$$H^1$$-conforming finite element cochain complexes and commuting quasi-interpolation operators on Cartesian meshes

A finite element cochain complex on Cartesian meshes of any dimension based on the $$H^1$$ H 1 -inner product is introduced. It yields $$H^1$$ H 1 -conforming finite element spaces with exterior derivatives in $$H^1$$ H 1 . We use a tensor product construction to obtain $$L^2$$ L 2 -stable projectors into these spaces which commute with the exterior derivative. The finite element complex is generalized to a family of arbitrary order.

Cohomology of BiHom-associative algebras

Bihom-associative algebras have been recently introduced in the study of group hom-categories. In this paper, we introduce a Hochschild type cohomology for bihom-associative algebras with suitable coefficients. The underlying cochain complex (with coefficients in itself) can be given the structure of an operad with a multiplication. Hence, the cohomology inherits a Gerstenhaber structure. We show that this cohomology also control corresponding formal deformations. Finally, we introduce bihom-associative algebras up to homotopy and show that some particular classes of these homotopy algebras are related to the above Hochschild cohomology.

Hochschild Cohomology, Monoid Objects and Monoidal Categories

This paper expands further on a category theoretical formulation of Hochschild cohomology for monoid objects in monoidal categories enriched over abelian groups, which has been studied in Hellstrøm-Finnsen (Commun Algebra 46(12):5202–5233, 2018). This topic was also presented at ISCRA, Isfahan, Iran, April 2019. The present paper aims to provide a more intuitive formulation of the Hochschild cochain complex and extend the definition to Hochschild cohomology with values in a bimodule object. In addition, an equivalent formulation of the Hochschild cochain complex in terms of a cosimplicial object in the category of abelian groups is provided.

Homological codes and abelian anyons

We study a generalization of Kitaev’s abelian toric code model defined on CW complexes. In this model, qudits are attached to [Formula: see text]-dimensional cells and the interaction is given by generalized star and plaquette operators. These are defined in terms of coboundary and boundary maps in the locally finite cellular cochain complex and the cellular chain complex. We find that the set of energy-minimizing ground states and the types of charges carried by certain localized excitations depends only on the proper homotopy type of the CW complex. As an application, we show that the homological product of a CSS code with the infinite toric code has excitations with abelian anyonic statistics.

Variations on the theme of the uniform boundary condition

The uniform boundary condition (UBC) in a normed chain complex asks for a uniform linear bound on fillings of null-homologous cycles. For the [Formula: see text]-norm on the singular chain complex, Matsumoto and Morita established a characterization of the UBC in terms of bounded cohomology. In particular, spaces with amenable fundamental group satisfy the UBC in every degree. We will give an alternative proof of statements of this type, using geometric Følner arguments on the chain level instead of passing to the dual cochain complex. These geometric methods have the advantage that they also lead to integral refinements. In particular, we obtain applications in the context of integral foliated simplicial volume.

Floer type cohomology on cosymplectic manifolds

We investigate a Floer type cohomology on cosymplectic manifolds M. To do this, we study a symplectic type action functional on the universal covering space of the loop space of contractible loops in M and the moduli space of gradient flow lines of the functional. The cochain complex induced by the critical points of the functional produces Floer type cohomology of M which is naturally isomorphic to a quantum type cohomology of M. We have an Arnold type theorem for Hamiltonian cosymplectomorphisms on compact semipositive cosymplectic manifolds. As an example, we consider the product of a Calabi–Yau 3-fold and the unit circle.

The Lifting-Extension Problem for Duchain Complexes of Dwyer and Kan

A duchain complex of W. Dwyer and D. Kan is a common extension of the notions of a chain complex and a cochain complex.Given a square commutative diagram of duchain complexes, the lifting-extension problem asks whether there exists a diagonal map making the two resulting triangles commute. Duchain complexes have a model category structure, and hence a lift exists if the left vertical map is a cofibration, the right vertical map is a fibration, and one of them is a weak equivalence.We show that it is possible to replace the two conditions above, by a countably infinite, bigraded, family of conditions which guarantee the existence of a lift.

Finite domination and Novikov rings: Laurent polynomial rings in two variables

Let C be a bounded cochain complex of finitely generated free modules over the Laurent polynomial ring L = R[x, x-1, y, y-1]. The complex C is called R-finitely dominated if it is homotopy equivalent over R to a bounded complex of finitely generated projective R-modules. Our main result characterizes R-finitely dominated complexes in terms of Novikov cohomology: C is R-finitely dominated if and only if eight complexes derived from C are acyclic; these complexes are C ⊗L R〚x, y〛[(xy)-1] and C ⊗L R[x, x-1]〚y〛[y-1], and their variants obtained by swapping x and y, and replacing either indeterminate by its inverse.