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.
The paper shows examples of 2-complexes that can't be embedded in R^4, hiding obstructions in higher Milnor invariants.
problem Embedding 2-complexes in R^4 with hidden obstructions.
method Provides examples of 2-complexes and families of PL immersions that hide embedding obstructions.
result Embedding obstructions vanish for the given examples, answering a question in Avramidi-Okun-Schreve's paper.
In the present paper, we construct a simple invariant which provides a sliceness obstruction for {\em free knots}. This obstruction provides a new point of view to the problem of studying cobordisms of curves immersed in 2-surfaces, a problem previously studied by Carter, Turaev, Orr, and others. The obstruction to sli…
New extremal Poincaré type metrics found via blow-up theorem.
problem Finding new extremal Poincaré type metrics.
method Proved Arezzo-Pacard-Singer blow-up theorem for Poincaré type metrics and applied to new examples.
result Found new examples of extremal Poincaré type metrics with an additional obstruction.
This work treats on the question whether a given map f: M -> B of smooth closed manifolds is homotopic to a smooth fiber bundle. We define a first obstruction in H^1(B;Wh(π_1(E))) and, provided that this obstruction vanishes and one additional condition is verified, a second obstruction in Wh(π_1(E)) >. Both elements v…
The paper proves conditions for Kähler-Einstein metrics on certain bundles.
problem Conditions for the existence of Kähler-Einstein metrics on unit sphere bundles.
method Analyzes curvature conditions and Ricci eigenvalues of Kähler manifolds.
result Conditions for obstruction flatness and existence of Kähler-Einstein metrics.
The paper studies Kähler-Einstein metrics on circle bundles and their obstruction flatness.
problem Understanding Kähler-Einstein metrics on circle bundles and their smoothness properties.
method Analyzing the obstruction flatness of hypersurfaces arising as unit circle bundles over Kähler manifolds.
result Complete Kähler-Einstein metrics on disk bundles are possible under certain conditions.
In this article, we derive a topological obstruction to the removal of a isolated degenerate complex tangent to an embedding of a 3-manifold into C3 (without affecting the structure of the remaining complex tangents). We demonstrate how the vanishing of this obstruction is both a necessary and sufficient co…
Minimal hypersurfaces can't always be connected by mean curvature flow.
problem Existence of connecting mean curvature flows for minimal hypersurfaces.
method Minimal hypersurface analogue of gradient flow trajectories between critical points.
result Additional topological and variational obstructions to connecting mean curvature flows.
The technique of generating families produces obstructions to the existence of embedded Lagrangian cobordisms between Legendrian submanifolds in the symplectizations of 1-jet bundles. In fact, generating families may be used to construct a TQFT-like theory that, in addition to giving the aforementioned obstructions, yi…
The purpose of this paper is to study finite dimensional equivariant moduli problems from the viewpoint of stratification theory. We show that there exists a stratified obstruction system for a finite dimensional equivariant moduli problem. In addition, we define a coindex for a G-vector bundle which is determined by t…
The paper defines a Chern-Simons invariant for stably trivial vector bundles and uses it to obstruct conformal immersions.
problem Obstructing conformal immersions of Riemannian manifolds.
method Defining a Chern-Simons invariant for stably trivial vector bundles and using it to derive an obstruction.
result An obstruction for conformally immersing a n-dimensional Riemannian manifold in a translation manifold of dimension n+1.
Study confoliations' symplectic fillability, finding obstructions.
problem Understanding confoliations' symplectic fillability in higher dimensions.
method Generalized Massot-Niederkrüger-Wendl's bordered Legendrian open book for confoliations.
result Obstructions to weak symplectic fillability for confoliations.
Improved bounds on embedding k-skeletons into 2k-manifolds.
problem Determine the smallest n for embedding k-skeleton of a simplex into a compact PL 2k-manifold. method New obstruction via intersection form and nullhomotopy condition.
result Improved bounds and conditions for embeddability of k-complexes into 2k-manifolds. In the present paper, we establish a gluing construction for the Nahm pole solutions to the Kapustin-Witten equations over manifolds with boundaries and cylindrical ends. Given two Nahm pole solutions with some convergence assumptions on the cylindrical ends, we prove that there exists an obstruction class for gluing t…
Link concordance equals homotopy for high-dimensional spheres.
problem Understanding when immersions of high-dimensional spheres are homotopically trivial.
method Developed stratified Morse theory for generic immersions, using gradient-like vector fields and Cerf theory.
result Every link of high-dimensional spheres is homotopically trivial, resolving a long-standing conjecture.
We define an algebraic group comprising symmetric chain complexes which captures the first two stages of the Cochran-Orr-Teichner solvable filtration of the knot concordance group in a single obstruction. To achieve this we impose additional structure on each chain complex which puts extra control on the fundamental gr…
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. 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.
Paper proves index theorem for self-adjoint elliptic boundary problems.
problem Proving index theorem for self-adjoint elliptic boundary problems.
method Topological and pseudo-differential methods, generalized Atiyah-Singer approach.
result Removed technical assumption to prove index theorem.
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.
We show that a complete submanifold M with tamed second fundamental form in a complete Riemannian manifold N with sectional curvature KN≤κ≤0 are proper, (compact if N is compact). In addition, if N is Hadamard then M has finite topology. We also show that the fundamental tone is an obstruction fo…
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.
To a closed wide Lie subgroupoid A of a Lie groupoid L, i.e. a Lie groupoid pair, we associate an Atiyah class which we interpret as the obstruction to the existence of L-invariant fibrewise affine connections on the homogeneous space L/A. For Lie groupoid pairs with…
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.
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…
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 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.
We summarize and expand known connections between the study of Dehn surgery on links and the study of trisections of closed, smooth 4-manifolds. In addition, we describe how the potential counterexamples to the Generalized Property R Conjecture given by Gompf, Scharlemann, and Thompson yield genus four trisections of t…
It has been shown in [Pa1] that on a simple, compact Riemannian 2-manifold the attenuated geodesic ray transform, with attenuation given by a connection and Higgs field, is injective on functions and 1-forms modulo the natural obstruction. Furthermore, the scattering relation determines the connection and Higgs field m…
The paper explores flat extensions of connections and their relation to Chern-Simons invariants.
problem Understanding flat extensions of principal connections and their implications.
method Introducing flat extensions and relating them to Chern-Simons invariants.
result Flat extensions of connections are linked to the vanishing of Chern-Simons invariants.
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.
Knot Floer homology detects ribbon concordance.
problem Detecting ribbon concordance between knots.
method Using knot Floer homology to study ribbon concordance.
result The map induced by ribbon concordance on knot Floer homology is injective.
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 introduces controllable principal connections and estimates distances between bundles and spaces.
problem Estimating distances between bundles and spaces using controllable connections.
method Combining orbit theorem, Ambrose-Singer theorem, and controllable principal connections.
result Proves convergence of metrics to normal reductive homogeneous spaces.
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
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. The study examines Bäcklund transformations for hyperbolic systems and their invariants.
problem Identifying pairs of hyperbolic Euler-Lagrange systems with Bäcklund transformations.
method Expressing obstructions in terms of local invariants and discovering a new class of transformations.
result A class of Bäcklund transformations between systems of different types.
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.