Barcelona Topology Workshop 2014

I will discuss various generalized bundle concepts and present a few speculations and conjectures.

I will present computations of the Picard groups of several spectra of topological modular forms, with and without level structures, periodic and non-periodic. The toolbox for these computations will consist of descent theory and a technical lemma allowing us to compare stable and unstable information in spectral sequences. This is joint work with Akhil Mathew.

Given a prime $p$, a (saturated) fusion system over a finite $p$-group $S$ is a category $\mathcal{F}$ which encode "conjugacy" relations among subgroups of $S$, and which is modelled on the fusion in a Sylow $p$-subgroup of a finite group. As in the case of a finite group, there exists a topological space $B\mathcal{F}$ which plays the same role of the Bousfield-Kan $p$-completion of the classifying space of a finite group.
In this work, we study when we can built $B\mathcal{F}$ from $BP$ by means of pointed homotopy colimits, where $P$ is a finite $p$-group.

Characteristic classes play an important role in the determination of bordism rings and provide an important link to representation theory. I will construct and compute characteristic classes for spin and string bundles with values in the cohomology theory of topological modular forms TMF_0(3). I will describe some applications for string bordism and for positive energy representations of loop groups.

A differential cohomology theory is differential geometric refinement of a generalized cohomology theory (in the sense of algebraic topology). Examples naturally arise in physics or in the study of secondary invariants (e.g. Chern-Simons invariants). We discuss this notion from a higher categorical point of view. This leads to a natural decomposition of any differential cohomology theory which we illustrate with many examples. Moreover we show how to obtain a good integration theory and a notion of twisted differential cohomology and discuss some aspects and examples.

We explain a result of global local rigidity for representations of a 3-manifold group in SL_2(C). More precisely, we show that all the representations of almost all the Dehn fillings of a given knot are infinitesimally rigid. The question arises from asymptotic problems in topological quantum field theory and uses deep result in diophantine geometry known as Zilber-Pink conjectures. This is joint work with G. Maurin.

$L^2$-Betti numbers were defined for discrete groups by Atiyah (1976) and Cheeger- Gromov (1986), for measurable equivalence relations by Gaboriau (2002), and for arbitrary unimodular locally compact groups by Petersen (2012) and Kyed-Petersen-Vaes (2013). In joint work with Henrik Petersen, we give a formula for computing the $n$-th $L^2$-Betti number of a type I, unimodular, locally compact group $G$: it is given as the integral over the dual $\tilde{G}$, with respect to Plancherel measure, of the usual dimension of the reduced $n$-cohomology of $G$ with coefficients in an irreducible representation. Since the latter has been much studied, we are able to compute the $L^2$-Betti numbers of real or p-adic semi-simple Lie groups, and automorphisms groups of locally finite trees that act transitively on the boundary.

Co-Kähler and Sasakian manifolds are two natural extensions of the notion of Kähler manifolds in odd dimensions. The corresponding notions for symplectic manifolds are those of co-symplectic and K-contact manifolds. We analyse the rational homotopy type of manifolds admitting such structures, in particular focusing on the property of formality.