arXiv research

New descriptions of a subgroup in mapping class groups.

The paper finds symplectic mapping class relations using pencil pairs.

Paper describes how to extend multiple conjugation quandles using maps.

The present research work proposes a new fast fixed-point averaging algorithm on the compact Stiefel manifold based on a mixed retraction/lifting pair. Numerical comparisons between fixed-point algorithms based on the proposed non-associated retraction/lifting map pair and two associated retraction/lifting pairs confir…

New findings on normal closure of maps for genus 0.

This paper centers around two basic problems of topological coincidence theory. First, try to measure (with help of Nielsen and minimum numbers) how far a given pair of maps is from being loose, i.e. from being homotopic to a pair of coincidence free maps. Secondly, describe the set of loose pairs of homotopy classes. …

Paper finds Hempel pairs distinguishable by Turaev-Viro invariants.

A quandle is an algebra whose axioms are motivated from knot theory. A linear extension of a quandle can be described by using a pair of maps called an Alexander pair. In this paper, we show that a linear extension of a multiple conjugation quandle can be described by using a pair of maps called an MCQ Alexander pair, …

Let {a,b} and {c,d} be two pairs of bounding simple closed curves on an oriented surface which intersect nontrivialy. We prove that if these pairs are invariant under the action of an orientation reversing involution, then the corresponding bounding pair maps generate a free group. This supports the conjecture stated b…

The paper solves conditions for non-singular extensions of fold maps.

A natural one-parameter family of Kähler quantizations of the cotangent bundle $T^*K$ of a compact Lie group $K$, taking into account the half-form correction, was studied in \cite{FMMN}. In the present paper, it is shown that the associated Blattner-Kostant-Sternberg (BKS) pairing map is unitary and coincides with the…

Surgery obstruction of a normal map to a simple Poincare pair $(X,Y)$ lies in the relative surgery obstruction group $L_*(π_1(Y)\toπ_1(X))$. A well known result of Wall, the so called $π$-$π$ theorem, states that in higher dimensions a normal map of a manifold with boundary to a simple Poincare pair with $π_1(X)\congπ_…

The paper classifies 10 antipodal pairings of self-dual maps.

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

The Torelli group, I(S_g), is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface. There are three types of elements that naturally arise in studying I(S_g): bounding pair maps, separating twists, and simply intersecting pair maps (SIP-maps). Historically the…

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.

A Lie group G in a group pair (D,G), integrating a Lie algebra g in a Manin pair (d,g) has a quasi-Poisson structure. We define the quasi-Poisson actions of such Lie groups G, that generalize the Poisson actions of Poisson Lie groups. We define and study the moment maps for those quasi-Poisson actions which are quasi-h…

New $L_\infty$ liftings derived from Chern-Simons classes for coherent sheaves.

The paper studies the moduli space of Higgs pairs and their geometric properties.

Differential forms on the Fréchet manifold F(S,M) of smooth functions on a compact k-dimensional manifold S can be obtained in a natural way from pairs of differential forms on M and S by the hat pairing. Special cases are the transgression map associating (p-k)-forms on F(S,M) to p-forms on M (hat pairing with a const…

In this paper, we the improve the bound for the moment map derivative proved by Donaldson in his recent proof of the Hilbert-Mumford stability of complex manifolds with constant scalar curvature. The proof depends on the identification of Donaldson's symplectic form with the curvature of a certain Deligne pairing.

Defines and computes geometric pairings for discrete groups using Baum-Connes assembly map.

The paper explores density of stable mappings and their properties.

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

Let $G$ be a complex Lie group and $ΛG$ denote the group of maps from the unit circle ${\mathbb S}^1$ into $G$, of a suitable class. A differentiable map $F$ from a manifold $M$ into $ΛG$, is said to be of \emph{connection order $(_a^b)$} if the Fourier expansion in the loop parameter $λ$ of the ${\mathbb S}^1$-family …

We study invertible generating pairs of fundamental groups of graph manifolds, that is, pairs of elements (g,h) for which the map g --> g^{-1}, h --> h^{-1} extends to an automorphism. We show in particular that a graph manifold is of Heegaard genus 2 if and only if its fundamental group has an invertible generating pa…

Computes minimal dilatation for Thurston maps on surfaces.

