arXiv research

Geometrically interprets a duality theorem linking cochain and chain complexes.

Simplicial versions of topological abelian gauge theories are constructed which reproduce the continuum expressions for the partition function and Wilson expectation value of linked loops, expressible in terms of R-torsion and linking numbers respectively. The new feature which makes this possible is the introduction o…

Discrete exterior calculus shows natural properties of wedge product and averaging.

In this paper we develop several algebraic structures on the simplicial cochains of a triangulated manifold that are analogues of objects in differential geometry. We study a cochain product and prove several statements about its convergence to the wedge product on differential forms. Also, for cochains with an inner p…

Characterizes Whitney forms on simplices and proves their uniqueness.

Develops combinatorial theory of vector bundles on simplicial complexes.

We propose a general notion of algebraic gauge theory obtained via extracting the main properties of classical gauge theory. Building on a recent work on transferring curved $A_{\infty}$-structures we show that, under certain technical conditions, algebraic gauge theories can be transferred along chain contractions. Sp…

Let X be a pseudomanifold. In this text, we use a simplicial blow-up to define a cochain complex whose cohomology with coefficients in a field, is isomorphic to the intersection cohomology of X, introduced by M. Goresky and R. MacPherson. We do it simplicially in the setting of a filtered version of face sets, also cal…

Let X be a topological space, and let C(X) be the complex of singular cochains on X with real coefficients. We denote by Cc(X) the subcomplex given by continuous cochains, i.e. by such cochains whose restriction to the space of simplices (endowed with the compact-open topology) defines a continuous real function. We pr…

Defines a simplicial operad related to Fulton-MacPherson.

Given a finite simplicial complex, a unimodular representation of its fundamental group and a closed twisted cochain of odd degree, we define a twisted version of the Reidemeister torsion, extending a previous definition of V. Mathai and S. Wu. The main tool is a complex of piecewise smooth currents, defined by J. Dupo…

We describe stable cup-i products on the cochain complex with $F^2$ coefficients of any augmented semi-simplicial object in the Burnside category. An example of such an object is the Khovanov functor of Lawson, Lipshitz and Sarkar. Thus we obtain explicit formulas for cohomology operations on the Khovanov homology of a…

We describe an $A_\infty$-quasi-equivalence of dg-categories between the first authors' $\mathcal{P}_{\mathcal{A}}$ ---the category of category of prefect $A^0$-modules with flat $\Z$-connection, corresponding to the de Rham dga $\mathcal{A}$ of a compact manifold $M$--- and the dg-category of \emph{infinity-local syst…

Geometrically solves differentiating simplicial manifolds.

The aim of this paper is to explain the relationship between the (co)homology of the free loop space and the Hochschild homology of its singular cochain algebra. We introduce all the relevant technical tools, namely simplicial and cyclic objects, and we provide the various steps of the proofs, which are scattered aroun…

The original de Rham cohomology due to Souriau and the singular cohomology in diffeology are not isomorphic to each other in general. This manuscript introduces a singular de Rham complex endowed with an integration map into the singular cochain complex which gives the de Rham theorem for every diffeological space. It …

We develop the intersection theory at relative chain-cochain level, and apply it along with the use of Seifert disks for an oriented link to give a combinatorial algorithm to compute Massey's higher order linking numbers. It is subtle to compute higher-order linking numbers, and it has been a folklore to use the inters…

New framework uses cohomology to analyze probabilistic distortions and arbitrage.

The uniform boundary condition in a normed chain complex asks for a uniform linear bound on fillings of null-homologous cycles. For the $\ell^1$-norm on the singular chain complex, Matsumoto and Morita established a characterisation of the uniform boundary condition in terms of bounded cohomology. In particular, spaces…

We describe algorithms for finding harmonic cochains, an essential ingredient for solving elliptic partial differential equations in exterior calculus. Harmonic cochains are also useful in computational topology and computer graphics. We focus on finding harmonic cochains cohomologous to a given cocycle. Amongst other …

New method compares geometric and standard cup products.

We propose a method for calculating cohomology operations for finite simplicial complexes. Of course, there exist well--known methods for computing (co)homology groups, for example, the reduction algorithm consisting in reducing the matrices corresponding to the differential in each dimension to the Smith normal form, …

The proof of Brouwer's fixed-point theorem based on Sperner's lemma is often presented as an elementary combinatorial alternative to advanced proofs based on algebraic topology. The goal of this note is to show that: (i) the combinatorial proof of Sperner's Lemma can be considered as a cochain-level version, written in…

New operations match Steenrod squares on Khovanov homology.

