Study on Mackenzie obstruction for transitive Lie algebroids, proving triviality for certain Lie algebras.
problem Calculating the Mackenzie obstruction for transitive Lie algebroids given a Lie algebra and tangent bundle.
method Proved the triviality of the Mackenzie obstruction for specific Lie algebras.
result The Mackenzie obstruction is trivial for Lie algebras that are direct sums of center and subalgebra without center.
Unified understanding of integrability obstructions for Lie algebroids.
problem Equivalence of integrability obstructions for transitive Lie algebroids.
method Unified approach using cohomological and homotopical methods.
result Unified understanding of integrability obstructions.
Paper traces origins of graded Lie brackets theory.
problem Understanding the development of graded Lie brackets theory.
method Historical review of publications by Schouten, Nijenhuis, Frölicher-Nijenhuis, Gerstenhaber, and Nijenhuis-Richardson.
result Relates the origins and early development of graded Lie brackets theory.
Double Lie algebroids were discovered by Kirill Mackenzie from the study of double Lie groupoids and were defined in terms of rather complicated conditions making use of duality theory for Lie algebroids and double vector bundles. In this paper we establish a simple alternative characterization of double Lie algebroids…
We give a simple characterization of Mackenzie's double Lie algebroids in terms of homological vector fields. Application to the `Drinfeld double' of Lie bialgebroids is given and an extension to the multiple case is suggested.
This text is meant to be a brief overview of the topics announced in the title and is based on my talk in Vienna (August/September 2007). It does not contain new results (except probably for a remark concerning Q-manifold homology, which I wish to elaborate elsewhere). "Mackenzie theory" stands for the rich circle of n…
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…
Lie theory for the integration of Lie algebroids to Lie groupoids, on the one hand, and of Poisson manifolds to symplectic groupoids, on the other, has undergone tremendous developements in the last decade, thanks to the work of Mackenzie-Xu, Moerdijk-Mrcun, Cattaneo-Felder and Crainic-Fernandes, among others. In this …
Due to a result by Mackenzie, extensions of transitive Lie groupoids are equivalent to certain Lie groupoids which admit an action of a Lie group. This paper is a treatment of the equivariant connection theory and holonomy of such groupoids, and shows that such connections give rise to the transition data necessary for…
This paper is concerned with the holonomy of a class of spaces which includes Landsberg spaces of Finsler geometry. The methods used are those of Lie groupoids and algebroids as developed by Mackenzie. We prove a version of the Ambrose-Singer Theorem for such spaces. The paper ends with a discussion of how the results …
We calculate the group of dualization operations for triple vector bundles, showing that it has order 96 and not 72 as given in Mackenzie's original treatment. The group is a nonsplit extension of S4 by the Klein group. Dualization operations are interpreted as functors on appropriate categories and are said to be equa…
It is well-known that a Lie algebroid A is equivalently described by a degree 1 Q-manifold M. We study distributions on M, giving a characterization in terms of A. We show that involutive Q-invariant distributions on M correspond bijectively to IM-foliations on A (the infinitesimal version of Mackenzie's ideal systems)…
We consider a simple instance of action up to homotopy. More precisely, we consider strict actions of DGLAs in degrees -1 and 0 on degree 1 NQ-manifolds. In a more conventional language this means: strict actions of Lie algebra crossed modules on Lie algebroids. When the action is strict, we show that it integrates to …
We approach Mackenzie's LA-groupoids from a supergeometric point of view by introducing Q-groupoids, which are groupoid objects in the category of Q-manifolds. There is a faithful functor from the category of LA-groupoids to the category of Q-groupoids. We associate to every Q-groupoid a double complex that provides a …
We use symplectic reduction to give a new construction of the core C of a symplectic double groupoid D as the common leaf space of characteristic foliations associated to various coisotropic submanifolds of D. In the case of the cotangent double groupoid of a Lie groupoid G, the canonical relations arising from…
The equivalence of principal bundles with transitive Lie groupoids due to Ehresmann is a well known result. A remarkable generalisation of this equivalence, due to Mackenzie, is the equivalence of principal bundle extensions with those transitive Lie groupoids over the total space of a principal bundle, which also admi…
Generalizes Lie bialgebroids to supermanifolds with homotopy Poisson structures.
