Computing KO^-1 of RP^3 without AHSS

Question

I wanted to compute $\mathit{KO}^{-1}(\mathbb{R}P^3)$ and regrettably I could only think of using the Atiyah Hirzebruch spectral sequence, which seemed like a big overkill but looking at similar computations of $K^0(\mathbb{R}P^n)$ done in Atiyah's book do not look particularly simpler as he uses equivariant $\mathit{K}$-theory to get some short exact sequences involving the representation ring $R(\mathbb{Z}/2\mathbb{Z})$, which means having to deal in our case with real representations...

So is there a very simple way of computing $\mathit{KO}^{-1}(\mathbb{R}P^3)$? If not perhaps there is some trick which greatly simplifies the AHSS computation?

Thanks

Answer 1

The $KO$-theory of all truncated real projective spaces (including the calculation you want) was carried out very systematically by Frank Adams in his famous paper on the vector fields on spheres: see section 7 of [J.F.Adams, Ann.Math. (1962)]. He also invented/constructed `Adams operations' (not his term!) in this paper, and carefully calculates what they are doing too. His calculation proceeds by going through complex K theory and complex projective spaces enroute, so one has a good sense of every element that is constructed.