Relative Group Cohomology and The Orbit Category

Let G be a finite group and ℱ be a family of subgroups of G closed under conjugation and taking subgroups. We consider the question whether there exists a periodic relative ℱ-projective resolution for ℤ when ℱ is the family of all subgroups H ≤ G with rk H ≤ rkG − 1. We answer this question negatively by calculating the relative group cohomology ℱH*(G, 𝔽2) where G = ℤ/2 × ℤ/2 and ℱ is the family of cyclic subgroups of G. To do this calculation …