Homotopy Structures in Barcelona Conference (HoStBCN)

Using rational algebraic models, we look at commutative ring structures on real and complex equivariant K-theory. Joint work with Bohmann, Hazel, Ishak, and Kędziorek showed uniqueness of both the naive and the genuine commutative ring structures on rational for G finite abelian. In joint work in progress with Bohmann and Kędziorek, we aim to extend these results to KOG.

A symmetric (simplicial) set is a presheaf on the category of nonempty finite sets and all functions. The Segal symmetric sets are just the nerves of groupoids. We consider two weakenings of the Segal condition for symmetric sets. The first defines partial groupoids, first thought up and studied by Chermak in p-local finite group theory. The second defines the d-Segal spaces of Dyckerhoff–Kapranov and (for d = 2) Gálvez-Carrillo–Kock–Tonks. This talk is about some useful tools to understand “how higher Segal” a partial group is. They are based on the discrete geometry of partial group actions and ultimately involve solving Helly type problems in abstract closure spaces. I’ll explain some of the computations we’ve done to compute the degree of Segality of interesting partial groups, including some computations that are still in progress for partial groups arising in p-local finite group theory. This is joint work with Philip Hackney.

For topological spaces with an action of a group G, an isovariant map is an equivariant map which preserves isotropy subgroups. Isovariant maps play an important role in equivariant surgery theory and equivariant h-cobordism theory. In this talk, I will discuss joint work with Sarah Yeakel, in which we study the homotopy theory of G-spaces with isovariant maps. Results discussed may include an isovariant Whitehead’s theorem, an application to isovariant fixed point theory, and the beginnings of isovariant stable homotopy theory.

• locally finite if every finitely generated subgroup of G is finite;
• a p-group (p prime) if every element of G has p-power order; and
• artinian if descending chains of subgroups of G become constant.

For a prime p, we say that G is p-artinian if every p-subgroup of G is artinian. If G is locally finite and p-artinian, then every p-subgroup of G is locally finite and artinian, and hence is discrete p-toral by standard results.
After recalling some of this background on fusion systems and their classifying spaces, I will describe some of our recent work, where we showed how to associate a fusion system to each locally finite p-artinian group G. The first problem was that such groups need not have Sylow p-subgroups, at least not in the strong sense: there need not be a p-subgroup of G that contains all others up to conjugacy. Instead, we defined the concept of a “weakly Sylow p-subgroup” of G, and showed it is unique up to an appropriately defined relation of “local” conjugacy. We then showed that if G is locally finite and p-artinian and S ≤ G is a weakly Sylow p-subgroup, then there is a saturated fusion system over S (explicitly defined), and its classifying space has the homotopy type of ​. The definitions of and are similar to those used for finite groups, but with extra complications caused by the weaker definitions of Sylow subgroups and conjugacy. To prove that ​, we needed to apply a recent theorem of Alex Gonzalez: a “stable elements theorem” that describes as a subgroup of .

We study homological and cohomological Bousfield classes in Bousfield localizations of rigidly-compactly generated tensor–triangulated categories via the homological spectrum. When applied to chromatic homotopy theory, our work gives a negative answer to a question of Wolcott.

In the 1980s, after having introduced Intersection Cohomology, M. Goresky and R. MacPherson proposed a number of problems and conjectures regarding homotopical foundations for this cohomological theory. Such an homotopical enhancement will have some potential applications to the study of singular sopaces in geometry.
In this talk I will give a survey of such an enhancement based on a simplicial approach to Intersection Cohomology and on stratified homotopy theory. In particular I will discuss a “motivic like” approach to the subject developped in collaboration with S. Douteau.

For a closed, topological $n$-manifold $M$, let $E(M)$ denote its space of pointed self-homotopy equivalences. In [Hambleton-Kreck, 2004] a braid of interlocking exact sequences was established in order to obtain new information about $Aut(M):=\pi_0(E(M))$, assuming that $M$ is a closed, oriented $4$-manifold and the self-equivalences are orientation-preserving. It seemed clear to the authors at the time that a similar braid should exist for higher dimensional manifolds.

In this project we carry out the details of this extension (at the space-level), and construct a homotopy (highly) cartesian square relating the space of homotopy self-equivalences of $M$ to an infinite loop space model for the associated bordism theory of the normal $k$-type of $M$.

This leads to a conceptual explanation for the existence of the Hambleton—Kreck braid in the $4$-manifold setting, and to a broad generalization of this tool to include information about the higher homotopy groups $\pi_k(E(M))$ and related variants.

This is joint work with Kursat Sozer (McMaster) and Robin Sroka (Muenster).

Baez and Dolan introduced in the late 90’s the notion of groupoid cardinality of finite groupoids, and more generally of π-finite spaces. Analogously to the Euler characteristic of finite spaces, which behaves “additively” under
finite (homotopy) colimits, the groupoid cardinality of π-finite spaces behaves multiplicatively under finite (homotopy) limits.
However, while the Euler characteristic of finite spaces can be seen as the shadow of a higher invariant (a map of spaces from the space of finite spaces to the algebraic K-theory space of finite spaces, aka the A-theory space of a point), Baez and Dolan’s groupoid cardinality has not been “homotopized”.
In this talk, I will propose a definition of higher groupoid cardinality, which would be to classical groupoid cardinality what the refinement to a point in the A-theory space of a point is to classical Euler characteristics, and show that this definition, while a priori “higher” actually produces a discrete object: there is no higher groupoid cardinality (and I will explain how this is relatively “robust”, i.e. not so dependent on the precise definition of higher groupoid cardinality).
Time permitting, I will discuss some speculations and conjectures about combining higher groupoid cardinality and euler characteristics in a single object.

Equivariant Loday constructions are a means to providing geometric interpretations of equivariant homology theories such as topological Real Hochschild homology. For the family of dihedral groups Angelini-Knoll, Gerhardt and Hill defined Real D2m-Hochschild homology groups for discrete Eσ-rings. In joint work with Ayelet Lindenstrauss and Foling Zou we show that these have an interpretation as the homotopy groups of an equivariant Loday construction where we consider a D2m-action on the 1-skeleton of a regular 2m-gon. To that end we need to generalize equivariant Loday constructions so that they only take into account the isotropy subgroups of the G-simplicial set. If time permits, we will also present a family of examples related to the symmetric group actions on permutohedra.

The Balmer spectrum of finite rational G-spectra is a space X(G) of conjugacy classes of subgroups. The space X(G) is a disjoint union of blocks V(G,H) of subgroups dominated by H, and there is a growing list of groups for which this known explicitly (for example 18 blocks when G=SU(3)). There are many cases where the algebraic model for rational G-spectra is known to be a category of sheaves over X(G). The talk will describe the general shape of V(G,H) and make it explicit in some examples. The algebraic models will be also be described in various cases.