315 research outputs found
Foliation groupoids and their cyclic homology
In this paper we study the Lie groupoids which appear in foliation theory. A
foliation groupoid is a Lie groupoid which integrates a foliation, or,
equivalently, whose anchor map is injective. The first theorem shows that, for
a Lie groupoid G, the following are equivalent:
- G is a foliation groupoid,
- G has discrete isotropy groups,
- G is Morita equivalent to an etale groupoid.
Moreover, we show that among the Lie groupoids integrating a given foliation,
the holonomy and the monodromy groupoids are extreme examples.
The second theorem shows that the cyclic homology of convolution algebras of
foliation groupoids is invariant under Morita equivalence of groupoids, and we
give explicit formulas. Combined with the previous results of Brylinski, Nistor
and the authors, this theorem completes the computation of cyclic homology for
various foliation groupoids, like the (full) holonomy/monodromy groupoid, Lie
groupoids modeling orbifolds, and crossed products by actions of Lie groups
with finite stabilizers. Some parts of the proof, such as the H-unitality of
convolution algebras, apply to general Lie groupoids.
Since one of our motivation is a better understanding of various approaches
to longitudinal index theorems for foliations, we have added a few brief
comments at the end of the second section.Comment: 18 page
Deformations of Lie brackets: cohomological aspects
We introduce a new cohomology for Lie algebroids, and prove that it provides
a differential graded Lie algebra which ``controls'' deformations of the
structure bracket of the algebroid. We also have a closer look at various
special cases such as Lie algebras, Poisson manifolds, foliations, Lie algebra
actions on manifolds.Comment: 17 pages, Revised version: small corrections, more references adde
On the developability of subalgebroids
In this paper, the Almeida-Molino obstruction to developability of
transversely complete foliations is extended to Lie groupoids
On the integrability of subalgebroids
Let G be a Lie groupoid with Lie algebroid g. It is known that, unlike in the
case of Lie groups, not every subalgebroid of g can be integrated by a
subgroupoid of G. In this paper we study conditions on the invariant foliation
defined by a given subalgebroid under which such an integration is possible. We
also consider the problem of integrability by closed subgroupoids, and we give
conditions under which the closure of a subgroupoid is again a subgroupoid
Representing topoi by topological groupoids
It is shown that every topos with enough points is equivalent to the classifying topos of a topological groupoid
General static spherically symmetric solutions in Horava gravity
We derive general static spherically symmetric solutions in the Horava theory
of gravity with nonzero shift field. These represent "hedgehog" versions of
black holes with radial "hair" arising from the shift field. For the case of
the standard de Witt kinetic term (lambda =1) there is an infinity of solutions
that exhibit a deformed version of reparametrization invariance away from the
general relativistic limit. Special solutions also arise in the anisotropic
conformal point lambda = 1/3.Comment: References adde
Relative compactness conditions for topos
In this paper a systematic study is made of various notions of proper map
in the context of toposes
Modulo some separation conditions a proper map Y X of spaces is generally understood to be a continuous function which preserves compactness of subspaces under inverse image and which therefore in particular has compact bers In this spirit a rst denition of proper map between toposes was put forward by Johnstone in
There a map of toposes fF E was called proper if fF is a compact lattice object in the topos E This is probably the most direct way of expressing that F is compact when viewed as a topos over the base E In fact Johnstone used the term perfect
rather than
proper
and developed the theory mostly in the context of localic maps between toposes se
Minimal fibrations of dendroidal sets
We prove the existence of minimal models for fibrations between dendroidal sets in the model structure for
∞–operads, as well as in the covariant model structure for algebras and in the stable one for connective spectra. We also explain how our arguments can be used to extend the results of Cisinski (2014) and give the existence of minimal fibrations in model categories of presheaves over generalized Reedy categories of a rather common type. Besides some applications to the theory of algebras over ∞–operads, we also prove a gluing result for parametrized connective spectra (or
Γ–spaces)
On the universal enveloping algebra of a Lie-Rinehart algebra
We review the extent to which the universal enveloping algebra of a
Lie-Rinehart algebra resembles a Hopf algebra, and refer to this structure as a
Rinehart bialgebra. We then prove a Cartier-Milnor-Moore type theorem for such
Rinehart bialgebras
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