arXiv research

New algebraic structures for topological pairs.

Disk pairings with zero signature are related to topological surfaces.

Gordon-Litherland pairing connects combinatorics and topology.

Study on the minimum length of curves on once-punctured hyperbolic surfaces.

New examples of Cappell-Shaneson knot pairs with same Alexander polynomial found.

Study conic line arrangements of degree 7, finding their topology and connected components.

New Bailey pairs derived for tetrahedron index, linking knot invariants.

The paper explores new phenomena in boundaries of relatively hyperbolic groups.

Researchers found multiple surfaces with same topological and symmetry properties.

Characterizes alternating links in thickened surfaces using Gordon-Litherland pairing.

This paper explores how pairs of multicurves can be realized as cylinders on translation surfaces.

We investigate some topological properties of a normal functor $H$ introduced earlier by Radul which is some functorial compactification of the Hartman--Mycielski construction HM. We prove that the pair ($HX$, HM$Y$) is homeomorphic to the pair $(Q,σ)$ for each nondegenerated metrizable compactum $X$ and each dense $σ$…

We will survey some aspects of the smooth topology, algebraic geometry, symplectic geometry and contact geometry of anti-canonical pairs in complex dimension two.

In string theory, the concept of T-duality between two principal T^n-bundles E_1 and E_2 over the same base space B, together with cohomology classes h_1\in H^3(E_1) and h_2\in H^3(E_2), has been introduced. One of the main virtues of T-duality is that h_1-twisted K-theory of E_1 is isomorphic to h_2-twisted K-theory o…

Graph Neural Networks solve topology problems in simple 3D models.

Two unique conic-line arrangements with degree 9 are found.

The article finds conditions for separating filling pairs on surfaces and constructs a Morse function.

Survey of invariants for knotted 2-spheres in 4-space.

Paper finds infinite pairs of fiber-type curves with same topology but different embeddings.

A $k$-Artal arrangement is a reducible algebraic curve composed of a smooth cubic and $k$ inflectional tangents. By studying the topological properties of their subarrangements, we prove that for $k=3,4,5,6$, there exist Zariski pairs of $k$-Artal arrangements. These Zariki pairs can be distinguished in a geometric way…

Study one-dimensional topological theories with linear generating functions.

We show examples of pairs of smooth, compact, homeomorphic 4-manifolds, whose diffeomorphism types are distinguished by the topology of the singular sets of smooth stable maps defined on them. In this distinction we rely on results from Seiberg-Witten theory.

Authors create non-isometric 3-orbifolds with identical topology and volume.

In string theory, the concept of T-duality between two principal U(1)-bundles E_1 and E_2 over the same base space B, together with cohomology classes $h_1\in H^3(E_1)$ and $h_2\in H^3(E_2)$, has been introduced. One of the main virtues of T-duality is that $h_1$-twisted K-theory of $E_1$ is isomorphic to $h_2$-twisted…

Let X be a locally compact Polish space and G a non-discrete Polish ANR group. By C(X,G), we denote the topological group of all continuous maps f:X \to G endowed with the Whitney (graph) topology and by C_c(X,G) the subgroup consisting of all maps with compact support. It is known that if X is compact and non-discrete…

Expands Euler-Poincare characteristic to supergeometry.

Measure homology was introduced by Thurston in his notes about the geometry and topology of 3-manifolds, where it was exploited in the computation of the simplicial volume of hyperbolic manifolds. Zastrow and Hansen independently proved that there exists a canonical isomorphism between measure homology and singular hom…

We give a necessary and sufficient condition for the mapping class group of the pair of the 3-sphere and a graph embedded in it to be isomorphic to the topological symmetry group of the embedded graph.

We investigate the spectra of a family of pairs (M_i,A_i) consisting of a complete Riemannian manifold M_i and a closed subset A_i and which converge in the Lipschitz topology to a pair (M,A). This is used to construct manifolds of bounded curvature, nonempty essential spectrum, infinitely many eigenvalues below the es…

New origamis found for surfaces with minimal intersections.

In this paper we systematically describe relations between various structure sets which arise naturally for pairs of compact topological manifolds with boundary. Our consideration is based on a deep analogy between the case of a compact manifold with boundary and the case of a closed manifold pair. This approach also g…

We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. This invariant is shown to be a generalization of the I-invariant of line arrangements developed by the first author with Artal and Florens. We give two practical tools for computin…

Suppose M is a non-compact connected smooth n-manifold. Let D(M) denote the group of diffeomorphisms of M endowed with the compact-open C^\infty-topology and D^c(M) denote the subgroup consisting of diffeomorphisms of M with compact support. Let D(M)_0 and D^c(M)_0 be the connected components of id_M in D(M) and D^c(M)…

The authors examine topological properties of the 7-dimensional Eschenburg biquotients diag(z^k1,z^k2,z^k3)\SU(3)/diag(z^l1,z^l2,z^l3). A subfamily of these spaces carry a 3-Sasakian metric. The authors show that among this subfamily there exist many 3-Sasakian spaces which are homeomorphic but not diffeomorphic. In ad…

This paper classifies and determines the length of the shortest filling pairs on a specific type of surface.

The study finds pairs of curves at distance 5 in surface curve graphs.

One can realize higher laminations as positive configurations of points in the affine building. The duality pairings of Fock and Goncharov give pairings between higher laminations for two Langlands dual groups $G$ and $G^{\vee}$. These pairings are a generalization of the intersection pairing between measured laminatio…

The study of topological groups with compact open subgroups and their geometric properties.

Quantum invariants are explained as intersections in configuration spaces.

The invariant $\mathcal{I}(\mathcal{A},ξ,γ)$ was first introduced by E. Artal, V. Florens and the author. Inspired by the idea of G. Rybnikov, we obtain a multiplicativity theorem of this invariant under the gluing of two arrangements along a triangle. An application of this theorem is to prove that the extended Rybnik…

We show that the topological groups $Diff_{+}^{1}(I)$ and $Diff_{+}^{1}(\mathbb{S}^1)$ of orientation-preserving $C^1$-diffeomorphisms of the interval and the circle, respectively, admit finitely generated dense subgroups. We also investigate the question of genericity (in the sense of Baire category) of such finite to…

Equivariant T-duality connects bundles with twists.

The paper explores conic-line arrangements via Poncelet's theorem and finds families of reducible curves.

The transition maps for a Sobolev $G$-bundle are not continuous in the critical dimension and thus the usual notion of topology does not make sense. In this work, we show that if such a bundle $P$ is equipped with a Sobolev connection $A$, then one can associate a topological isomorphism class to the pair $\left( P, A\…

We study codimension one foliations with singularities defined locally by Bott-Morse functions on closed oriented manifolds. We carry to this setting the classical concepts of holonomy of invariant sets and stability, and prove a stability theorem in the spirit of the local stability theorem of Reeb. This yields, among…

Given a representation of a link group, we introduce a trilinear form, as a topological invariant. We show that, if the link is either hyperbolic or a knot with malnormality, then the trilinear form equals the pairing of the (twisted) triple cup product and the fundamental relative 3-class. Further, we give some exampl…

We propose an intuitive interpretation for nontrivial $L^2$-Betti numbers of compact Riemann surfaces in terms of certain loops in embedded pairs of pants. This description uses twisted homology associated to the Hurewicz map of the surface, and it satisfies a sewing property with respect to a large class of pair-of-pa…

New algorithm optimizes DAGs by swapping node pairs to avoid cycles.