arXiv research

The Goresky-Hingston coproduct was first introduced by D. Sullivan and later extended by M. Goresky and N. Hingston. In this article we give a Morse theoretic description of the coproduct. Using the description we prove homotopy invariance property of the coproduct. We describe a connection between our Morse theoretic …

The paper studies obstructions to homotopy invariance of loop coproducts.

Proves loop coproduct invariance under simple homotopy equivalences.

String topology coproduct and Turaev cobracket computed for surfaces.

Finite spaces can be or not coproducts of subspaces.

Study path spaces and their homology, extending loop products and coproducts.

We give a variant of Naef's formula for the failure of invariance of the string topology coproduct under homotopy equivalences, using an obstruction class build from the higher homotopy data one can associate to a homotopy equivalence as well as the ``fake diagonal''. The vanishing of our obstruction class can be seen …

Let M be a closed Riemannian manifold. We extend the product of Goresky-Hingston, on the cohomology of the free loop space of M relative to the constant loops, to a nonrelative product. It is graded associative and commutative, and compatible with the length filtration on the loop space, like the original product. We p…

Study string topology on symmetric spaces, showing non-triviality and nilpotency results.

Introduces a new triple coproduct for knots on surfaces, preserving local crossing patterns.

The paper finds non-isotopic Legendrian unit conormal bundles in high dimensions.

A modular object in a symmetric monoidal bicategory is a Frobenius algebra object whose product and coproduct are biadjoint, equipped with a braided structure and a compatible twist, satisfying rigidity, ribbon, pivotality, and modularity conditions. We prove that the oriented 3-dimensional bordism bicategory of 1-, 2-…

The goal of this paper is to obtain restrictions on the prime to p quotient of the étale fundamental group of a smooth projective variety in characteristic $p\ge 0$. The results are analogues some theorems in the study of Kähler groups. Our first main result is that such groups are indecomposable under coproduct. The s…

Generalizations in many directions of the contraction procedure for Lie algebras introduced by E.J.Saletan are proposed. Products of arbitrary nature, not necessarily Lie brackets, are considered on sections of finite-dimensional vector bundles. Saletan contractions of such infinite-dimensional algebras are obtained vi…

We give a homotopy invariant construction of the Reidemeister trace for the coincidence of two maps between closed manifolds of not necessarily the same dimensions. It is realized as a homology class of the homotopy equalizer, which coincides with the Hurewicz image of Koschorke's stabilized bordism invariant. To defin…

We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and show that it satisfies analogues of the Eilenberg-Steenrod axioms except for the dimension axiom. The resulting long exact sequence of a pair generalizes various earlier long exact sequences such as the handle attaching sequence, …

In this paper, we investigate the behaviour of the Serre spectral sequence with respect to the algebraic structures of string topology in generalized homology theories, specificially with the Chas-Sullivan product and the corresponding coproduct and module structures. We prove compatibility for two kinds of fibre bundl…

The paper is devoted to introduce some notions extending the unique path lifting property from a homotopy viewpoint and to study their roles in the category of fibrations. First, we define some homotopical kinds of the unique path lifting property and find all possible relationships between them. Moreover, we supplemen…

New type of spaces with tangent structures for analysis.

We explore the graded and filtered formality properties of finitely generated groups by studying the various Lie algebras over a field of characteristic 0 attached to such groups, including the Malcev Lie algebra, the associated graded Lie algebra, the holonomy Lie algebra, and the Chen Lie algebra. We explain how thes…

Proof of injection from double shuffle to Kashiwara-Vergne Lie algebra.

Extends Lannes-Quillen theorem to all profinite groups.

Jacobi/Poisson algebras are algebraic counterparts of Jacobi/Poisson manifolds. We introduce representations of a Jacobi algebra $A$ and Frobenius Jacobi algebras as symmetric objects in the category. A characterization theorem for Frobenius Jacobi algebras is given in terms of integrals on Jacobi algebras. For a vecto…

Let $P$ be a Poisson algebra, $E$ a vector space and $π: E \to P$ an epimorphism of vector spaces with $V = {\rm Ker} (π)$. The global extension problem asks for the classification of all Poisson algebra structures that can be defined on $E$ such that $π: E \to P$ becomes a morphism of Poisson algebras. From a geometri…

Let $\mathfrak{L}$ be a Leibniz algebra, $E$ a vector space and $π: E \to \mathfrak{L}$ an epimorphism of vector spaces with $\mathfrak{g} = {\rm Ker} (π)$. The global extension problem asks for the classification of all Leibniz algebra structures that can be defined on $E$ such that $π: E \to \mathfrak{L}$ is a morph…

Developed a theory of stated SL(n)-skein modules for 3-manifolds.