Barcelona Topology Workshop 2015

I will sketch a short proof of Waldhausen Additivity for Waldhausen quasicategories. This proof is entirely in the world of quasicategories and simplicial sets. As a special case, Waldhausen K-theory sends every split exact sequence of stable quasicategories A ⟶ E ⟶ B to a stable equivalence of spectra K(A) v K(B) ⟶ K(E). I will also sketch how S_•^∞ is an (∞,2)-functor.

I will talk about the following theorem. Let E be an oriented vector bundle over a compact oriented manifold M, and assume that the Euler number of E is nonzero. Then there exists some constant C such that: for any odd prime p and any smooth action of a finite p-group G on M which lifts to an action on E there is some point in M whose stabiliser has index at most C in G. For p=2 the same statement holds with the extra requirement that the action leaves invariant an almost complex structure on M. I will explain some consequences of this theorem on actions of arbitrary finite groups on symplectic manifolds.

While the concept of exotic p-local finite group is clearly defined, in the compact case there are several families of groups giving rise to p-local compact groups, blurring this way the condition of being exotic. In this talk we will present two families of p-local compact groups which are exotic in the sense that they do not come from compact Lie groups nor p-compact groups.

In a recent set of collaborations with C. Blanchet, N. Geer, B. Patureau, we developed a new family of so-called "non semi-simple" topological quantum field theories which have remarkable properties and differences with respect to the famous Reshetikhin-Turaev TQFTs. In this talk I will first recall the general machine of the "universal construction" which allows to produce topological quantum field theories out of invariants of three-manifolds. Then I will outline how to get invariants of manifolds out of the category of representations of the quantum group U_q(sl₂), comparing the Reshetikhin-Turaev case with the non semi-simple one. I will point out some of the key aspects of the invariants of three manifolds obtained this way, and I will explain why these points induce remarkable properties on the TQFTs in particular for what concerns representations of mapping class groups. If time permits I will also comment on the relation existing between these TQFTs and the volume conjecture.

The AKSZ construction, as implemented by Pantev-Toën-Vaquié-Vezzosi in the context of derived algebraic geometry, gives a symplectic structure on the derived stack of maps from an oriented compact manifold to a symplectic derived stack. I will discuss how this gives rise to a family of extended topological field theories valued in higher categories of symplectic derived stacks, with the higher morphisms given by a notion of iterated Lagrangian correspondences. This is ongoing work with Damien Calaque and Claudia Scheimbauer.

In this talk we discuss categories of (colored) wheeled properads over a suitable monoidal model category (e.g. simplicial sets, symmetric spectra, chain complexes over a characteristic zero field). These categories, with both fixed and varying color sets, admit Quillen model structures. In the latter case, this model structure is an extension of the known model structures on small enriched categories. It turns out that these model structures are not left proper, but they satisfy a weaker version. Finally, we discuss change of base category.

A (complex) homotopy representation of a compact connected Lie group G is a map f : BG → BU(d). For every homotopy representation f : BG → BU(n), we can assign its character — an element χ(f) ∈ R(G) ≃ R(T)^W represented by a homomorphism φ : T → U(d) such that Bφ ≃ f|_BT. The natural question is which virtual characters α ∈ R(G) are homotopy characters, i.e., the characters of homotopy representations. The obvious constraint, implied by the Dwyer-Zabrodsky-Notbohm theorem, is that the restriction of α to any p-toral subgroup of G is the character of a representation; such characters we call P-characters. We say that a P-character μ has the splitting property if it satisfies the following condition. For every P-character ν such that μ + ν is a homotopy character, also ν is a homotopy character. I will prove that the character of the identity representation id : U(n) → U(n) has the splitting property. Next, I will apply this result to construct new examples of homotopy representations of unitary groups. Finally, I will classify homotopy representations of U(n) having dimensions less than ½ n³ .

(joint with M. Anel, E. Finster, and A. Joyal) Factorization hierarchies axiomatize a proof of the Blakers-Massey theorem found by Finster/Lumsdaine in the language of Homotopy Type Theory. We show that the Goodwillie tower yields a factorization hierarchy. As a consequence we can prove a Blakers-Massey analogue in the context of calculus of homotopy functors conjectured by Goodwillie.