In this work, we ask the following question: Can visual analogies, learned in an unsupervised way, be used in order to transfer knowledge between pairs of games and even play one game using an agent trained for another game? We attempt to answer this research question by creating visual analogies between a pair of game…

Call a periodic map $h$ on the closed orientable surface $Σ_g$ extendable if $h$ extends to a periodic map over the pair $(S^3, Σ_g)$ for possible embeddings $e: Σ_g\to S^3$. We determine the extendabilities for all periodical maps on $Σ_2$. The results involve various orientation preserving/reversing behalves of the p…

A lamination of a graph embedded on a surface is a collection of pairwise disjoint non-contractible simple closed curves drawn on the graph. In the case when the surface is a sphere with three punctures (a.k.a. a pair of pants), we first identify the lamination space of a graph embedded on that surface as a lattice pol…

Constructs Chern-Weil classes for Cartan geometries.

Let $S_{g}$ denote the closed orientable surface of genus $g$. We construct exponentially many mapping class group orbits of pairs of simple closed curves which fill $S_{g}$ and intersect minimally, by showing that such orbits are in correspondence with the solutions of a certain permutation equation in the symmetric g…

Defines a new invariant for 4-manifolds with boundary.

The problem of splitting a homotopy equivalence along a submanifold is closely related to the surgery exact sequence and to the problem of surgery of manifold pairs. In classical surgery theory there exist two approaches to surgery in the category of manifolds with boundaries. In the $rel \partial$ case the surgery on…

This paper is a rigorous study of two dual pairs of momentum maps arising in the context of fluid equations whose configuration Lie group is the group of automorphism of a trivial principal bundle, generically called here non-abelian fluids. It is shown that the actions involved are mutually completely orthogonal, whic…

In this note, we unveil homotopy-rich algebraic structures generated by the Atiyah classes relative to a Lie pair $(L,A)$ of algebroids. In particular, we prove that the quotient $L/A$ of such a pair admits an essentially canonical homotopy module structure over the Lie algebroid $A$, which we call Kapranov module.

New bicombings found for mapping class groups and Teichmüller spaces.

A linking pairing is a symetric bilinear pairing lambda: GxG --> Q/Z on a finite abelian group. The set of isomorphism classes of linking pairings is a non-cancellative monoid E under orthogonal sum, which is infinitely generated and infinitely related. We propose a new presentation of E that enables one to detect whet…

Paired autoencoders solve inverse problems using latent space projections.

Given any connected compact orientable surface, a pair of mapping classes are said to be procongruently conjugate if they induce a conjugate pair of outer automophisms on the profinite completion of the fundamental group of the surface. For example, this occurs if they induce conjugate outer automorphisms on every char…

Given a null-homologous knot $K$ in a rational homology 3-sphere $M$, and the standard infinite cyclic covering $\tilde{X}$ of $(M,K)$, we define an invariant of triples of curves in $\tilde{X}$, by means of equivariant triple intersections of surfaces. We prove that this invariant provides a map $φ$ on $\Al^{\otimes 3…

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…

The paper introduces a group $LSP$ of obstructions for splitting a homotopy equivalence along a pair of submanifolds. We develop exact sequences relating the $LSP$-groups with various surgery obstruction groups for manifold triple and structure sets arising from triples of manifolds. The natural map from the surgery ob…

We describe for any Riemannian manifold a certain infinitesimal neighbourhood of the diagonal. Semi-conformal maps are analyzed as those that preserve such neighbourhoods; harmonic maps are analyzed as those that preserve mirror image formation for pairs of points in such neighbourhoods.

Alternative proof classifies Kleinian groups with two parabolics.

The note proves positive currents induced by VKE with mixed singularities.

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…

Method learns to map dynamics of different systems.