The spectrum ko has a C_2 action coming from "taking the dual real vector bundle" (is this trivial?). Let E denote the homotopy fixed points. I suspect that pi_{-4}(E) = Z, up to 2-adic completion. Is there a good way of seeing this? I am particularly interested in the action of adams operations on this group...

@TomBachmann If ko = K(vect), then "taking the dual" is a symmetric monoidal functor vect -> vect^op. i can't think of a way to unwind this to a self-action of C_2 that doesn't essentially make use of an equivalence like K(C) ~ K(C^op) that uses the factorization through C^≃ being equivalent to (C^op)^≃ or the like. but the net effect of this on (⊔ BGL_n) is "inverse transpose" and that restricts to the identity on (⊔ BO_n). maybe you had something different in mind?

Seems like the only other reasonable option would be the map induced just by “inverse” instead of “inverse transpose” which isn’t the identity any more

(Or just by “transpose”, but as you say those restrict to the same thing on BO)

ah, right. sorry, i'm being slow today

I would like to believe that this action is trivial, yes.

If it’s trivial then you’re just asking for a computation of KO^*(RP^{infty}) and its Adams operations

Which goes back to Adams ;)

I feel stupid now ^^. But also delighted.

Haha that was not my intention! But anyway now you’ve got me interested in what the answer is for the nontrivial action

i'm still being slow. i see a self-map of K-theory spaces that I can deloop once but don't see the infinite delooping

Which paper of adams am I looking for?

“Vector fields on spheres”

@TylerLawson hmm I see what you’re saying. Maybe you’re right and we have to use a self equivalence with the opposite category which brings us back to the identity. I guess we know all the self maps of ko so we could probably check directly that we can’t do this...

i guess it’s E-infty but wrt the other E-infty structure... and the logarithm would just tell us to use (-1) instead as our automorphism. So maybe that’s the other interesting automorphism of ko? I wonder what its htpy fixed points are

@DylanWilson If you have looked at this paper before, would you mind letting me know how to read off ko^4(RP^oo) from Adams' computation?

Oh no! Adams only gives you KO^0. Try this: core.ac.uk/download/pdf/35274365.pdf

(You can also find it in Mahowald-Milgram “on operations which detect Sq^4...”)

I cannot find what I want in either paper. But I think I see how to deduce the result by comparison with ku.

Thanks!

Isn’t it in Theorem 1? In the row for i=4 mod 8? (Let m go to infty and you get the 2-adics)

For a third (fourth?) proof, the Atiyah-Segal completion theorem tells you you can compute the ring of virtual quaternionic representations of C_2 and group complete

I see. Great!