The paper refines cohomology of VB-groupoids with homogeneous cochains.
problem Cohomology of VB-groupoids with linear structure.
method Refine cohomology by considering k-homogeneous cochains.
result Van Est theorem for k-homogeneous cochains on VB-groupoids.
Proves integrability of strict Lie 2-algebras using cohomological methods.
problem Integrability of strict Lie 2-algebras.
method Van Est theorems relating cohomologies of Lie 2-groups and algebras.
result Proves integrability of Lie 2-algebras.
Revisits Van Est theory for Lie groupoids using homotopy inverses.
problem Relating Lie groupoids and their Lie algebroids cohomology.
method Using the Perturbation Lemma from homological algebra to construct homotopy inverses.
result Constructs homotopy inverses to the van Est differentiation maps at the cochain level.
This paper belongs to a series devoted to the study of the cohomology of classifying spaces. Generalizing the Weil algebra of a Lie algebra and Kalkman's BRST model, here we introduce the Weil algebra W(A) associated to any Lie algebroid A. We then show that this Weil algebra is related to the Bott-Shulman-Stasheff…
Generalizes van Est map to geometric stacks and homotopy theory.
problem Computing cohomology of geometric stacks and Lie algebroids.
method Generalizes van Est map to stacks and foliations, using modules instead of representations.
result Derives new cohomology results and unifies differentiable stacks, Lie algebroids, and homotopy theory.
The Van Est homomorphism for a Lie groupoid G⇉M, as introduced by Weinstein-Xu, is a cochain map from the complex C∞(BG) of groupoid cochains to the Chevalley-Eilenberg complex C(A) of the Lie algebroid A of G. It was generalized by Weinstein, Mehta, and Abad-Crainic to a morphism from…
Study vector bundles over Lie groupoids, controlling their deformations.
problem Understanding deformations of vector bundles over Lie groupoids.
method Attach cochain complexes to VB-groupoids to control deformations, discuss Morita invariance and van Est theorem.
result Fundamental features of VB-groupoids' deformations, including Morita invariance and van Est theorem.
Global homotopies upgrade classical map in differential geometry.
problem Upgrade classical Hochschild-Kostant-Rosenberg map to a deformation retract.
method Combining symbol calculus and coalgebraic van Est theorem.
result Develop deformation retracts in various settings.
Geometrically solves differentiating simplicial manifolds.
problem Differentiating simplicial manifolds.
method Establishes a normal form theorem, identifies a differentiating ideal, proves quotient semi-freeness, interprets as Chevalley-Eilenberg algebra of higher Lie algebroid.
result Introduces higher van Est map and proves van Est isomorphism theorem.
Analyses cohomology relations for moving frames and coframes.
problem Relating Hopf cyclic cohomology of moving frames and coframes.
method Uses van Est analogy for DG Hopf algebras.
result Establishes cohomology isomorphism for DG Hopf algebras.
Generalizes van Est map to sheaves of sections taking values in G-modules.
problem Classifying geometric structures involving Lie groupoids and stacks.
method Infinitesimal description of G-modules and generalized van Est map. result Definition and study of the generalized van Est map.
Maps Lie 2-groups to Weil algebras, showing cohomology isomorphisms.
problem Cohomology of strict Lie 2-groups.
method Constructs van Est map using double complex and Weil algebra.
result Induces isomorphisms in cohomology under connectedness.
Paper generalizes representations of Lie algebroids to weighted Lie algebroids.
problem Representations of Lie algebroids and their generalizations.
method Introducing and studying weighted Lie algebroids, showing relations to VB-algebroids and generalizing the van Est theorem.
result New natural examples of higher term representations up to homotopy of Lie algebroids uncovered.
We define the "localized index" of longitudinal elliptic operators on Lie groupoids associated to Lie algebroid cohomology classes. We derive a topological expression for these numbers using the algebraic index theorem for Poisson manifolds on the dual of the Lie algebroid. Underlying the definition and computation of …
Groupoids help define Riemann sums on manifolds.
problem Defining Riemann sums on compact manifolds.
method Using groupoids and the van Est map.
result Riemann sums converge to the usual integral.
Lie algebra cohomology reformulated in a topological framework.
problem Classical Lie algebra cohomology results.
method Topological reformulation of Lie algebra cohomology.
result Shapiro lemma and van Est isomorphism generalize to topological setting.
We show that representations up to homotopy can be differentiated in a functorial way. A van Est type isomorphism theorem is established and used to prove a conjecture of Crainic and Moerdijk on deformations of Lie brackets.
