Operations in equivariant ℤ/p-cohomology

Abstract

If G is a compact Lie group and M a Mackey functor, then Lewis, May and McClure [4] define an ordinary cohomology theory H*_G(−; M) on G-spaces, graded by representations. In this article, we compute the ℤ/p-rank of the algebra of integer-degree stable operations [Ascr ]_M, in the case where G=ℤ/p and M is constant at ℤ/p. We also examine the relationship between [Ascr ]_M and the ordinary mod-p Steenrod algebra [Ascr ]_p.

The main result implies that while [Ascr ]_M is quite large, its image in [Ascr ]_p consists of only the identity and the Bockstein. This is in sharp contrast to the case with M constant at ℤ/p for q≠p; there [Ascr ]_M≅[Ascr ]_q.