problem Relating Lie bialgebroids to homotopy Poisson structures on supermanifolds.
method Introduces L-infinity bialgebroids and higher Koszul brackets to connect these structures.
result Shows that (TM,T∗M) has an L-infinity bialgebroid structure for homotopy Poisson structures. The abstract discusses a spectral sequence for Lie algebroids.
problem The abstract tackles the spectral sequence of Lie algebroids.
method The abstract presents a spectral sequence for Lie algebroids, generalizing classical constructions.
result The spectral sequence converges to Lie algebroid cohomology for wide Lie subalgebroids and to formal Lie algebroid cohomology for Lie subalgebroids over proper submanifolds.
This paper studies torsion obstructions to complex sections on manifolds.
problem Torsion obstructions to finding complex sections on almost complex manifolds.
method Calculations using the Adams-Novikov spectral sequence for Thom spectra.
result Torsion obstructions for finding r complex sections of order p vanish for r<p2−p. Describes Wall's finiteness obstruction.
problem Finiteness obstruction in algebraic topology.
method Self-contained description of Wall's finiteness obstruction.
result Comprehensive explanation of Wall's finiteness obstruction.
New obstructions for knots in high dimensions prevent double sliceness.
problem Preventing double sliceness of knots in high dimensions.
method Developed L(2)-signature obstructions for metabelian knot groups. result Found infinite families of knots not obstructed by previous invariants.
Fewer obstructions for small graphs in knotless embedding.
problem Finite obstructions for knotless embedding of small graphs.
method Analysis of graph minor obstructions and knotless embedding.
result There are only three obstructions for size 23 graphs, significantly fewer than previously known.
3028 obstructions found for embedding without knots.
problem Finding obstructions for knotless embedding.
method Surveying recent work, updating obstructions, and proposing new questions.
result New obstructions with μ=6 and insights into connectivity.
Obstructing deformations of vector forms on Kähler manifolds.
problem Deformations of vector forms on compact Kähler manifolds.
method Annihilating obstruction classes by cohomology classes.
result Obstruction classes are annihilated by cohomology classes.
Study calculates Euler obstruction for determinantal varieties with ICIS singular set.
problem Calculating Euler obstruction for determinantal varieties with ICIS singular set.
method Obtained formulas to calculate Euler obstruction for determinantal varieties with ICIS singular set.
result Formulas to calculate the Euler obstruction for determinantal varieties with ICIS singular set.
The paper develops obstructions for embedding 2D complexes into 4D space.
problem Embedding 2D complexes into 4D space and understanding obstructions.
method Uses Goodwillie-Weiss calculus and intersections of Whitney disks.
result Two approaches to obstructions lead to the same result.
Global obstructions found for conformally Einstein metrics in 6D.
problem Obstructing the existence of conformally Einstein metrics in six dimensions.
method Presentation of global conformal invariants.
result Nontrivial global conformal invariant obstructing conformally Einstein metrics.
Complete surgery obstructions for manifolds with finite fundamental group, disproving a conjecture.
problem Disproving the Oozing Conjecture for manifolds with finite fundamental group.
method Description and calculation of surgery obstructions up to homotopy equivalence.
result New obstructions found for Arf invariant product formulas in codimensions ≥ 4, disproving the Oozing Conjecture.
Lie 2-algebra structure on vector fields of geometric stacks proved.
problem Proving Lie 2-algebra structure on vector fields of geometric stacks.
method Using Lie groupoid presentation, constructing Lie 2-algebra from known Lie brackets and Lie algebroid sections.
result Lie 2-algebra structure on vector fields of geometric stacks is well-defined.
Given a polarized manifold there are obstructions for asymptotic Chow semistability described as integral invariants. One of them is an obstruction to the existence for the first Chern class of the polarization to admit a constant scalar curvature Kähler (cscK) metric. A natural question is whether or not the other obs…
The article classifies cubiquitous sublattices and applies them to branched covers.
problem Understanding cubiquitous sublattices as obstructions to rational homology 4-balls.
method Developed a geometric Wu obstruction to classify cubiquitous sublattices and applied it to branched covers.
result Completely classified which sublattices with orthogonal bases are cubiquitous.
