Obstructions for gluing biset functors

TL;DR

This paper introduces an obstruction theory for the gluing problem in biset functors, utilizing cohomology groups of a specialized category, and applies it to compute obstructions for the Dade group of odd p-groups.

Contribution

It develops a new obstruction theory for the gluing problem in biset functors and applies it to compute specific obstruction groups.

Findings

Obstruction groups are reduced cohomology groups of a category ${ ext{\mathcal D}_G$.

Calculated the obstruction group for the Dade group of odd p-groups.

Provided a framework for analyzing existence and uniqueness in gluing problems.

Abstract

We develop an obstruction theory for the existence and uniqueness of a solution to the gluing problem for a destriction functor and apply it to some well-known biset functors. The obstruction groups for this theory are reduced cohomology groups of a category ${\mathcal D}_G$, whose objects are the sections $(U,V)$ of $G$ with $V\neq 1$, and whose morphisms are defined as a generalization of morphisms in the orbit category. Using this obstruction theory, we calculate the obstruction group for the Dade group of a $p$-group when $p$ is odd.