A split exact sequence of Mackey functors (Q756208)

The result of \textit{A. W. M. Dress} [Algebraic K-Theory II, Lect. Notes Math. 342, 183-240 (1973; Zbl 0331.18016)] shows that relative projectivity of a Mackey functor with respect to some subgroups allows to compute the functor in terms of those subgroups. As a refinement of this result, the author proves a theorem which also allows to compute a Mackey functor on a finite group G in terms of its values on certain subgroups of G using an action of G on a simplicial complex, as well as projectivity of the functor relative to certain subgroups. The theorem is also an extension of \textit{K. S. Brown}'s work [Homological Group Theory, Lond. Math. Soc. Lect. Note Ser. 36, 27-70 (1979; Zbl 0426.20038)] which describes the cohomology of a group in terms of its action on the simplicial complex of p-subgroups. The author of the paper presents many situations in which his theorem can be applied in computing the values of specific Mackey functors, and also deducing a structure theorem for the chain complex of Brown's simplicial complex of p-subgroups.