Recovering a cohomological Mackey Functor from its restriction to Sylow subgroups

TL;DR

This paper demonstrates how to reconstruct the top level of a cohomological Mackey functor from its restrictions to Sylow subgroups and applies this to compute G-equivariant homology groups for specific groups.

Contribution

It introduces a method to recover the top level of a cohomological Mackey functor from Sylow subgroup restrictions and applies it to compute equivariant homology groups for groups of order pq and A4.

Findings

Reconstruction of Mackey functor from Sylow subgroups

Calculation of G-equivariant homology for groups of order pq

Extension of methods to the group A4

Abstract

We explain how to recover the top level of a cohomological G-Mackey functor $\underline{M}$ from the restriction of $\underline{M}$ to each of the Sylow subgroups of G. As an application, we compute the Mackey functor valued G-equivariant homology groups of a point with constant $\mathbb{Z}$-coefficients when G has order pq for odd primes p<q. We also indicate how the calculation goes for $G=A_4$.