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, 𝔽₂) 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.

Key Words:

• Group cohomology
• Higher limits
• Orbit category

2010 Mathematics Subject Classification:

• Primary: 20J06
• Secondary: 55N25

ACKNOWLEDGMENTS

The material in the first two sections of the paper is part of the first author's Ph.D. dissertation. The first author thanks her thesis advisor Ian Hambleton for his constant support during her Ph.D. Both authors thank Ian Hambleton for his support which made it possible for the authors to meet at McMaster University where the computations appearing in Section 4 were done. The material appearing in the last two sections is derived from the second author's discussions with Peter Symonds during one of his visits to Bilkent University. The second author thanks TÜB\. ITAK for making this visit possible and Peter Symonds for introducing him to the connections between higher limits and relative group cohomology.

The second author is partially supported by TÜB\. ITAK-TBAG/110T712 and by TÜB\. ITAK-B\. IDEB/2221 Visiting Scientist Program.

Notes

Communicated by P. Tiep.