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In Caruso's paper, "Operations in equivariant $Z/p$-cohomology," http://www.ams.org/mathscinet-getitem?mr=1684248, he shows that the integer-graded stable cohomology operations in $RO(\mathbb{Z}/p)$-graded cohomology (with constant $\mathbb{Z}/p$ Mackey functor coefficients) have image in the usual (non-equivariant) mod-$p$ Steenrod algebra consisting of only the identity and Bockstein operations.

I would like to know:

  1. What do we know (if anything) about the rest of the cohomology operations, i.e. the ones which correspond to non-integer gradings and the unstable operations?

  2. What do we know about the image of the $RO(\mathbb{Z}/p)$-graded (un)stable operations in the Steenrod algebra for Bredon cohomology?

  3. Who is currently thinking about this stuff? (Besides me.)

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    $\begingroup$ There have been no responses, but many votes. Maybe a comment will get a reaction. There is a paper by Oruc (ams.org/mathscinet-getitem?mr=1003301) called "The Equivariant Steenrod Algebra." The gist of the paper is that given a mod $p$ Mackey functor field $\mathcal{F}$, the computation of the Hopf algebroid structure of the equivariant Steenrod algebra $H\mathcal{F}^*(H\mathcal{F})$ reduces to computing those structures on $HF^*(HF)$, where $F=\mathcal{F}(G/G)$ is a finite field of characteristic $p$. $\endgroup$
    – Bill Kronholm
    Apr 27, 2010 at 19:47
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    $\begingroup$ In Caruso's paper, and in my own work, the coefficient Mackey functor ring is constant $\mathbb{Z}/p$ coefficients ($p=2$ for me), which is NOT a mod $p$ Mackey functor field as it has non-trivial Mackey functor ideals. What mod $p$ Mackey functor fields are relevant for applications? $\endgroup$
    – Bill Kronholm
    Apr 27, 2010 at 19:58
  • $\begingroup$ Hu and Kriz compute the “dual” $\underline{\pi}_\star(H\mathbf{Z}/2 \wedge H\mathbf{Z}/2)$ in their paper on the $\mathbf{Z}/2$-equivariant Adams-Novikov spectral sequence (here $H\mathbf{Z}/2$ is the Eilenberg-Mac Lane spectrum associated to the constant Mackey functor on $\mathbf{Z}/2$). $\endgroup$
    – Sam Isaacson
    May 22, 2010 at 18:18

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