Groupoids and Noncommutative Geometry Seminar 

I will explain the notion of extension for Lie algebroids, and introduce Ehresmann connections in this context. The approach includes non-abelian extensions, and Lie algebroids over different bases. Particular attention will be payed to examples. The talk will mainly rely on recent work that can be found at the following link: http://eprintweb.org/S/article/math/0810.1462

I will explain the notion of extension for Lie algebroids, and introduce Ehresmann connections in this context. The approach includes non-abelian extensions, and Lie algebroids over different bases. Particular attention will be payed to examples. The talk will mainly rely on recent work that can be found at the following link: http://eprintweb.org/S/article/math/0810.1462

Haefliger's classification theory of foliations on open manifolds, based on Gromov's h-principle, will be described. Time allowing, we will also focus on problems of integrability of almost symplectic and almost complex structures on open manifolds.

References:

• Haefliger, «Feuilletages sur les varietes ouvertes», Topology 9 (1970), 183-194
• Haefliger, «Homotopy and Integrability», Manifolds — Amsterdam 1970, Springer LNM 197 (1971)
• Haefliger, «Lectures on the theorem of Gromov», Symposium on singularities, Liverpool 1970, Springer LNM 209 (1971)
• Gromov, «Stable mappings of foliations into manifolds», Izv. Akad. Nauk USSR 33 (1969), No. 4, 671-693
• Gromov, «Transversal mappings of foliations», Dokl. Akad. Nauk USSR 182 (1968), No. 2, 1126-1129
• Landweber, «Complex structures on open manifolds», Topology 13 (1974), 69-75
• Lawson, «The quantitative theory of foliations», AMS Regional conference series in Mathematics No. 27

We will do a short review of Dirac geometry and introduce coupling Dirac structures. Then we will show how to integrate the so called Yang-Millls-Higgs settings "by hands", using basic techniques involving transitive groupoids and inner group actions on algebroids.

The main concern will be to describe the presymplectic groupoid integrating a coupling Dirac structure, with a particular attention to the so-called Yang-Mills-Higgs settings. Dirac structures are half way between symplectic and Poisson structures since they can be thought of as a foliation by presymplectic leaves. I will recall these notions and introduce the coupling version of a Dirac structure; these are strongly related with symplectic and Poisson fibrations. A particularly interesting situation they apply to is in a geometric description of the neighborhood of a coadjoint orbit.

In this lecture we focus on cohomology of etale groupoids (without smoothness assumptions) and sheaves on these etale groupoids (not representations as before). We give a minimal(/crash) introduction to simplicial objects and geometric realization and apply this to categories and groupoids. Next we make the link to the first series of lectures in this series by explaning the relation to semigroup cohomology (following Renault). If time allows, we also discuss a related approach by Moerdijk using the so-called embedding category of a groupoid.

In this lecture we focus on cohomology of etale groupoids (without smoothness assumptions) and sheaves on these etale groupoids (not representations as before). We give a minimal(/crash) introduction to simplicial objects and geometric realization and apply this to categories and groupoids. Next we make the link to the first series of lectures in this series by explaning the relation to semigroup cohomology (following Renault). If time allows, we also discuss a related approach by Moerdijk using the so-called embedding category of a groupoid.

Every etale topological groupoid $G$ gives rise to an inverse semigroup equipped with a natural representation on the space of units of $G$. The germs of such representation can be given the structure of an etale groupoid which turns out to be isomorphic to $G$. We extend this construction to `wide' inverse semigroups over a topological space, which allows one to effectively construct etale groupoid extensions by extending or modifying the underlying inverse semigroup. We use this machinery in order to provide a simpler construction of Paterson's universal groupoid of an inverse semigroup, and also of the \'etale groupoid that arises by applying Stone--Cech compactification to the unit space of the pair groupoid on a set, which is the translation groupoid of Skandalis, Tu, and Yu used in their study of the Novikov conjecture by coarse geometric methods.

