Barcelona Topology Workshop 2012

One of the method to solve Deligne's conjecture is to give a description of the little disc operad in terms of compactification of points in the plane followed by a filtration on this compactification. This has been suggested by Getzler and Jones and "corrected" by Voronov.
In this talk we will first give some details on the classical case, the little disc operad and the Deligne's conjecture. We will then define the Swiss-Cheese operad, compute its homology and explain why it is not so easy to adapt the ideas of Getzler and Jones to prove the Swiss-Cheese conjecture. This is a joint work with E. Hoefel.

I’ll discuss some Galois theoretic and Mackey functor properties of Lubin-Tate theory and Morava K-theory of classifying spaces.
The Good: in certain circumstances the covering EG to BG induces faithful extensions of commutative E_n or K_n-algebras. This is opposite to the situation with HF_p. The Bad: these are not Galois extensions even though the analogous extensions are Galois for HF_p.
The Mackey: the Green functor K_n^*(B(-)) has interesting properties which I’ll discuss. In particular, it extends to saturated fusion systems and is also a global Mackey functor with induction defined for all homomorphisms of finite groups.

This is joint work with Emmanuel Dror Farjoun. The question which lies at the origin of this project is the following: start with an extension of groups and suppose we know that the quotients by the lower central series form again an extension of (nilpotent) groups. Now pull the original extension back, along any group homomorphism.
What can be said about the nilpotent quotient of this extension? We provide complete answers for this question, as well as for the homotopical analogous question hinted at in the title.

Jim Stasheff gave two apparently distinct definitions for A-infinity form in his papers in '63 and '70. The papers claim that the two definitions are equivalent, while it is not apparently clear. In this talk, we show that it is actually true in a slightly weaker form under small additional assumptions. To treat units in A-infinity form defined in '63, we need to construct Associahedra and Multiplihedra as convex polytopes with piecewise linear decompositions, which is performed in my master thesis in '83 or a paper with Mimura in '89, while we do not know much about the construction of Associahedra given in a unpublished work by Haiman in '84. We shall follow the construction of Associahedra and Multiplihedra in '89.

Given discrete groups Q and G, and an action of Q on G, the homotopy fixed point set of BG by the action of Q can be easily described as a disjoint union of classifying spaces of subgroups of G. In a joint work with Carles Broto, we study the situation where G is replaced by a p-local finite group and Q is replaced by a p-group.
The language of p-local finite groups was developed by C. Broto, R. Levi and B. Oliver (among others) in order to study the homotopy type of p-completed classifying spaces of finite groups, where a p-completed classifying space is a space whose cohomology is concentrated at the prime p. In a rather loose way, a p-local finite group can be thought of as a category satisfying a certain set of axioms, and its classifying space is thus the p-completion of the nerve of such category.
The language of localities was developed more recently by A. Chermak as a tool to prove a conjecture regarding p-local finite groups, and in developing this language, he has given the theory of p-local finite groups a flavor closer to group theory. One can think of a locality as a set with a multiplication, but where some of the products are not defined, plus a certain set of axioms.
Combining these two languages and the tools that both of them provide, we can then describe to homotopy fixed point set of the classifying space of a p-local finite group by a p-group as a disjoint union of classifying spaces of p-local finite subgroups, just as it happens in the case of discrete groups.

Sufficiently nice schemes can be reconstructed from their symmetric monoidal categories of quasicoherent sheaves. One approach to noncommutative geometry exploits this and works with module categories in place of actual noncommutative geometric objects. In this world, there is a “Zariski” topology given by localization, as well as a stabilization akin to that of Morel-Voevodsky in the commutative case. In this talk, we will consider the functors represented by the stabilizations of some standard noncommutative varieties, such as affine space, projective space, and the multiplicative group.

Units have often been neglected when dealing with strongly homotopy associative algebras, a.k.a. A-infinity algebras. We are allowed to do this in classical algebra because being unital is a property there. However, for A-infinity algebras being unital is a structure. The existence of a model structure on non-symmetric operads allows the construction of moduli spaces of unital and non-unital A-infinity algebras, as well as a comparison morphism between them, modelling the obvious forgetful functor. We will show under mild assumptions that the comparison morphism is essentially an inclusion of connected components.
We will also present a generalization of a result due to Rezk which allows the construction of moduli spaces of algebras over an operad in a homotopical algebraic geometry context in the sense of Toën and Vezzosi. The result mentioned in the previous paragraph can be interpreted in this context as follows: The moduli stack of unital A-infinity algebras over a fixed object is a formal Zariski open subset of the corresponding moduli stack of non-necessarily unital A-infinity algebras. The analogue for associative algebra structures on a finite-dimensional vector space was long ago noticed by Gabriel.