depend on the classification of the finite simple groups. If time permits, I will explain how Chermak's method of proof and an old result of Glauberman help to shed some additional light on automorphisms of linking systems.

## Rémi Molinier: Cohomology with twisted coefficients of linking systems and stable elements

A theorem of Boto, Levi and Oliver describes the cohomology of the geometric realization of a linking system, with trivial coefficients, as the submodule of stable elements in the cohomology of the Sylow. When we are looking at twisted coefficients, the formula can not be true in general as pointed out by Levi and Ragnarsson but we can try to understand under which condition it holds. In this talk we will see some conditions under which we can express the cohomology of a linking system as stable elements.

Video recording [Unfortunately there is no sound starting from minute 30]

## Bob Oliver: Local structure of finite groups and of their p-completed classifying spaces

I will describe the close connection between the homotopy theoretic properties of the p-completed classifying space of a finite group G and the p-local group theoretic properties of G. One way in which this arises is in the following theorem originally conjectured by Martino and Priddy: for finite groups G and H, BG^{v}_{p} ≃ BH^{v}_{p} if and only if G and H have the same p-local structure (the same conjugacy relations among p-subgroups). Another involves a description, in terms of the p-local properties of G, of the group Out(BG^{v}_{p}) of homotopy classes of self equivalences of the space BG^{v}_{p}.

After stating some general results, I'll give a few examples and applications of both of these, especially in the case where G and H are finite simple groups of Lie type.

## Sejong Park: Double Burnside rings and Mackey functors with applications to fusion systems

(Globally defined) Mackey functors appear naturally as, for example, cohomology and representation rings for finite groups. They can be viewed as additive functors defined on certain categories of finite groups whose endomorphism rings of objects are double Burnside rings. Mackey functors can be defined for fusion systems; also fusion systems can be viewed as idempotents in the double Burnside rings of finite p-groups. Using Mackey functors for fusion systems we will extend Dwyer's sharpness result on homology decomposition of classifying spaces of finite groups to some exotic fusion systems (joint with Antonio Díaz). Also, we will study the structure of the double Burnside rings of some finite groups with "ghost maps", identifying their idempotents and simple and projective modules (joint with Goetz Pfeiffer).

## Radu Stancu: Fusion systems: survival kit

For p a prime number, fusion systems on a finite p-group were introduced by Puig, as an axiomatization of the p-local structure of a finite group and of a block algebra of a finite group. Broto, Levi and Oliver, aiming to solve the Martino-Priddy conjecture, independently developed the notion of fusion systems and constructed their homotopy theory. With this new approach, that helped in reformulating the conjecture, and using the classification of finite simple groups, Oliver succeeded in proving the Martino-Priddy conjecture. Since then, works by Chermak, Oliver and, jointly, by Glauberman and Lynd removed the dependence of the proof on the classification. Widely speaking, the Martino-Priddy conjecture-now-theorem claims that the p-local structure of a finite group G, i.e. the fusion system on a Sylow p-subgroup given by the conjugations in G is equivalent to the p-local structure on the classifying space BG, i.e. its p-completion.

In this introductory talk we define the fusion systems and their saturation and aim to give, through examples, their basic properties and some topological sides of the story.

## Radu Stancu: Saturation and the double Burnside ring

When creating the homotopy theory of fusion systems Broto, Levi and Oliver introduced the notion of a characteristic biset of a fusion system. As a basic example, a finite group G with Sylow p-subgroup S is a characteristic (S,S)-biset for the fusion system of the group G on S. In general, such a characteristic biset always exists for a saturated fusion system, even though it need not be unique. If one allows p-local coefficients, Ragnarsson constructed a characteristic idempotent in the double Burnside ring, and proved it is unique. In fact, the saturation of a fusion system and the existence of a characteristic biset are equivalent, as showed in a joint work with Ragnarsson.

In this talk we'll introduce the notion of double Burnside ring and try to explain the strong connection between this ring and the saturation of a fusion system.

## Nathaniel Stapleton: Transchromatic character theory for fusion systems

The transchromatic character maps for Morava E-theory are a generalization of the classical character map from the representation ring of a finite group to class functions. In this talk I will present joint work with Sune Precht Reeh and Tomer Schlank extending the Morava E-theory transchromatic character maps from finite groups to fusion systems. One of the key technical ingredients is a functorial evaluation map from the free loop space of a fusion system times the circle back to the fusion system.

## Ergün Yalcin: Representation rings for fusion systems and dimension functions

I will give a talk on recent joint work with Sune Precht Reeh. In this work, we define the representation ring of a saturated fusion system ℱ as the Grothendieck ring of the semiring of ℱ-stable representations, and study the dimension functions of ℱ-stable representations using the transfer map induced by the characteristic idempotent of ℱ. We find a list of conditions for an ℱ-stable super class function to be realized as the dimension function of an ℱ-stable virtual representation. The main motivation for studying this problem is to find new methods for constructing finite group actions on homotopy spheres with a given isotropy type.