  We give homotopy invariant definitions corresponding to three well known
properties of complete intersections, for the ring, the module theory and the
endomorphisms of the residue field, and we investigate them for the mod p
cochains on a space, showing that suitable versions of the second and third are
equivalent and that the first is stronger. We are particularly interested in
classifying spaces of groups, and we give a number of examples.
  This paper follows on from arXiv:0906.4025 which considered the classical
case of a commutative ring and arXiv:0906.3247 which considered the case of
rational homotopy theory.