In the first section we discuss Morita invariance of differentiable/algebroid cohomology. In the second section we present an extension of the van Est isomorphism to groupoids. This immediately implies a version of Haefliger's conjecture for differentiable cohomology. As a first application we clarify the connection be…
Introduces new cohomology theories for Lie 2-algebras and groups.
problem Classical cohomology theories do not extend to Lie 2-algebras and groups.
method Develops new cohomology theories and uses them to prove integrability.
result New cohomology theories classify extensions and prove integrability of Lie 2-algebras.
Given a proper, cocompact action of a Lie groupoid, we define a higher index pairing between invariant elliptic differential operators and smooth groupoid cohomology classes. We prove a cohomological index formula for this pairing by applying the van Est map and algebraic index theory. Finally we discuss in examples th…
Maps geometric deformations to algebraic classes in Lie groupoids and algebroids.
problem Deformation theory of Lie groupoids and algebroids.
method Defining a morphism between deformation complexes and Hochschild complexes, applying to adiabatic groupoids.
result Induced van Est map from geometric to algebraic deformation cohomology.
We study deformations of Lie groupoids by means of the cohomology which controls them. This cohomology turns out to provide an intrinsic model for the cohomology of a Lie groupoid with values in its adjoint representation. We prove several fundamental properties of the deformation cohomology including Morita invariance…
Develops relative cohomology for Lie groupoids and algebroids.
problem Lack of relative cohomology theory in Lie groupoids and algebroids.
method Structural theory development, van Est maps relation, intrinsic characteristic classes definition.
result Provides an intrinsic definition of characteristic classes using relative cohomology.
This paper extends a category equivalence to an A-infinity quasi-equivalence for compact Lie groups.
problem Equivalence of DG categories for smooth singular chains on Lie groups.
method Construction of A-infinity quasi-isomorphisms and use of Van Est map, De Rham theorem.
result Extension of equivalence to A-infinity quasi-equivalence for compact Lie groups.
We prove a cyclic cohomological analogue of Haefliger's van Est-type theorem for the groupoid of germs of diffeomorphisms of a manifold. The differentiable version of cyclic cohomology is associated to the algebra of transverse differential operators on that groupoid, which is shown to carry an intrinsic Hopf algebraic…
Study vector fields and derivations on differentiable stacks.
problem Understanding structures on differentiable stacks.
method Introduced module structures on dgla of multiplicative vector fields and graded algebra of functions on Lie groupoids.
result Associated structure of a graded Lie-Rinehart algebra on vector fields is Morita invariant.
Direct proofs of implications between three theorems on maps of simplex.
problem Understanding relations between three theorems on maps of simplex.
method Direct proofs using interesting relations between van Kampen and Conway-Gordon-Sachs numbers.
result Exhibited relations and direct proofs of implications between the theorems.
New theorem shows embedding restrictions for manifold skeletons.
problem Embedding restrictions for triangulated manifolds.
method Proves van Kampen-Flores theorem for manifolds with specific Stiefel-Whitney classes.
result Triangulated manifolds with non-trivial Stiefel-Whitney classes cannot embed into R2d. Review and generalize Haefliger's differentiable cohomology for diffeomorphisms and flat Cartan groupoids.
problem Define and investigate differentiable cohomology for diffeomorphisms and flat Cartan groupoids.
method Define Haefliger's differentiable cohomology for diffeomorphisms, investigate its structure, and generalize to flat Cartan groupoids.
result Define characteristic maps for geometric structures on manifolds associated to flat Cartan groupoids.
The paper studies deformations of symplectic groupoids using cohomology and spectral sequences.
problem Deformations of symplectic groupoids and their cohomology.
method Deformation cohomology, Moser path methods, Lie groupoids, multiplicative forms, de Rham models, spectral sequences.
result Computations and constructions of deformation cohomology for various types of symplectic groupoids.
The paper derives Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.
problem Computing curvature and geodesic curvature for surfaces and curves in affine and rigid motions groups.
method Defined deformed Schouten-Van Kampen connections, computed Gaussian curvature limits, and signed geodesic curvature.
result Derived Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.