Every etale topological groupoid $G$ gives rise to an inverse semigroup equipped with a natural representation on the space of units of $G$. The germs of such representation can be given the structure of an etale groupoid which turns out to be isomorphic to $G$. We extend this construction to `wide' inverse semigroups over a topological space, which allows one to effectively construct etale groupoid extensions by extending or modifying the underlying inverse semigroup. We use this machinery in order to provide a simpler construction of Paterson's universal groupoid of an inverse semigroup, and also of the étale groupoid that arises by applying Stone-Cech compactification to the unit space of the pair groupoid on a set, which is the translation groupoid of Skandalis, Tu, and Yu used in their study of the Novikov conjecture by coarse geometric methods.

Every etale topological groupoid $G$ gives rise to an inverse semigroup equipped with a natural representation on the space of units of $G$. The germs of such representation can be given the structure of an etale groupoid which turns out to be isomorphic to $G$. We extend this construction to `wide' inverse semigroups over a topological space, which allows one to effectively construct etale groupoid extensions by extending or modifying the underlying inverse semigroup. We use this machinery in order to provide a simpler construction of Paterson's universal groupoid of an inverse semigroup, and also of the étale groupoid that arises by applying Stone-Cech compactification to the unit space of the pair groupoid on a set, which is the translation groupoid of Skandalis, Tu, and Yu used in their study of the Novikov conjecture by coarse geometric methods.

We define the Brauer group \(\operatorname{Br}(G)\) of a locally compact groupoid \($G\) to be the set of Morita equivalence classes of pairs \((\mathcal{A},\alpha)\)consisting of an elementary C*-bundle \(\mathcal{A}\) over \(G^{(0)}\) satisfying Fell\'s condition and an action \(\alpha\) of \(G\) on \(\mathcal{A}\) by \(\ast\)-isomorphisms. When \(G\) is a transformation groupoid, then \(\operatorname{Br}(G)\) is the equivariant Brauer group of [1].

It is shown that \(\operatorname{Br}(G)\) is isomorphic to \(\operatorname{Ext}(G,\boldsymbol{T})\), as defined by Renault. It follows that if \(G\) and \(H\) are equivalent groupoids, then \(\operatorname{Br}(G)\) and \(\operatorname{Br}(H)\) are isomorphic.

If \(G\) is étale, then \(\operatorname{Br}(G) \cong H^2(G, \boldsymbol{T})\), where \(H^\ast(G, \cdot)\) denotes the the natural extension of Grothendieck's equivariant sheaf cohomology to étale groupoids. The assignment of such a cohomology class to a pair \((\mathcal{A},\alpha)\) may be viewed as a generalized Dixmier--Douady invariant.

1. A. Kumjian, P. Muhly, J. Renault, D. Williams. The Brauer group of a locally compact groupoid, Amer. J. Math. 120 (1998) 901—954.

I will explain what is a calculus of fractions in a category. This has to do with inverting some elements in a category. Then I'll show how $\mathrm{KK}$ and $E$-theory for $C^{*}$-algebras can be seen as categories of fractions, and derive some consequences.

The notion of textile system was introduced by M. Nasu in order to analyze endomorphisms of topological Markov chains. It consists of two graphs $G$ and $H$ and two morphisms $p,q:G\to H$, with some extra properties. In the case $p$ and $q$ have the path lifting property, we prove that they induce groupoid morphisms $\pi ,\rho :\Gamma (G)\to \Gamma (H)$ between the corresponding étale groupoids of $G$ and $H$. This way, $C^{*}(G)$ is the C*-algebra of two different Fell bundles over $\Gamma (H)$. It turns out that a textile system determines a first quadrant two-dimensional subshift of finite type, via a collection of Wang tiles, and conversely, any such subshift is conjugate to a textile shift. Our groupoid morphisms and C*-algebras encode the complexity of these two-dimensional subshifts. Several concrete examples will be considered.

I will explain how bitorsors can act on extensions. This enables us, using categorical language, to see the restriction functor introduced in the previous talk as an almost principal bundle. Besides, we will see that its lack of surjectivity can be measured by a 2-cohomology class.

We will explain how Moerdijk tackles this problem in his paper Lie Groupoids, Gerbes and Non-Abelian Cohomology. We will introduce the notion of bitorsors and the way they are involved.