Saturated Fell bundles and their classification
Natã Machado, Stefan Wagner
TL;DR
The paper develops a comprehensive framework for classifying saturated Fell bundles over locally compact groups by encoding symmetry through a Picard homomorphism $\psi:G\to \mathrm{Pic}(B)$ and refining it with factor systems $(B_g,\Psi_{g,h})$. It proves that factor systems yield Fell bundles with unit fiber $B$ and that Fell bundles determine factor systems, enabling existence and classification results via cohomology: in the discrete case, equivalence classes are governed by $H^2(G,UZ(B))_{\psi}$ modulo coboundaries with a vanishing obstruction $\chi(\psi)\in H^3(G,UZ(B))_{\psi}$; in the topological setting, a similar $H^2_c(G,UZ(B))_{\psi}$-action describes equivalence classes up to a fundamental section. The framework also identifies crossed product bundles as a tractable class realized by a pair $(S,\omega)$ encoding twisted action and cocycle data, and it outlines extensions to groupoids and spectral-geometry aspects. Collectively, the results provide a robust cohomological classification of saturated Fell bundles, linking noncommutative principal bundles, Morita theory, and topological group cohomology with practical descriptions via factor systems and crossed products.
Abstract
We present a comprehensive classification theory for saturated Fell bundles over locally compact groups, utilizing data associated with their base group and unit fiber. This framework offers a unified approach to understanding the structure and properties of such bundles, providing key insights into their classification.