A fundamental theorem of Wolfe isometrically identifies the space of flat differential forms of dimension $m$ in $\mathbb{R}^n$ with the space of flat $m$-cochains, that is, the dual space of flat chains of dimension $m$ in $\mathbb{R}^n$. The main purpose of the present paper is to generalize Wolfe's theorem to the se…

The paper discusses $q$-deformations of the Aomoto complex.

In this paper, we consider the concept of connection cochain of central extensions introduced by Moriyoshi and apply it to the abelian case. We will show the relationship between connection cochain and connection $1$-form of a principal bundle whose structure group is abelian.

The aim of the present paper is to define a notion of weakly differentiable cochain in the generality of metric measure spaces and to study basic properties of such cochains. Our cochains are (sub-)linear functionals on a subspace of chains, and a suitable notion of chains in metric spaces is given by Ambrosio-Kirchhei…

We construct a pair of transverse genuine laminations on an atoroidal 3-manifold admitting transversely orientable uniform 1-cochain. The laminations are induced by the uniform 1-cochain and they are indeed the "straightening" of the coarse laminations defined in [Ca], by using minimal surface techniques. Moreover, whe…

We give a detailed and easily accessible proof of Gromov's Topological Overlap Theorem. Let $X$ be a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension $d$. Informally, the theorem states that if $X$ has sufficiently strong higher-dimensional expansion properties (which generali…

Paper constructs Thom-Smale complex using instantons from Morse functions.

VB-groupoids define a special class of Lie groupoids which carry a compatible linear structure. In this paper, we show that their differentiable cohomology admits a refinement by considering the complex of cochains which are k-homogeneous on the linear fiber. Our main result is a Van Est theorem for such cochains. We a…

Paper proves all Lagrangians unobstructed if one is, using non-archimedean analytic structure.

This paper studies covariant derivatives for Lie groupoids with representation-valued forms.

The paper computes KV cochain differentials and their geometric implications.

Paper bridges matching rules and height functions in aperiodic tilings.

This work redefines data-centric AI by unifying categorical and cochain notions.

We prove that the algebra of singular cochains on a smooth manifold, equipped with the cup product, is equivalent to the A-infinity structure on the Lagrangian Floer cochain group associated to the zero section in the cotangent bundle. More generally, given a pair of smooth manifolds of the same dimension with embeddin…

(1) For a compact Riemannian manifold without boundary $(M,g)$ containing $n+1$ points $p_i$ and the $n$-dimensional standard simplex $Δ$, the miniser of \[ E: M \times Δ\to {\mathbf R}, (a,λ) \mapsto λ^0 d^2(a,p_0) + \dots + λ^n d^2(a,p_n) \] is considered as point with "barycentric coordinates" $λ_i$ within the so-ca…

New complexes derived from any filtered cochain complex compute the same cohomology.

Invariants measure letter interleaving in groups, detecting group dimensions.

The abstract describes a strategy to construct reduced Khovanov homology for links in lens spaces.

Many quantum groups and quantum spaces of interest can be obtained by cochain (but not cocycle) twist from their corresponding classical object. This failure of the cocycle condition implies a hidden nonassociativity in the noncommutative geometry already known to be visible at the level of differential forms. We exten…

Let $G$ be a group and $N$ be a normal subgroup of $G$. There exists the group extension $G$ of $G/N$ by $N$. For a $G$-module $A$ which $N$ acts on trivially and a $G$-invariant homomorphism on $N$ to $A$, we obtain a central extension of $G/N$ by $A$. By using connection cochains, we exhibit the formula of its extens…

We elaborate on an idea of M. Abouzaid of equipping the Morse cochain complex of a smooth Morse function on a closed oriented manifold with the structure of an $A_\infty$-algebra. This is a variation on K. Fukaya's definition of Morse-$A_\infty$-categories for closed oriented manifolds involving families of Morse funct…

The classical Van Est theory relates the smooth cohomology of Lie groups with the cohomology of the associated Lie algebra, or its relative versions. Some aspects of this theory generalize to Lie groupoids and their Lie algebroids. In this paper, continuing an idea from [18], we revisit the van Est theory using the Per…

Seidel and Smith have constructed an invariant of links as the Floer cohomology for two Lagrangians inside a complex affine variety Y. This variety is the intersection of a semisimple orbit with a transverse slice at a nilpotent in the Lie algebra $sl_{2m}.$ We exhibit bijections between a set of generators for the Sei…

Geometric cohomology model uses co-oriented maps to define a product structure.