On Serre Microfibrations and a Lemma of M. Weiss

Raptis, George

doi:10.1017/s0017089516000458

Cited by 2 publications

(2 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, in order for many of the same properties to hold one should put some restrictions on the source and target maps. A nice and fairly large class of maps, which include both fibrations and topological submersions, are homotopic submersions (see [54]), or microfibrations (see [68]). Noohi discusses some appropriate classes of maps to use in [61].…”

Section: Properties Of the Category Of La-groupoidsmentioning

confidence: 99%

The van Est Map on Geometric Stacks

Lackman¹

2022

Preprint

View full text Add to dashboard Cite

We generalize the van Est map and isomorphism theorem in three ways, and we propose a category of Lie algebroids and LA-groupoids (with equivalences). First, we generalize the van Est map from a comparison map between Lie groupoid cohomology and Lie algebroid cohomology to a (more conceptual) comparison map between the cohomology of a stack G and the foliated cohomology of a stack H Ñ G mapping into it. At the level of Lie groupoids, this amounts to describing the van Est map as a map from Lie groupoid cohomology to the cohomology of a particular LA-groupoid. We do this by associating to any (nice enough) homomorphism of Lie groupoids f : H Ñ G a natural foliation of the stack rH 0 {Hs . In the case of a wide subgroupoid H ãÝ Ñ G , this foliation can be thought of as equipping the normal bundle of H with the structure of an LA-groupoid. In particular, this generalization allows us to derive classical results, including van Est's isomorphism theorem about the maximal compact subgroup, which we generalize to proper subgroupoids, as well as the Poincaré lemma; it also gives a new method of computing Lie groupoid cohomology.Secondly, we generalize the functions that we can take cohomology of in the context of the van

show abstract

Section: Properties Of the Category Of La-groupoidsmentioning

confidence: 99%

The van Est Map on Geometric Stacks

Lackman¹

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Note that the fiber over a ∈ Ψ(Q ⊂ M ) is given by Ψ(I k × I n−k rel a), and similarly for Φ, and by assumption the induced map on fibers is a weak equivalence. Now apply the result in [Rap17], which says that if we have a commutative diagram…”

Section: Lemma 315 a Flexible Sheaf γ Satisfies Conditions (H) And (W)mentioning

confidence: 99%

Three applications of delooping to h-principles

Kupers¹

2018

Geom Dedicata

View full text Add to dashboard Cite

In this paper we give three applications of a method to prove h-principles on closed manifolds. Under weaker conditions this method proves a homological h-principle, under stronger conditions it proves a homotopical one. The three applications are as follows: a homotopical version of Vassiliev's h-principle, the contractibility of the space of framed functions, and a version of Mather-Thurston theory.source of the "categorical approach" to h-principles known to the author is a collection of lecture notes due to Michael Weiss [Wei15].1.2. Applications. The goal of this paper is to give three applications of Theorem A. The first is the contractibility of the space of framed functions, see Corollary 6.4. It was originally proven by Eliashberg-Mishachev [EM12], and Galatius used techniques similar to ours in an unpublished proof. Corollary B. The space G fr (M rel b) of framed functions is contractible for all smooth manifolds M and boundary conditions b near ∂M . Our second application is a generalization of Vassiliev's h-principle [Vas92], see Corollary 5.12. It will be a consequence of Thom's jet transversality theorem and implies the hprinciple for generalized Morse functions proven in [EM00].Corollary C. Suppose Z is a smooth manifold and D is a closed Diff-invariant stratified subset of codimension at least n + 2 of the space of r-jets of smooth functions R n → Z. Let F (−, D) denote the space of functions to Z with r-jet avoiding D, then the mapis a homology equivalence for all smooth manifolds M of dimension n and boundary conditions b near ∂M . If D is additionally Diff(Z)-invariant, then this map is in fact a weak equivalence.Finally, we give short proof of a version of Mather-Thurston theory for foliations. Let Fol CAT (−) denote the space of codimension n CAT-foliations as in Definition 7.2. There is a unique such foliation F 0 on a single manifold M , which we can take as a boundary condition near ∂M .Corollary D. The map j : Fol CAT (M rel F 0 ) → Fol f CAT (M rel j(F 0 )) is a homology equivalence for all CAT-manifolds M of dimension n.

show abstract

On Serre Microfibrations and a Lemma of M. Weiss

Cited by 2 publications

References 9 publications

The van Est Map on Geometric Stacks

The van Est Map on Geometric Stacks

Three applications of delooping to h-principles

Contact Info

Product

Resources

About