Relative Group Cohomology and The Orbit Category

Abstract

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, F{double-struck}2) where G = ℤ/2 × ℤ/2 and ℱ is the family of cyclic subgroups of G. To do this calculation we first observe that the relative group cohomology ℱH*(G, M) can be calculated using the ext-groups over the orbit category of G restricted to the family ℱ. In second part of the paper, we discuss the construction of a spectral sequence that converges to the cohomology of a group G and whose horizontal line at E 2 page is isomorphic to the relative group cohomology of G. © 2014 Copyright Taylor & Francis Group, LLC

On the sectional category of subgroup inclusions and Adamson cohomology theory

“…We note that this result has been recently derived with different methods in [3], and also in [22] when G is a finite group.…”

Equivariant dimensions of groups with operators

“…For such coefficient modules, our definition agrees with that of Inassaridze [24], who defines the equivariant group cohomology of π with coefficients in N to be H * (π ⋊ G, G; N ), the relative group cohomology in the sense of Hochschild [23] and Adamson [1] (see also Benson [4,Section 3.9]). As observed in [36,Section 2], one has an isomorphism…”

Equivariant dimensions of groups with operators

“…Here we have switched notation from the expected H i ([K : F ]; M ) to be consistent with our references [36,37].…”

“…The coinduction functor is also known as the fixed-points functor. It is shown in [36,37,5] that applying restriction to an O F (K)-projective resolution…”

“…It is known that Adamson cohomology can be viewed as a special case of Bredon cohomology, where we take coefficients in the L-fixed points of a K-module [36,37,8,5]. Therefore cd([K : L]) ≤ cd L (K) always holds, and in the context of equivariant group cohomology we have cd([G ⋊ Γ : Γ]) ≤ cd G (G ⋊ Γ).…”