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Another (nicer) way to get the product in Tate cohomology, from a pairing of (co-)chain complexes, appears in Definition 4.1.5 of Hesselholt, Lars; Madsen, Ib On the K-theory of local fields. Ann. of Math. (2) 158 (2003), no. 1, 1–113.
This is not what you ask for, but perhaps it is worth mentioning that there is an $E_\infty$-algebra $gr_{ev}^* S/I_\infty$ in $gr_{ev}^* S$-modules, corresponding to the even commutative $MU_* MU$-comodule algebra $MU_*/I_\infty$, so $gr_{ev}^* MU/I_\infty$ exists as an $E_\infty$-$MU$-algebra in that category.
The formula is usually given for $\zeta(1-n)$, with $K_{2n-2}$ in the numerator and $K_{2n-1]$ in the denominator. When written as a formula for $\zeta(-n)$, the denominator must involve $K_{2n+1}$.
@FShrike Yes, Fritsch and Puppe (1967) proved that for each simplicial set $X$ there exists a homeomorphism $|sd(X)| \cong |X|$, where sd is Kan's normal subdivision. See e.g. section 2.3 in mn.uio.no/math/personer/vit/rognes/papers/aoms186-nocrop.pdf for more history, references, and generalizations. We write Sd and B for your sd and S.
The construction for Witt vectors of finite fields was effectively given in "Schwänzl, R.; Vogt, R. M.; Waldhausen, F. Adjoining roots of unity to $\mathbb{E}_\infty$ ring spectra in good cases—a remark. Homotopy invariant algebraic structures (Baltimore, MD, 1998), 245–249, Contemp. Math., 239, Amer. Math. Soc., Providence, RI, 1999", and these authors also proved that $\mathbb{Z}[i]$ cannot be lifted to the sphere.
Are any of the slightly more recent references in Section 6 of Sullivan, Dennis René Thom's work on geometric homology and bordism. Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 3, 341–350 ams.org/journals/bull/2004-41-03/S0273-0979-04-01026-2 of use to you?
From the chromatic point of view, the case p=2 is perhaps a bit surprising, since \pi_1 of the K-local sphere is (Z/2)^2, and only one of these comes from \pi_1 of the sphere. The definition of the connective image-of-J spectrum j involves some tweaking (compared to taking the connective cover of L_K(S)), precisely to get rid of that extra Z/2.
For \pi_3^S, do you use the binary tetrahedral group as a superperfect cover of A_4? Why is this already in the stable range for H_3 (which I expect starts at A_8)?
Isn't Nakaoka's integral (co-)homological stability range for \Sigma_p --> \Sigma_\infty more like * \le p/2 than * \le 2p-2 ? The mod p stability range is better, as you write.
