I don't know if this counts as homotopy theory, but I thought I'd give it a shot.

Has $\Omega^{Spin}_{10}$ been computed? I know it surjects onto $\mathbb{Z}_2$, but I don't know what it is. Is it $\mathbb{Z}_2$?

@MichaelAlbanese Does Theorem 3.1 here help: map.mpim-bonn.mpg.de/Spin_bordism

I am not sure. To be honest, I understand very little of what is written there.

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)

That's OK, I was hoping it was $\mathbb{Z}_2$, but unfortunately that's not the case.

Yes I think that's what I'm saying. But with the caveat that I'm also not totally sure if I interpret their result correctly.

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 believe this clarifies things a bit: map.mpim-bonn.mpg.de/…

@EricPeterson aren't we seeing pi_10 KO = Z/2 in all J summands (unless we see zero)?

I think there should be no $\mathbb{Z}$ summands.

oh, of course, you’re right: ko<10> starts in 10 but not with Z

Thanks for your help Tom.

Sure thing, glad I could help.

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)?

Intuitively it should be oplax symmetric monoidal (at least if the ⊗ on C commutes with colimits in each variable)

Yeah I've proven that actually, haha. But... It's not going to take algebras to algebras I don't think, right?

you want a map (colim_X A_x)⊗(colim_X B_x)=colim_{X×X}A_x⊗B_y→colim_X (A_x⊗B_x). So you need some kind of transfer along the diagonal X→X×X, I'd think

Hm

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?