Hopf Algebra Conjecture

[Mike Hill]

So I thought I would put this out there. Andre and I at one point
worked out the following:

\pi_*(HZ_2 ^_{ko} HZ_2)=A(1)_*
\pi_*(HZ_2 ^_{eo2}HZ_2)=A(2)_*
\pi_*(HZ_3 ^_{eo2}HZ_3)=A(1)_*\otimes E(a_2)

These are Hopf algebras which allow us to compute quickly and easily
the homotopies of the ground ring spectra or of the ground ring
spectrum homology of any space using an Adams spectral sequence.

We also conjectured the following:

\pi_*(HZ_p ^_{eo(p-1)} HZ_p)=A(1)_*\otimes E(a_2,...,a_{p-1})

I can show that it should contain this A(1)_*, based on some sort of
sketchy, moral arguments. The rest should follow from the homotopy and
the roles of the additional elements.

This again gives a nice finite Hopf algebroid for computing Ext and
other stuff. Haynes said that he and Mike knew that there would be such
a thing, they just didn't know how to compute it.

Here is my conjecture:

\pi_*(HZ_p ^_{eo(p^k(p-1))} HZ_p)=A(k+1)_*\otimes E(a_2,...,a_{p-1})

Of course, this is really conjectural. Perhaps more of a shot in the
dark. I should say also that by eo(p^k(p-1)), I mean an appropriate
connective cover / ring of integers of EO_{p^k(p-1)} where we take the
maximal finite subgroup that contains Z_{p^{k+1}}.

Thoughts?

[Mike Hill]

One correction, on second thought. I think that the conjectural Hopf algebra should actually be A(p^k)_* times the rest, not A(k+1)_*. This seems to have more of the v_is in it, making it a better candidate.

--
Mais il avait compté sans une force qui, si elle est nourrie d'abord par vanité, vainc le dégoût, le mépris, l'ennui même : c'est l'habitude.