Valerio Proietti
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View article: Chern character for torsion-free ample groupoids
Chern character for torsion-free ample groupoids Open
For an ample groupoid with torsion-free stabilizers, we construct a Chern character map going from the domain of the Baum-Connes assembly map of G to the groupoid homology groups of G with rational coefficients. As a main application, assu…
View article: A geometric Elliott invariant and noncommutative rigidity of mapping tori
A geometric Elliott invariant and noncommutative rigidity of mapping tori Open
We prove a rigidity property for mapping tori associated to minimal topological dynamical systems using tools from noncommutative geometry.More precisely, we show that under mild geometric assumptions, a leafwise homotopy equivalence of tw…
View article: Homology and K-theory of dynamical systems IV. Further structural results on groupoid homology
Homology and K-theory of dynamical systems IV. Further structural results on groupoid homology Open
We consider the homology theory of étale groupoids introduced by Crainic and Moerdijk [A homology theory for étale groupoids. J. Reine Angew. Math. 521 (2000), 25–46], with particular interest to groupoids arising from topological dynamica…
View article: A CATEGORICAL APPROACH TO THE BAUM–CONNES CONJECTURE FOR ÉTALE GROUPOIDS
A CATEGORICAL APPROACH TO THE BAUM–CONNES CONJECTURE FOR ÉTALE GROUPOIDS Open
We consider the equivariant Kasparov category associated to an étale groupoid, and by leveraging its triangulated structure we study its localization at the ‘weakly contractible’ objects, extending previous work by R. Meyer and R. Nest. We…
View article: Hofstadter-Toda spectral duality and quantum groups
Hofstadter-Toda spectral duality and quantum groups Open
The Hofstadter model allows to describe and understand several phenomena in condensed matter such as the quantum Hall effect, Anderson localization, charge pumping, and flat-bands in quasiperiodic structures, and is a rare example of fract…
View article: Homology and K-theory of dynamical systems. IV. Further structural results on groupoid homology
Homology and K-theory of dynamical systems. IV. Further structural results on groupoid homology Open
We consider the homology theory of étale groupoids introduced by Crainic and Moerdijk, with particular interest to groupoids arising from topological dynamical systems. We prove a Künneth formula for products of groupoids and a Poincaré-du…
View article: Homology and $K$-theory of dynamical systems II. Smale spaces with totally disconnected transversal
Homology and $K$-theory of dynamical systems II. Smale spaces with totally disconnected transversal Open
We apply our previous work on the relation between groupoid homology and K -theory to Smale spaces. More precisely, we consider the unstable equivalence relation of a Smale space with totally disconnected stable sets and prove that the ass…
View article: A geometric Elliott invariant and noncommutative rigidity of mapping tori
A geometric Elliott invariant and noncommutative rigidity of mapping tori Open
We prove a rigidity property for mapping tori associated to minimal topological dynamical systems using tools from noncommutative geometry. More precisely, we show that under mild geometric assumptions, an orientation-preserving leafwise h…
View article: Homology and K-theory of dynamical systems III. Beyond stably disconnected Smale spaces
Homology and K-theory of dynamical systems III. Beyond stably disconnected Smale spaces Open
We study homological invariants of étale groupoids arising from Smale spaces, continuing on our previous work, but going beyond the stably disconnected case by incorporating resolutions in the space direction. We show that the homology gro…
View article: Categorical approach to the Baum-Connes conjecture for étale groupoids
Categorical approach to the Baum-Connes conjecture for étale groupoids Open
We consider the equivariant Kasparov category associated to an étale groupoid, and by leveraging its triangulated structure we study its localization at the "weakly contractible" objects, extending previous work by R. Meyer and R. Nest. We…
View article: Homology and K-theory of dynamical systems. II. Smale spaces with totally disconnected transversal
Homology and K-theory of dynamical systems. II. Smale spaces with totally disconnected transversal Open
We apply our previous work on the relation between groupoid homology and K-theory to Smale spaces. More precisely, we consider the unstable equivalence relation of a Smale space with totally disconnected stable sets, and prove that the ass…
View article: Groups with Spanier–Whitehead duality
Groups with Spanier–Whitehead duality Open
Building on work by Kasparov, we study the notion of Spanier–Whitehead [math] -duality for a discrete group. It is defined as duality in the [math] -category between two [math] -algebras which are naturally attached to the group, namely th…
View article: Homology and K-theory of dynamical systems. I. torsion-free ample groupoids
Homology and K-theory of dynamical systems. I. torsion-free ample groupoids Open
Given an ample groupoid, we construct a spectral sequence with groupoid homology with integer coefficients on the second sheet, converging to the K-groups of the (reduced) groupoid C*-algebra, provided the groupoid has torsion-free stabili…
View article: Homology and $K$-Theory of Torsion-Free Ample Groupoids and Smale Spaces
Homology and $K$-Theory of Torsion-Free Ample Groupoids and Smale Spaces Open
Given an ample groupoid, we construct a spectral sequence with groupoid homology with integer coefficients on the second sheet, converging to the $K$-groups of the groupoid C*-algebra when the groupoid has torsion-free stabilizers and sati…
View article: Index theory on the Mi\v{s}\v{c}enko bundle
Index theory on the Mi\v{s}\v{c}enko bundle Open
We consider the assembly map for principal bundles with fiber a countable\ndiscrete group. We obtain an index-theoretic interpretation of this\nhomomorphism by providing a tensor-product presentation for the module of\nsections associated …