Obstructions for gluing biset functors (Q1999352)

Let \(G\) be a finite group. A section of \(G\) is a pair \((U,V)\) of subgroups of \(G\) with \(V\) normal in \(U\), and we call subquotient of \(G\) the quotient \(U/V\). The normaliser of such section is the group \(N_G(U,V)=N_G(U)\cap N_G(V)\). Given finite groups \(H,K\), a \((K,H)\)-biset is a finite set equipped with a left action of \(K\) and a right action of \(H\) such that both actions commute. These bisets are by definition the morphisms in the biset category \(\mathcal B\), whose objects are all the finite groups. Given a commutative (unital) ring \(R\), a biset functor is an \(R\)-linear functor from \(R\mathcal B\) to the category of (finitely generated) \(R\)-modules. In this article, the authors investigate the gluing problem for biset functors on the subquotients of a finite group \(G\). This question, in the context of biset functors, was raised in the study of endo-permutation modules for finite \(p\)-groups. More generally, every \((K,H)\)-biset can be expressed as a composition of five types of bisets, called induction, inflation, isogation, deflation, and restriction bisets. For instance, given a section \((U,V)\) of \(G\), the destriction functor \(\operatorname{Defres}^G_{U/V}\) gives a morphism \(G\to U/V\) in \(\mathcal B\), and using only destrictions and isogation functors, we obtain the destriction functors. A gluing data for a biset or destriction functor \(F(G)\) on \(G\) is a sequence \((f_H)_{1< H\leq G}\) of elements \(f_H\in F(N_G(H)/H)\) satisfying two compatibility conditions. The gluing problem for the biset (or destriction) functor \(F\) at \(G\) is the problem of finding an element \(f\in F(G)\) such that for all nontrival subgroups \(H\) of \(G\), the destriction of \(f\) to \(N_G(H)/H\) is \(f_H\). In this article, the authors give a solution to the gluing problem in the form of an exact sequence exhibiting the gluing data (as an inverse limit) and a certain obstruction group to finding a solution to the gluing problem for \(F\) at \(G\). Partial results about the obstruction groups for specific biset functors have been obtained in previous works, and in the present article, the authors introduce a common framework in which all these obstruction groups can be calculated as the zero-th reduced cohomology group of a certain orbit category of sections. Thus they first re-state the gluing problem in terms of this new category. They also relate the obstruction groups to the reduced cohomology of the Quillen category when \(G\) is a finite \(p\)-group. In the last two sections of the paper, they specialise to the so-called rhetorical \(p\)-biset functors.