New obstructions found for smooth desingularization of compact Einstein orbifolds.
problem Finding obstructions to desingularizing compact Einstein orbifolds.
method Identifying new obstructions specific to compact Einstein 4-orbifolds. result Almost all flat orbifold metrics on T4/Z2 are not limits of Ricci-flat metrics. Study uses braid conjugacy class invariants to find ribbon obstructions for knots.
problem Finding effective obstructions for slice knots to be ribbon.
method Apply Rudolph's work and Khovanov-Lee theory to braid conjugacy class invariants.
result Cannot obtain effective ribbon obstructions using Khovanov-Lee theory.
For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S^3, we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundam…
Study surgery obstructions for Seifert fibered homology spheres using knot and Heegaard Floer homology.
problem Determining which knots can yield Seifert fibered homology spheres.
method Utilized Heegaard Floer and Knot Floer homology, along with the mapping cone formula, to find obstructions.
result Showed that genus one knots cannot yield Seifert fibered homology spheres with six or more singular fibers.
Study identifies obstructions for solving a 4th-order boundary problem.
problem Solving a 4th-order boundary problem with specific curvature conditions.
method Derived Kazdan-Warner type identities using variational formulation and conformal variations.
result Obtained nontrivial integral obstructions to solvability.
Developing a singular dimension descent method for positive scalar curvature obstructions
problem Positive scalar curvature obstructions in arbitrary dimensions
method Schoen--Yau type singular dimension descent method
result Proving obstructions to positive scalar curvature on enlargeable manifolds
Study Euler obstruction of 1-forms on determinantal singularities.
problem Understanding the Euler obstruction of 1-forms on determinantal singularities.
method Investigation of connections between local Euler obstruction and PHN index.
result Explicit computations of Euler obstruction for specific singularities.
The paper explores properties of CR hypersurfaces and their flatness.
problem Analyzing the flatness properties of CR hypersurfaces.
method Investigation of local and global properties of strongly pseudoconvex CR hypersurfaces.
result Existence of obstruction flat points on compact CR hypersurfaces.
Proves Massey's theorems on complex structure obstructions.
problem Finding complex structures on real vector bundles.
method Fractional Stiefel-Whitney classes and combinations of Pontryagin, Chern, and Euler classes.
result Determines the second obstruction for rank six bundles.
New algebra structure derived from Lie pairs.
problem Constructing A∞-algebras from Lie pairs. method Using homotopy equivalence and Lie algebroids.
result Chevalley-Eilenberg cohomology gains an associative algebra structure.
Study on a specific obstruction in four-dimensional geometry.
problem Singular Yamabe obstruction of a hypersurface.
method Derived various explicit formulas from the original definition.
result Relate formulas to literature and prove elementary.
The paper explores obstructions for symplectic Lie algebroids on surfaces.
problem Obstacles to the existence of symplectic Lie algebroids.
method Analysis through characteristic classes of symplectic Lie algebroids.
result Full obstructions for surfaces to carry symplectic Lie algebroids.
Introduce generalized Ueda obstruction classes for line bundles and apply them to non-semi-positivity.
problem Non-semi-positivity of line bundles
method Introduce generalized Ueda obstruction classes and use Dolbeault resolution
result Recover classical examples and provide new examples of nef but not semi-positive line bundles
We define \textit{graded manifolds} as a version of supermanifolds endowed with an additional Z-grading in the structure sheaf, called \textit{weight} (not linked with parity). Examples are ordinary supermanifolds, vector bundles over supermanifolds, double vector bundles, iterated constructions like TTM, e…
Surgery obstructions extended to integer homology spheres using Heegaard Floer homology.
problem Obstructing knots in integer homology spheres using surgery.
method Extending Heegaard Floer homology obstructions to all integer homology spheres for both positive and negative surgeries.
result Deduced a lower bound on b2(W) for smooth cobordism between integer homology spheres. Extended solitons show constant curvature on compact manifolds.
problem Understanding self-similar solutions to geometric flows.
method Analyzing extended solitons and applying integral inequalities.
result Closed ambient obstruction solitons are ambient obstruction flat and have constant scalar curvature.
Study the topological information of map germs using Euler obstruction.
problem Capturing topological information from map germs with complex singular spaces.
method Investigates the Euler obstruction and its relation to local Euler obstruction and Brasselet number.
result Relates Chern number to the number of cusps in a perturbed map-germ.