arXiv research

The paper characterizes Whitney and contact Whitney spheres in complex and Sasakian space forms.

New proof of Kondo-Tanaka theorem using geometric measure theory.

No Hantzsche-Wendt manifolds over 3D admit spin^c structures.

Characterizes Whitney forms on simplices and proves their uniqueness.

Real Bott manifolds is a class of flat manifolds with holonomy group $\mathbb Z_2^k$ of diagonal type. In this paper we want to show how we can compute even Stiefel - Whitney classes on real Bott manifolds. This paper is an answer to the question of professor Masuda if is it possible to extend A. Gąsior "Spin-structure…

It is shown that an equivariant Lagrangian sphere with a positivity condition on its Ricci curvature develops a type-II singularity under the Lagrangian mean curvature flow that rescales to the product of a grim reaper with a flat Lagrangian subspace. In particular this result applies to the Whitney spheres.

New forms generalize Whitney forms with rational coefficients for numerical analysis.

A new type of knot energy is presented via real life experiments involving a thin resilient metallic tube. Knotted in different ways, the device mechanically acquires a uniquely determined (up to isometry) normal form at least when the original knot diagram has a small number of crossings, thus outperforming the famous…

The Whitney sphere has a unique energy gap for a specific equation.

Extends Young integral to Hölder differential forms in arbitrary dimensions.

The first part of this paper completes the classification of Whitney towers in the 4-ball that was started in three related papers. We provide an algebraic framework allowing the computations of the graded groups associated to geometric filtrations of classical link concordance by order n (twisted) Whitney towers in th…

The first part of this paper exposits a simple geometric description of the Kirby-Siebenmann invariant of a 4--manifold in terms of a quadratic refinement of its intersection form. This is the first in a sequence of higher-order intersection invariants of Whitney towers studied by the authors, particularly for the 4--b…

For each integer $d$ at least two, we construct non-spin closed oriented flat manifolds with holonomy group $\mathbb Z_2^d$ and with the property that all of their finite proper covers have a spin structure. Moreover, all such covers have trivial Stiefel-Whitney classes.

Let $X$ be a real-analytic manifold and $g\colon X\to{\mathbf R}^n$ a proper triangulable subanalytic map. Given a subanalytic $r$-form $ω$ on $X$ whose pull-back to every non singular fiber of $g$ is exact, we show tha $ω$ has a relative primitive: there is a subanalytic $(r-1)$-form $Ω$ such that $dgΛ(ω-dΩ)=0$. The p…

We express any Courant algebroid bracket by means of a metric connection, and construct a Courant algebroid structure on any orthogonal Whitney sum $E\oplus C$ where E is a given Courant algebroid and C is a flat, pseudo- Euclidean vector bundle. Then, we establish the general expression of the bracket of a transitive …

We establish the equivalence of the Tuynman midpoint area formula for a spherical triangle to the classical area formulas of Euler and of Cagnoli. The derivation also yields a variant of the Cagnoli formula in terms of the medial triangle. We introduce the three barycentric coordinates of a point within the spherical t…

We compute the group of link homotopy classes of link maps of two 2-spheres into 4-space. It turns out to be free abelian, generated by geometric constructions applied to the Fenn-Rolfsen link map and detected by two self-intersection invariants introduced by Paul Kirk in this setting. As a corollary, we show that any …

The paper finds Riemannian metric representatives for Stiefel-Whitney classes.

We relate certain abelian invariants of a knot, namely the Alexander polynomial, the Blanchfield form, and the Arf invariant, to intersection data of a Whitney tower in the 4-ball bounded by the knot. We also give a new 3-dimensional algorithm for computing these invariants.

The paper identifies 2^(k+1) distinct components of hyperbolic representations.

Study geometric properties of S1 singularities and their deformations.

Paper calculates homology and intersection form of trisected 4-manifolds with boundary.

In this paper, we prove a conjecture of Friedl and Powell that their Casson-Gordon type invariant of 2-component link with linking number one is actually an obstruction to being height 3.5 Whitney tower/grope concordant to the Hopf Link. The proof employs the notion of solvable cobordism of 3-manifolds with boundary, w…

Study the geometry of a surface formed by extending a Whitney umbrella.

A canonically defined mod 2 linear dependency current is associated to each collection of m sections of a real rank n vector bundle. This current is supported on the linear dependency set of the collection of sections. It is defined whenever the collection satisfies a weak measure theoretic condition called "atomicity"…

Extends rigidity results for Whitney spheres in higher dimensions.

In this paper we define a Poincaré-Reidemeister scalar product on the determinant line of the cohomology of any flat vector bundle over a closed orientable odd-dimensional manifold. It is a combinatorial "torsion-type" invariant which refines the PR-metric, introduced earlier by the first author, and contains an additi…

3D Schoenflies theorem for simply-connected 2-complexes.

The study of Seifert linking forms for punctured n-manifolds in (2n-1)-space.

Integrates rough geometric forms on manifolds.

Non-trivial Clifford bundle from loop space tangent bundle.

Stiefel-Whitney classes of moment-angle manifolds are trivial.

A criterion for Whitney disks connects intersections in 3-manifold homology.

A real Bott manifold is the total space of iterated RP^1 bundles starting with a point, where each RP^1 bundle is projectivization of a Whitney sum of two real line bundles. We prove that two real Bott manifolds are diffeomorphic if their cohomology rings with Z/2 coefficients are isomorphic. A real Bott manifold is a …

Notes on Whitney towers in 4-manifolds, focusing on local surface manipulations and invariants.

In this paper we show that certain generalizations of the $C^r$-Whitney topology, which include the Hölder-Whitney and Sobolev-Whitney topologies on smooth manifolds, satisfy the Baire property, to wit, the countable intersection of open and dense sets is dense.

Study on curves around a Whitney umbrella focusing on geodesic and normal curvatures.

This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new qua…

We study the structure of the exteriors of gropes and Whitney towers in dimension 4, focusing on their fundamental groups. In particular we introduce a notion of unknottedness of gropes and Whitney towers in the 4-sphere. We prove that various modifications of gropes and Whitney towers preserve the unknottedness and do…

Many open problems and important theorems in low-dimensional topology have been formulated as statements about certain 2--complexes called gropes. This paper describes a precise correspondence between embedded gropes in 4--manifolds and the failure of the Whitney move in terms of iterated `towers' of Whitney disks. The…

In 1976, Dodziuk and Patodi employed Whitney forms to define a combinatorial codifferential operator on cochains, and they raised the question whether it is consistent in the sense that for a smooth enough differential form the combinatorial codifferential of the associated cochain converges to the exterior codifferent…

We classify the Lagrangian orientable surfaces in complex space forms with the property that the ellipse of curvature is always a circle. As a consequence, we obtain new characterizations of the Clifford torus of the complex projective plane and of the Whitney spheres in the complex projective, complex Euclidean and co…

Sobolev mappings preserve the Rumin complex on contact manifolds.

New hyperbolic manifolds with diverse features created.

Link concordance and Whitney towers linked to Milnor invariants.

This paper describes grope and Whitney tower filtrations on the set of concordance classes of classical links in terms of class and order respectively. Using the tree-valued intersection theory of Whitney towers, the associated graded quotients are shown to be finitely generated abelian groups under a (surprisingly) we…

We introduce a notion of symmetric Whitney tower cobordism between bordered 3-manifolds, aiming at the study of homology cobordism and link concordance. It is motivated by the symmetric Whitney tower approach to slicing knots and links initiated by Cochran, Orr, and Teichner. We give amenable Cheeger-Gromov rho-invaria…

Geometric trick simplifies link homotopy and concordance.