In this paper we define new K-theoretic secondary invariants attached to a Lie groupoid G. The receptacle for these invariants is the K-theory of Cr⁎(Gad∘) (where Gad∘ is the adiabatic deformation G ...restricted to the interval 0,1)). Our construction directly generalises the cases treated in 29,30, in the setting of the Coarse Geometry, to more involved geometrical situations, such as foliations. Moreover we tackle the problem of producing a wrong-way functoriality between adiabatic deformation groupoid K-groups associated to transverse maps. This extends the construction of the lower shriek map in 6. Furthermore we prove a Lie groupoid version of the Delocalized APS Index Theorem of Piazza and Schick. Finally we give a product formula for secondary invariants.
Multiplicative Ehresmann connections Loja Fernandes, Rui; Mărcuţ, Ioan
Advances in mathematics (New York. 1965),
08/2023, Volume:
427
Journal Article
Peer reviewed
Open access
We develop the theory of multiplicative Ehresmann connections for Lie groupoid submersions covering the identity, as well as their infinitesimal counterparts. We construct obstructions to the ...existence of such connections, and we prove existence for several interesting classes of Lie groupoids and Lie algebroids, including all proper Lie groupoids. We show that many notions from the theory of principal bundle connections have analogues in this general setup, including connections 1-forms, curvature 2-forms, Bianchi identity, etc. In 19 we provide a non-trivial application of the results obtained here to construct local models in Poisson geometry and to obtain linearization results around Poisson submanifolds.
Let Σ be a compact connected and oriented surface with nonempty boundary and let G be a Lie group equipped with a bi-invariant pseudo-Riemannian metric. The moduli space of flat principal G-bundles ...over Σ which are trivialized at a finite subset of ∂Σ carries a natural quasi-Hamiltonian structure which was introduced by Li-Bland and Ševera. By a suitable restriction of the holonomy over ∂Σ and of the gauge action, which is called a decoration of ∂Σ, it is possible to obtain a number of interesting Poisson structures as subquotients of this family of quasi-Hamiltonian structures. In this work we use this quasi-Hamiltonian structure to construct Poisson and symplectic groupoids in a systematic fashion by means of two observations:(1)gluing two copies of the same decorated surface along suitable subspaces of their boundaries determines a groupoid structure on the moduli space associated to the new surface, this procedure can be iterated by gluing four copies of the same surface, thereby inducing a double Poisson groupoid structure;(2)on the other hand, we can suppose that G is a Lie 2-group, then the groupoid structure on G descends to a groupoid structure on the moduli space of flat G-bundles over Σ. These two observations can be combined to produce up to three distinct and compatible groupoid structures on the associated moduli spaces. We illustrate these methods by considering symplectic groupoids over Bruhat cells, twisted moduli spaces and Poisson 2-groups besides the classical examples.
In this paper we study the notion of representation up to homotopy of a Lie groupoid and the resulting derived category, and show that the adjoint representation is well defined as a representation ...up to homotopy. As an application, we extend Bottʼs spectral sequence converging to the cohomology of classifying spaces of Lie groups to the case of Lie groupoids. We explain the relation of this construction with the models of Cartan and Getzler for equivariant cohomology. Our work is closely related to and inspired by Behrendʼs 3, Bottʼs 4, and Getzlerʼs 9.
A stratified space is a kind of topological space together with a partition into smooth manifolds. These kinds of spaces naturally arise in the study of singular algebraic varieties, symplectic ...reduction, and differentiable stacks. In this paper, we introduce a particular class of stratified spaces called stratified vector bundles, and provide an alternate characterization in terms of monoid actions. We will then provide large families of examples coming from the theory of Whitney stratified spaces, singular foliation theory, and equivariant vector bundle theory. Finally, we extend functorial properties of smooth vector bundles to the stratified case.
Cosymplectic groupoids Loja Fernandes, Rui; Iglesias Ponte, David
Journal of geometry and physics,
10/2023, Volume:
192
Journal Article
Peer reviewed
Open access
A cosymplectic groupoid is a Lie groupoid with a multiplicative cosymplectic structure. We provide several structural results for cosymplectic groupoids and we discuss the relationship between ...cosymplectic groupoids, Poisson groupoids of corank 1, and oversymplectic groupoids of corank 1.
In this paper, we introduce a notion of a principal 2-bundle over a Lie groupoid. For such principal 2-bundles, we have produced a short exact sequence of VB-groupoids, namely, the Atiyah sequence. ...Two notions of connection structures viz. strict connections and semi-strict connections on a principal 2-bundle arising respectively, from a retraction of the Atiyah sequence and a retraction up to a natural isomorphism have been introduced. We have constructed a class of principal G=G1⇉G0-bundles and connections from a given principal G0-bundle E0→X0 over X1⇉X0 with connection. An existence criterion for the connections on a principal 2-bundle over a proper, étale Lie groupoid is proposed. We have studied the action of the 2-group of gauge transformations on the category of strict and semi-strict connections. Finally, we have observed an extended symmetry of the category of semi-strict connections.
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