  For certain contractible G-CW-complexes and F a family of subgroups of G, we
construct a spectral sequence converging to the F-Bredon cohomology of G with
E1-terms given by the F-Bredon cohomology of the stabilizer subgroups. As
applications, we obtain several corollaries concerning the cohomological and
geometric dimensions of the classifying space for the family F. We also
introduce a hierarchically defined class of groups which contains all countable
elementary amenable groups and countable linear groups of characteristic zero,
and show that if a group G is in this class, then G has finite F-Bredon
(co)homological dimension if and only if G has jump F-Bredon (co)homology.
