Cohomology Theories of Partial Groups
Sandro Pfammatter
TL;DR
This work initiates a cohesive program for cohomology of partial groups by developing two parallel frameworks: an algebraic cohomology theory extending Chermak’s partial group structure and a simplicial-local-coefficients theory. It proves these approaches are equivalent and then applies the resulting cohomology to the extension theory of partial groups, including explicit computations for extensions of free partial groups. The paper also demonstrates compatibility with the classical theory by translating group extensions into the partial-group setting via bar constructions. Overall, it provides robust invariants for partial groups and connects their cohomology to traditional homological algebra and obstruction theory.
Abstract
We initiate a systematic study of cohomology theories for partial groups, algebraic structures introduced by Chermak that generalize groups by allowing only partially defined products. Inspired by classical group cohomology, we develop two parallel approaches - an algebraic theory based on Chermak's framework and a simplicial-set-based theory using local coefficient systems - and show that they coincide. As an application, we illustrate how the extension theory of partial groups, as developed by Broto and Gonzalez, can be interpreted and computed using our cohomology theory, including explicit examples such as extensions of free partial groups, and compare these results with classical group extensions.