On the conjectural cohomology for groups

TL;DR

This paper explores the properties of rack cohomology in relation to conjectural Leibniz cohomology for groups, establishing structural similarities and constructing algebraic morphisms.

Contribution

It demonstrates that rack cohomology shares key properties with conjectural Leibniz cohomology and introduces new algebraic structures and morphisms.

Findings

Rack cohomology has properties close to conjectural Leibniz cohomology.

Existence of a graded dendriform algebra structure on rack cohomology.

Construction of an injective algebra morphism from group cohomology to rack cohomology.

Abstract

The goal of this paper is to present results which are consistent with conjectures about the Leibniz (co)homology for discrete groups stated by J. L. Loday. We show that rack cohomology has properties very close to the properties expected for the conjectural Leibniz cohomology. In particular, we prove the existence of a graded dendriform algebra structure on rack cohomology, and we construct a graded associative algebra morphism from group cohomology to rack cohomology which is injective in degree 1.