Dans les années 1940-1970, Alexandrov et l'"École de Leningrad" ont développé une théorie très riche des surfaces singulières. Il s'agit de surfaces topologiques, munie d'une métrique intrinsèque pour laquelle on peut définir une notion de courbure, qui est une mesure de Radon. Cette classe de surfaces a de bonnes prop…
Q-groupoids and Q-algebroids are, respectively, supergroupoids and superalgebroids that are equipped with compatible homological vector fields. These new objects are closely related to the double structures of Mackenzie; in particular, we show that Q-groupoids are intermediary objects between Mackenzie's LA-groupoids a…
The paper calculates curvature limits and Gauss-Bonnet theorems in the Heisenberg group.
problem Computing curvature limits and Gauss-Bonnet theorems in the Heisenberg group.
method Sub-Riemannian limits of Gaussian curvature, Schouten-Van Kampen affine connections, and adapted connections.
result Gauss-Bonnet theorems associated with Schouten-Van Kampen affine connections in the Heisenberg group.
Paper develops Lie theory for Rota-Baxter operators on Lie algebras and groups.
problem Cohomology of relative Rota-Baxter operators on Lie algebras and groups.
method Cohomology construction, infinitesimal deformations study, differentiation and integration of Rota-Baxter operators.
result Integration of Rota-Baxter operators on Lie groups and Lie algebras.
Formalizes quantum path integrals using groupoids and differential forms.
problem Formalizing Feynman's path integral in quantum mechanics.
method Shifted focus to pair groupoid, using van Est map and piecewise linear structures.
result Developed a coordinate-free approach to integration of differential forms.
A new geometric definition of integration for differential forms.
problem Standard integration definitions are coordinate-dependent and not suitable for certain contexts.
method Uses triangulations and cochains on the pair groupoid to define integration.
result Natural definition in Lie algebroids, stochastic integration, and quantum field theory.
Paper calculates homotopy types of non-compact surfaces using groupoids.
problem Computing homotopy types of non-compact foliated surfaces.
method Application of van Kampen theorem for groupoids.
result Computation of homotopy types for specific non-compact surfaces.
In this paper we present short algebraic proofs of the Linear Conway--Gordon--Sachs and the Linear van Kampen--Flores theorems in the spirit of the Radon theorem on convex hulls. {\bf Theorem.} {\it Take any n+3 general position points in Rn. If n is odd, then there are two linked (n+1)/2-simplices wi…
Computes knot groups for torus links using Seifert-van Kampen theorem.
problem Computing knot groups for torus links.
method Groupoid version of the Seifert-van Kampen theorem.
result Economical presentations of knot groups for torus links.
Characterizes simplicial complexes embedding into spheres with few vertices.
problem Characterizing simplicial complexes that embed into spheres with few vertices.
method Simple characterization using non-face families and analogy with Fáry's theorem.
result Recovery of van Kampen--Flores theorem and Erd\H os--Ko--Rado theorem.
Study measures on differentiable stacks, proving invariance and formulas.
problem Defining and studying measures on differentiable stacks.
method Generalizing Haefliger's approach to leaf spaces and transverse measures for foliations.
result Proved Morita invariance, a Stokes formula, and Van Est isomorphism.
Given a front projection of a Legendrian knot K in R3 which has been cut into several pieces along vertical lines, we assign a differential graded algebra to each piece and prove a van Kampen theorem describing the Chekanov-Eliashberg invariant of K as a pushout of these algebras. We then use this the…
Solves local minima problems on smooth manifolds.
problem Local minima issues on smooth manifolds.
method Introducing valley functions and applying Morse's lemma.
result Eliminates critical points and reduces to 1D.
This thesis studies deformations of VB-algebroids and VB-groupoids in Lie algebroid and groupoid categories.
problem Deformations of VB-algebroids and VB-groupoids in Lie algebroid and groupoid categories.
method Attach cochain complexes to VB-algebroids and VB-groupoids, equip them with DGLA structures, discuss their properties and relationships with deformation complexes of total and base spaces.
result Linear van Est theorem and Morita invariance theorem for VB-groupoids.
Flat systems of up to 2 dimensions have flat subsystems.
problem Characterizing flat subsystems in flat systems of differential dimension 2.
method Analyzing subsystems of a flat system of differential dimension at most 2.
result Flat subsystems of a flat system of differential dimension at most 2 exist and can have independent time-uniform outputs.
New simplicial complexes show unavoidable link of spheres in high dimensions.
problem Finding unavoidable links of spheres in high-dimensional spaces.
method Simple argument in piecewise linear topology and application of the van Kampen--Flores theorem.
result Existence of additional simplicial complexes with unavoidable links of spheres.
A new algebra for Legendrian graphs defined combinatorially.
problem Defining an algebra for Legendrian graphs and tangles.
method Combinatorial approach using Legendrian contact homology.
result Shows a van Kampen type theorem for differential graded algebras.