I believe it says (in particular) that Omega^Spin_10 contains a copy of Z/2, corresponding to pi_10(ko<8>) = pi_10(KO) = Z/2, coming from the first summand and taking the multi-index J=(2).
It seems to me that the second summand can contribute when J=(3) and give you another Z/2
There might conceivably be more Z/2's, corresponding to the last summand (but they don't say in which dimension the classes z_i live - you would have to check the reference)
the J = (3) summand looks to me like it contributes a Z, not a Z/2. also, though they don’t specify the exact formulas for the z’s, they say they’re all ≥ 20
I just realized that Tate valued Frobenius doesn't "commute" with multiplications. More precisely, given a commutative ring spectrum A, there are two maps from the tensor $A\otimes\dots\otimes A$ to $A^{tC_p}$: either firstly multiply then apply the Tate valued Frobenius, or firstly apply the Tate valued Frobenii componentwise then multiply - seemingly in general they don't coincide. Is there a good explanation?
If A is a spherical monoidal ring (commutative) then they coincide, because the space level versions (for commutative monoids) of these maps coincides.
Elementary question: if we have two functors f and g in Fun(X,C) for C symmetric monoidal, is the colimit functor (lax?) symmetric monoidal with respect to the pointwise monoidal structure on Fun(X,C)?
Is a modification in Fun(C, D) where C, D are bicategories (or really (oo,2)-categories) (and Fun means pseudofunctors with pseudonatural transformations) an equivalence if it is so pointwise?