arXiv research

We consider $k$-dimensional random simplicial complexes that are generated from the binomial random $(k+1)$-uniform hypergraph by taking the downward-closure, where $k\geq 2$. For each $1\leq j \leq k-1$, we determine when all cohomology groups with coefficients in $\mathbb{F}_2$ from dimension one up to $j$ vanish and…

Random branched covers of groups are homotopy equivalent to geometrically small cancellation complexes.

Given an edge-independent random graph G(n,p), we determine various facts about the cohomology of graph products of groups for the graph G(n,p). In particular, the random graph product of a sequence of finite groups is a rational duality group with probability tending to 1 as n goes to infinity. This includes random ri…

The study explores finite quotients of 3-manifold groups and their existence and non-existence.

We prove a motivic stabilization result for the cohomology of the local systems on configuration spaces of varieties over $\mathbb{C}$ attached to character polynomials. Our approach interprets the stabilization as a probabilistic phenomenon based on the asymptotic independence of certain *motivic random variables*, an…

The vanishing of reduced $\ell^2$-cohomology for amenable groups can be traced to the work of Cheeger & Gromov. The subject matter here is reduced $\ell^p$-cohomology for $p \in ]1,\infty[$, particularly its vanishing. Results showing its triviality are obtained, for example: when $p \in ]1,2]$ and $G$ is amenable; whe…

We prove a rigidity theorem for the geometry of the unit ball in random subspaces of the scl norm in B_1^H of a free group. In a free group F of rank k, a random word w of length n (conditioned to lie in [F,F]) has scl(w)=log(2k-1)n/6log(n) + o(n/log(n)) with high probability, and the unit ball in a subspace spanned by…

New approach to Yang-Mills measure on surfaces via Morse theory.

We prove geometric and cohomological stabilization results for the universal smooth degree $d$ hypersurface section of a fixed smooth projective variety as $d$ goes to infinity. We show that relative configuration spaces of the universal smooth hypersurface section stabilize in the completed Grothendieck ring of variet…

We obtain sharp estimates on the growth rate of stable commutator length on random (geodesic) words, and on random walks, in hyperbolic groups and groups acting nondegenerately on hyperbolic spaces. In either case, we show that with high probability stable commutator length of an element of length $n$ is of order $n/\l…

We find new bounds on the conformal dimension of small cancellation groups. These are used to show that a random few relator group has conformal dimension 2+o(1) asymptotically almost surely (a.a.s.). In fact, if the number of relators grows like l^K in the length l of the relators, then a.a.s. such a random group has …

Extends cohomology theory for infinite volume transformation groups.

Generics extended to new cohomologies.

In this short note we define a new cohomology for a Lie algebroid $\mathcal{A}$, that we call the \emph{twisted cohomology} of $\mathcal{A}$ by an odd cocycle $θ$ in the Lie algebroid cohomology of $\mathcal{A}$. We proof that this cohomology only depends on the Lie algebroid cohomology class $[θ]$ of the odd cocycle $…

The article examines twisted cohomologies on algebraic and analytic varieties.

We view Dolbeault-Morse-Novikov cohomology H^{p,q}_η(X) as the cohomology of the sheaf Ω_{X,η}^p of η-holomorphic p-forms and give several bimeromorphic invariants. Analogue to Dolbeault cohomology, we establish the Leray-Hirsch theorem and the blow-up formula for Dolbeault-Morse-Novikov cohomology. At last, we conside…

Study cohomology of hemistrict Lie 2-algebras, proving isomorphic results.

New cohomology theories for heaps and ternary operations linked to group cohomology.

Study characterizes cohomology of Vaisman manifolds, linking Bott-Chern and Dolbeault numbers.

Proves a vanishing property for symplectic manifold cohomology.

De Rham theorem extended to Orlicz cohomology.

In this paper we define a new cohomology of a smooth manifold called Lichnerowicz type cohomology attached to a function. Firstly, we study some basic properties of this cohomology as: a de Rham type isomorphism, dependence on the function, singular forms, relative cohomology, Mayer-Vietoris sequence, homotopy invarian…

New proof of blow-up formula for Morse-Novikov cohomology.

Compute local cohomology of vector fields on manifolds.

Researchers calculate cohomological dimensions of manifold configuration spaces, proving arithmeticity and providing bounds.

The paper categorifies matroid characteristic polynomials using cohomology.

Deligne cohomology can be viewed as a differential refinement of integral cohomology, hence captures both topological and geometric information. On the other hand, it can be viewed as the simplest nontrivial version of a differential cohomology theory. While more involved differential cohomology theories have been expl…

Develops calculus for random submanifolds using zonoids.

Unified theory of orbifolds and cohomology.

New cohomology theory for diffeological spaces developed.

This study introduces a unified cohomology theory for braided algebras.

Investigates Künneth formula for foliated de Rham cohomology, overcoming non-Hausdorff issues.

The blow-down map is studied in Lie algebroid cohomology.

We give a definition of differentiable cohomology of a Lie group G (possibly infinite-dimensional) with coefficients in any abelian Lie group. This differentiable cohomology maps both to the cohomology of the group made discrete and to Lie algebra cohomology. We show that the secondary characteristic classes of Beilins…

New cohomological obstruction found for astheno-Kahler metrics.

Researchers redefine $\ell^\infty$-cohomology for groups and spaces, linking it to amenability, hyperbolicity, and algorithmic undecidability.

Inequalities for symplectic cohomology groups are derived.

Extends Adams' theorem to periodic cohomology.

Study shows Dolbeault cohomology is unchanged by complex Lie group actions.

We relate $L^{q,p}$-cohomology of bounded geometry Riemannian manifolds to a purely metric space notion of $\ell^{q,p}$-cohomology, packing cohomology. This implies quasi-isometry invariance of $L^{q,p}$-cohomology together with its multiplicative structure. The result partially extends to the Rumin $L^{q,p}$-cohomolog…

Analyses cohomology relations for moving frames and coframes.

Defines log Floer cohomology for symplectic surfaces with a degenerate part.

We study the tangential Poisson cohomology (TP-cohomology) of regular Poisson manifolds, first defined by Lichnerowicz using contravariant tensor fields. We show that for a regular Poisson manifold M, the TP-cohomology coincides with the leafwise de Rham (or Cech) cohomology of the symplectic foliation of M. Its comput…

Cohomology fractals illustrate complex 3-manifold properties.

Study on twisted Dolbeault cohomology in Kähler foliations.

In this article, we introduce a new cohomology theory associated to a Lie 2-algebras. This cohomology theory is shown to extend the classical cohomology theory of Lie algebras; in particular, we show that the second cohomology group classifies an appropriate type of extensions.

Cohomology defines hyperbolic spaces and their subgraphs.

New links show arbitrarily large torsion in Khovanov cohomology.