Homotopy Structures in Barcelona Conference (HoStBCN)

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.