@TomBachmann Ok, I'm not sure this is an "easy" proof but I think it is as easy as it could be. The Grothendieck construction on the functor i→I_{/i} is Tw I→I^{op} and the cocartesian edges are exactly those squares that are sent to equivalences by the target map. So it suffices to show that the target map Tw I → I is a localization. I will prove more generally that every cocartesian fibration with weakly contractible fibers is a localization.

So, let p:E→B be a cocartesian fibration with weakly contractible fibers. I need to prove that for every category C the functor $p^*:Fun(B,C)→Fun(E,C)$ is fully faithful with essential image the functors that invert the arrows in the fibers of E. Let me first assume that C has $\kappa$-small colimits, where $\kappa$ is a regular cardinal such that all the fibers of $p$ are $\kappa$-small. Then there is a left Kan extension functor $p_!:Fun(E,C)→Fun(B,C)$, left adjoint to $p^*$.

Moreover $p^*$ is fully faithful iff $p_!p^*→1$ is an equivalence. But since $p_!$ is obtained by taking colimits along the fibers the map is indeed an equivalence, since the fibers are weakly contractible. Moreover the essential image of $p^*$ consists of those functors such that $F→p^*p_!F$ is an equivalence. This is obviously contained in those functors that invert the arrows in the fibers, but if $F$ inverts the arrows in the fibers, then $F→p^*p_!F$ is an equivalence, again by the fact that

the lke is obtained by taking the colimit along the fibers and that the fibers are weakly contractible. So we just need to reduce to the case when $C$ has enough colimits. But note that the property is preserved by restrictions to full subcategories, so it is enough to show that every $C$ can be embedded in a category with enough colimits. But to do this it suffices to embed $C$ in a subcategory of $P(C)$ closed under $\kappa$-small colimits.

[You probably need to be more careful than I've been with set theoretic issues if the fibers of p are not small categories]

Part of Adams' classical work on the J-homomorphism is the construction of elements \mu_{8k+1} in the stable stems which are detected in K-theory, not in the image of J, and generated a Z/2-summand. He furthermore shows that \eta*\mu_{8k+1} has the same properties.

Multiplying with \eta once more (\eta^2*\mu_{8k+1}) lets one land in the image of J (see e.g. math.cornell.edu/~hatcher/stemfigs/stems.html). To whom should one attribute the latter fact? Is it already contained in Adams' work? I skimmed through his papers, but could not find it.

Anyone know a cite-able reference for the homotopy groups of SU(n)? At least, the ones with the nice pattern?

I feel like... maybe some old Seminaire Henri Cartan or something?

@PeterNelson I use \newcommand{\morph}{\mathop{\longrightarrow}\limits}, and then write A \morph^f B. Doesn't scale to the name of the morphism though.