Orbispaces as differentiable stratified spaces

Crainic, Marius; Mestre, João Nuno

doi:10.1007/s11005-017-1011-6

Cited by 21 publications

(40 citation statements)

References 51 publications

(123 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…2.25 are surjective submersions with the following condition: these maps are transversal to the orbits of the Lie groupoids (G and H respectively) and meet every orbit. This fact can be found in [17] and [8], and follows also from proposition A.5.…”

Section: A Second Look At Hausdorff Morita Equivalence Of Singular Fosupporting

confidence: 70%

Hausdorff Morita equivalence of singular foliations

Garmendia

Zambon

2018

Ann Glob Anal Geom

View full text Add to dashboard Cite

We introduce a notion of equivalence for singular foliations -understood as suitable families of vector fields -that preserves their transverse geometry. Associated to every singular foliation there is a holonomy groupoid, by the work of Androulidakis-Skandalis. We show that our notion of equivalence is compatible with this assignment, and as a consequence we obtain several invariants. Further, we show that it unifies some of the notions of transverse equivalence for regular foliations that appeared in the 1980's. A Appendix 34 A.1 Morita equivalence for open topological groupoids and Lie groupoids . . . . 34 Definition 1.12. Given a Lie groupoid G ⇒ M and a surjective submersion π :is the space of arrows of a Lie groupoid over P . (The source and target maps are the first and third projections and the multiplication is induced by the multiplication in G). This Lie groupoid is called the pullback groupoid of G by π.Definition 1.13. Given a Lie algebroid A over a manifold M with anchor # : A → T M , and a surjective submersion π : P → M , one checks that π −1 A := π * (A) # × dπ T P is the total space of a vector bundle over P . It has a natural Lie algebroid structure, with anchor# := pr 2 : π −1 A → T P being the second projection. The Lie bracket is determined by its restriction to "pullback sections", which is given by the Lie brackets in X(P ) and Γ(A). We call this Lie algebroid the pullback algebroid of A over π.These two definitions are nicely related by the following lemma:Lemma 1.14. Consider a surjective submersion π : P → M .(i) Let G be a Lie groupoid over M , denote by A its Lie algebroid. The Lie algebroid of the Lie groupoid π −1 G is π −1 A.(ii) Let A be an integrable Lie algebroid over M , denote by G the source simply connected Lie groupoid integrating it. If the map π has simply connected fibers, then the source simply connected Lie groupoid integrating π −1 A is π −1 G.Proof. The proof of part (i) can be found in [17, §4.3], so we address only the proof of part (ii). The Lie groupoid π −1 G integrates π −1 A by part (i). Therefore we need to only show that π −1 G is source simply connected. Take p ∈ P . Its source fiber is s −1 (p) = {(q, g, p) : π(p) = s(g) and π(q) = t(g)} P π × t s −1 (π(p)).The canonical submersion s −1 (p) → s −1 (π(p)) has simply connected fibers, since the πfibers are simply connected. Using that s −1 (π(p)) is simply connected and lemma 1.11 we conclude that s −1 (p) is simply connected.

show abstract

Section: A Second Look At Hausdorff Morita Equivalence Of Singular Fosupporting

confidence: 70%

Hausdorff Morita equivalence of singular foliations

Garmendia

Zambon

2018

Ann Glob Anal Geom

View full text Add to dashboard Cite

show abstract

“…inherits from X × R N a structure of a C ∞ -manifold. A good reference for this is the nice extended survey of Crainic and Mestre, [9], that clarifies and explains very interesting results on Lie groupid theory that were confusing in the litterature, in particular they explain the role of the linearization theorem for proper Lie groupoids (theorem 2 in ref.cit.) on the local structure of such groupoids and on their orbit spaces.…”

Section: The Orbit Space O X As a Manifold With Cornersmentioning

confidence: 89%

On Fredholm boundary conditions on manifolds with corners, I: Global corner cycles obstructions

2021

View full text Add to dashboard Cite

Given a connected manifold with corners of any codimension there is a very basic and computable homology theory called conormal homology defined in terms of faces and orientations of their conormal bundles, and whose cycles correspond geometrically to corner's cycles. Our main theorem is that, for any manifold with corners X of any codimension, there is a natural and explicit morphismoperators for X and the periodic conormal homology group with rational coeficients, and that T is a rational isomorphism. As shown by the first two authors in a previous paper this computation implies that the rational groups H pcn ev (X, Q) provide an obstruction to the Fredholm perturbation property for compact connected manifold with corners. The difference with respect to the previous article of the first two authors in which they solve this problem for low codimensions is that we overcome in the present article the problem of computing the higher spectral sequence K-theory differentials associated to the canonical filtration by codimension by introducing an explicit topological space whose singular cohomology is canonically isomorphic to the conormal homology and whose K-theory is naturally isomorphic to the K−theory groups of the algebra K b (X).

show abstract

“…We construct a continuous version of the space described in Example 4.14. Following [10], starting with the principal Z/2-bundle given by the double cover S 1 → S 1 /(Z/2), we construct a transitive groupoid G with G 0 = S 1 /(Z/2) and G 1 = (S 1 × S 1 )/(Z/2), the orbit space of the diagonal Z/2-action on…”

Section: Bredon Homology and Cohomologymentioning

confidence: 99%

Classifying spaces and Bredon (co)homology for transitive groupoids

Farsi

Scull

Watts

2020

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

We define the orbit category for transitive topological groupoids and their equivariant CWcomplexes. By using these constructions we define equivariant Bredon homology and cohomology for actions of transitive topological groupoids. We show how these theories can be obtained by looking at the action of a single isotropy group on a fiber of the anchor map, extending equivariant results for compact group actions. We also show how this extension from a single isotropy group to the entire groupoid action can be applied to the structure of principal bundles and classifying spaces.Date: June 5, 2019.

show abstract

Orbispaces as differentiable stratified spaces

Cited by 21 publications

References 51 publications

Hausdorff Morita equivalence of singular foliations

Hausdorff Morita equivalence of singular foliations

On Fredholm boundary conditions on manifolds with corners, I: Global corner cycles obstructions

Classifying spaces and Bredon (co)homology for transitive groupoids

Contact Info

Product

Resources

About