arXiv research

Unified Lie structures in homotopy and isotopy calculus.

Secondary Calculus formalizes PDEs using cohomology, simplifying their study.

The paper classifies links up to link-homotopy using claspers.

We describe a calculus of moves for modifying a framed flow category without changing the associated stable homotopy type. We use this calculus to show that if two framed flow categories give rise to the same stable homotopy type of homological width at most three, then the flow categories are move equivalent. The proc…

The paper focuses on various properties and applications of the homotopy operator, which occurs in the Poincaré lemma. In the first part, an abstract operator calculus is constructed, where the exterior derivative is an abstract derivative and the homotopy operator plays the role of an abstract integral. This operator …

Groups $Π_k(X;σ)$ of "flagged homotopies" are introduced of which the usual (abelian for $k>1$) homotopy groups $π_k(X;p)$ is the limit case for flags $σ$ contracted to a point $p$. Calculus of exterior forms with values in algebra $A$ is developped of which the limit cases are differential forms calculus (for $A=\bb R…

Global homotopies upgrade classical map in differential geometry.

We give an overview of how calculus of the embedding functor can be used for the study of long knots and summarize various results connecting the calculus approach to the rational homotopy type of spaces of long knots, collapse of the Vassiliev spectral sequence, Hochschild homology of the Poisson operad, finite type k…

The paper studies formal geometry of dg manifolds and proves isomorphism of their calculi.

We study the spaces of string links and homotopy string links in an arbitrary manifold using multivariable manifold calculus of functors. We construct multi-cosimplicial models for both spaces and deduce certain convergence properties of the associated Bousfield-Kan homotopy and cohomology spectral sequences when the a…

Study calculates homotopy groups and derivatives for disc diffeomorphisms.

This paper investigates the space of codimension zero embeddings of a Poincare duality space in a disk. One of our main results exhibits a tower that interpolates from the space of Poincare immersions to a certain space of "unlinked" Poincare embeddings. The layers of this tower are described in terms of the coefficien…

This thesis constructs families of arcs in 4-manifolds and analyzes their homotopy properties.

Two graph homologies help compute embedding space.

Families of objects appear in several contexts, like algebraic topology, theory of deformations, theoretical physics, etc. An unified coordinate-free algebraic framework for families of geometrical quantities is presented here, which allows one to work without introducing ad hoc spaces, by using the language of differe…

New research finds 145 infinite families of CS spheres are standard.

We study primary and secondary invariants of leafwise Dirac operators on foliated bundles. Given such an operator, we begin by considering the associated regular self-adjoint operator $D_m$ on the maximal Connes-Skandalis Hilbert module and explain how the functional calculus of $D_m$ encodes both the leafwise calculus…

The Lipshitz-Sarkar stable homotopy link invariant defines Steenrod squares on the Khovanov cohomology of a link. Lipshitz-Sarkar constructed an algorithm for computing the first two Steenrod squares. We develop a new algorithm which implements the flow category simplification techniques previously defined by the autho…

The Farey tree helps embed rational balls and lens spaces into complex projective space.

We develop a calculus for diagrams of knotted objects. We define Arrow presentations, which encode the crossing informations of a diagram into arrows in a way somewhat similar to Gauss diagrams, and more generally w-tree presentations, which can be seen as `higher order Gauss diagrams'. This Arrow calculus is used to d…

Study of embedding spaces using homotopy theory and operads.

We prove that in dimensions not equal to 4, 5, or 7, the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely generated in each degree. The proof uses homological stability, embedding calculus and the arithmeticity of mapping class …

The complex of "stable forms" on supermanifolds is studied. Stable forms on $M$ are represented by certain Lagrangians of "copaths" (formal systems of equations, which may or may not specify actual surfaces) on $M\times\mathbb R^D$. Changes of $D$ give rise to stability isomorphisms. The Cartan--de Rham complex made of…

This work continues the study of a homotopy-theoretic construction of the author inspired by the Bott-Taubes integrals. Bott and Taubes constructed knot invariants by integrating differential forms along the fiber of a bundle over the space of knots. Their techniques were later used by Cattaneo et al. to construct real…

Homotopy theory of differentiable sheaves connects manifold properties to underlying homotopy types.

The paper connects diffeomorphism groups and sphere embeddings, proving a group structure.

Detecting exotic spheres involves analyzing framed configuration spaces.

We give a complete obstruction to turning an immersion of an m-dimensional manifold M in Euclidean n-space into an embedding when 3n>4m+4. It is a secondary obstruction, and exists only when the primary obstruction, due to Haefliger, vanishes. The obstruction lives in a twisted cobordism group, and its vanishing implie…

Framed flow categories were introduced by Cohen-Jones-Segal as a way of encoding the flow data associated to a Floer functional. A framed flow category gives rise to a CW-complex with one cell for each object of the category. The idea is that the Floer invariant should take the form of the stable homotopy type of the r…

We study geometric variational problems for a class of effective models in quantum field theory known as Faddeev-Skyrme models. Mathematically one considers minimizing an energy functional on homotopy classes of maps from closed 3-manifolds into homogeneous spaces of compact Lie groups. The energy minimizers known as H…

We prove the existence and uniqueness of a projectively equivariant symbol map (in the sense of Lecomte and Ovsienko) for the spaces $D_p$ of differential operators transforming p-forms into functions. These results hold over a smooth manifold endowed with a flat projective structure. As an application, we classify the…

Abstract: Rational decomposition of homeomorphism spaces for manifolds.

Akbulut has recently shown that an infinite family of Cappell-Shaneson homotopy 4-spheres is diffeomorphic to the standard 4-sphere. In the present paper, a strictly larger family is shown to be standard by a simpler method. This new approach uses no Kirby calculus except through the relatively simple 1979 paper of Akb…

Paper addresses hidden faces in configuration space integrals for embeddings.

New algebraic structures for Lie 2-algebroids and their connections.

This article surveys the use of configuration space integrals in the study of the topology of knot and link spaces. The main focus is the exposition of how these integrals produce finite type invariants of classical knots and links. More generally, we also explain the construction of a chain map, given by configuration…

The paper explores deep learning through algebra and geometry, highlighting geometric structures and differential processes.

The LS-category of a topological space is a numerical homotopy invariant, introduced originally in a course on the global calculus of variations by Lyusternik and Schnirelmann, to estimate the number of critical points of a smooth function. When the topological space is a smooth manifold equipped with a proper action o…

Study embedding calculus and link invariants using functor calculus.

Algorithm solves covariant exterior derivative equations in small regions.

Embedding calculus proves convergence for surfaces.

Study geometric characterization of asymptotic pseudodifferential calculus on spinor bundles.

We explain that general differential calculus and Lie theory have a common foundation: Lie Calculus is differential calculus, seen from the point of view of Lie theory, by making use of the groupoid concept as link between them. Higher order theory naturally involves higher algebra (n-fold groupoids).(conceptual, topol…

In arXiv:1207.0332 [cs.LO] was proposed a graphic lambda calculus formalism, which has sectors corresponding to untyped lambda calculus and emergent algebras. Here we explore the sector covering knot diagrams, which are constructed as macros over the graphic lambda calculus.

Extends differential calculus to triole algebras.

A diagrammatic language for 3D manifolds with boundary.

Introduces tractors for basic examples and modern differential calculus.

We examine the N-Koszul calculus for the N-symmetric algebras. The case N=2 corresponds to the Elie Cartan calculus. We conjecture that, as in the case N=2, the N-Cartan calculus extends to manifolds when N>2, which would provide a new type of noncommutative differential geometry.