The Orbit Space and Basic Forms of a Proper Lie Groupoid

Watts, Jordan

doi:10.1007/978-3-030-87502-2_52

Cited by 6 publications

(1 citation statement)

References 14 publications

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“…To complete the proof of part (c), we are left to show that the symplectic structure on the pair groupoid descends to a symplectic structure on , and that integrates . To see that the restriction of to descends to a -form on , recall that this is equivalent to asking that is basic with respect to the -action on (in the sense of [PPT14, Wat19, Yud16]), which means that , where denote the target and source map of the action groupoid. This equality is readily verified.…”

Section: The Canonical Hamiltonian Stratificationmentioning

confidence: 99%

Stratification of the transverse momentum map

Mol

2024

Compositio Math.

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Given a Hamiltonian action of a proper symplectic groupoid (for instance, a Hamiltonian action of a compact Lie group), we show that the transverse momentum map admits a natural constant rank stratification. To this end, we construct a refinement of the canonical stratification associated to the Lie groupoid action (the orbit type stratification, in the case of a Hamiltonian Lie group action) that seems not to have appeared before, even in the literature on Hamiltonian Lie group actions. This refinement turns out to be compatible with the Poisson geometry of the Hamiltonian action: it is a Poisson stratification of the orbit space, each stratum of which is a regular Poisson manifold that admits a natural proper symplectic groupoid integrating it. The main tools in our proofs (which we believe could be of independent interest) are a version of the Marle–Guillemin–Sternberg normal form theorem for Hamiltonian actions of proper symplectic groupoids and a notion of equivalence between Hamiltonian actions of symplectic groupoids, closely related to Morita equivalence between symplectic groupoids.

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Section: The Canonical Hamiltonian Stratificationmentioning

confidence: 99%

Stratification of the transverse momentum map

Mol

2024

Compositio Math.

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Diffeological coarse moduli spaces of stacks over manifolds

Watts,

Wolbert

2024

Contemporary Mathematics

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In this paper, we consider diffeological spaces as stacks over the site of smooth manifolds, as well as the “underlying” diffeological space of any stack. More precisely, we consider diffeological spaces as so-called concrete sheaves and show that the Grothendieck construction sending these sheaves to stacks has a left adjoint: the functor sending any stack to its diffeological coarse moduli space. As an application, we restrict our attention to differentiable stacks and examine the geometry behind the coarse moduli space construction in terms of Lie groupoids and their principal bundles. Within this context, we define a “gerbe”, and show when a Lie groupoid is such a gerbe (or when a stack is represented by one). Additionally, we define basic differential forms for stacks and confirm in the differentiable case that these agree (under certain conditions) with basic differential forms on a representative Lie groupoid. These basic differentiable forms in turn match the diffeological forms on the orbit space.

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Singular foliations through diffeology

Miyamoto

2024

Contemporary Mathematics

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A singular foliation is a partition of a manifold into leaves of perhaps varying dimension. Stefan and Sussmann carried out fundamental work on singular foliations in the 1970s. We survey their contributions, show how diffeological objects and ideas arise naturally in this setting, and highlight some consequences within diffeology. We then introduce a definition of transverse equivalence of singular foliations, following Molino’s definition for regular foliations. We show that, whereas transverse equivalent singular foliations always have diffeologically diffeomorphic leaf spaces, the converse holds only for certain classes of singular foliations. We finish by showing that the basic cohomology of a singular foliation is invariant under transverse equivalence.

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The Orbit Space and Basic Forms of a Proper Lie Groupoid

Cited by 6 publications

References 14 publications

Stratification of the transverse momentum map

Stratification of the transverse momentum map

Diffeological coarse moduli spaces of stacks over manifolds

Singular foliations through diffeology

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