5:22 AM
so here's something that might be a neat topic for discussion, if anyone ever comes around and wants to talk: recall the spectra $X(k)$, which are the Thom spectra of the maps $\Omega SU(k)\to \Omega SU\simeq BU$. This is pretty much all I talk about these days... sorry.
Well, here's a neat fact, if $E$ is complex oriented then $E^\ast(\mathbb{C}P^k)\cong E_\ast[x]/(x^{k+1})$ and $E_\ast[b_1,\ldots,b_{k-1}]$, sooooo part of me is wondering whether or not one might compare $X(k)$ to $\mathbb{A}^k$, and the notion of an $n$-orientation is sort of similar to the cogroup structures I described above.
whoops, that should say $E_\ast(X(k))\cong E_\ast[b_1,\ldots,b_{k-1}]$
in other words, one notes that $\pi_\ast(HR\wedge X(k))$ is the affine $R$-line, so one might hope that $\pi_ast(X(k))$ is something like.... the affine $\mathbb{S}$-line. LOL
erm, affine $R$-$k-1$-space, or whatever