This result reflects a geometric construction of complex K-theory and suggests that a group theoretic description of a certain cohomology theory for BG might eventually lead to a geometric construction of the cohomology theory.
Let [Mathematical Expression Omitted] denote the cohomology theory, obtained by the Baas-Sullivan construction, with coefficient [Mathematical Expression Omitted] whose mod p reduction is the Morava K-theory K(n).
Having the right cohomology theory, the proof that [sub.2][[kappa].sup.4] is an invariant is almost cut and paste from [EM].
In [CCG], the authors introduced a cohomology theory for crossed modules (and implicitly for algebraic 3-types).
In general we expect that the cohomology theory at step n is a twist between the cohomology from step n - 1 with an appropriate cohomology theory that depends only on two groups.
(11)_____, On the cohomology theory
for associative algebras, Ann.