I think I’m my question got answered (but tbh this isn’t the first time I’ve thought “ah, good, finally this is settled”)

Thanks, ch@

Does anyone know a reference for the mackey functor structure on the RO(C_2)-graded cohomology of a point with coefficients in F_2?

Do you just want the answer or do you want to cite something? Does Ferland-Lewis not give the Mackey functor structure? I thought they did.

If you just want the answer: you know the fixed point part, and you know the underlying piece is F_2 exactly when the topological dimension is 0. On that line, there are three types of behavior: constant Mackey functor, Mackey functor with tr=id, Mackey functor with zero on the fixed part. The first happens in negative weight, the third happens at that one place where the fixed part is zero, and the second happens at the rest of the points with positive weight

(here "weight" means "number of copies of the sign representation")

oh you said cohomology not homology: so reverse "positive" and "negative" in my answer

Thanks Dylan. I had forgotten to look there. Maybe you know a counterexample to the following. Suppose I have a free module over the RO(G)-graded cohomology of a point. Is that then free as a Mackey functor? I would think the answer is yes. But maybe there is a counter example.

When G=C_2 something like this is probably true. Clover May and I were just talking about this the other day; I think you use the fact that the Mackey functor structure is essentially determined by the action of a_{\sigma}, because the image of a_{\sigma} is the kernel of the restriction and the kernel is the image of the transfer. So if you knew the RO(G)-graded cohomology and the underlying cohomology you'd be good because exact sequences of vector spaces split, I think, so you can choose

a basis of the underlying thing where you know what the transfer does, or you know it comes from the restriction of something

but you need to know the underlying thing, because otherwise you can't tell the difference between a point and C_{2+}

(that second kernel above refers to the kernel of multiplication by a_{\sigma})

this is very special to C_2 though, and certainly false even for C_p when p is odd.

actually, sorry I take that back- I guess when things are free maybe it's okay; I was thinking of the more complicated question of recovering the Mackey functor structure just from the RO(G)-graded cohomology, in general, even when you don't have a free thing.

@DylanWilson How about we switch to email...

when things are free, can't you choose generators $\{x_i\}$ and represent them by maps $S^V \to H \wedge X$, then maybe choose some extra generators $y_i$ if you have C_2-free summands in underlying cohomology and represent those by generators $y_i: C_{2+} \wedge S^k \to H \wedge X$. Then smash all the sources with H, by H-linearity, and wedge them all together. By assumption you get an equivalence on underlying and geometric fixed pieces, so you're good

and the same argument, slightly souped up, works for C_p, at least...

oh- sure, sorry!

@Aaron: if you wanted to recover maps X -> Y you should be looking something more like symmetric monoidal left adjoints QC(Y) -> QC(X), generalizations of tannaka reconstruction will get you this sort of thing. exact functors form an additive category so not something that looks much like a set of maps between schemes

@QiaochuYuan ah good point, thanks -- the additive structure is a dead giveaway