The third partial cohomology group and existence of extensions of semilattices of groups by groups

TL;DR

This paper explores the third partial cohomology group and its role in determining the existence of extensions of semilattices of groups by groups, providing a cohomological framework for classification and obstructions.

Contribution

It introduces the concept of a partial abstract kernel and links the third cohomology group to extension existence and classification.

Findings

H^3(G,C(A)) relates to obstructions for extensions

Extensions, if they exist, are classified by H^2(G,C(A))

Provides a cohomological approach to semilattice of groups extensions

Abstract

We introduce the concept of a partial abstract kernel associated to a group G and a semilattice of groups A and relate the partial cohomology group H^3(G,C(A)) with the obstructions to the existence of admissible extensions of A by G which realize the given abstract kernel. We also show that if such extensions exist then they are classified by H^2(G,C(A)).