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Denis Nardin
Denis Nardin
10:45 AM
@RuneHaugseng Sure, in fact I believe you can even work with a multicategory of symmetric sequences and it'll work (I suspect this is more or less what Ching is doing). Still, it does require a little bit of care
 
 
7 hours later…
Urs Schreiber
Urs Schreiber
5:40 PM
Hey, I have a basic example in genuine G-spectra that I'd like to better understand. If anyone has an idea, I'd be grateful. Might be trivial, or not:
I have two finite groups G1 and G2 and consider genuine equivariant spectra under their direct product G = G1 x G2.
Then I have a representation V of G1 x G2,
subject to the condition that both G1 and G2 fix only the origin, hence that for the representation sphere (S^V)^G1 = ((S^V)^G2) = (S^V)^{G1 x G2} = S^0
Now I want to consider the G1 x G2-equivariant stable cohomotopy of the point,
but in RO-degree V.
By applying the partial fixed point spectrum functor corresponding to the two normal subgroup inclusions Gi --> G1 x G2 we can extract from any cocycle in RO-degree V, a pair of cocycles in degree 0.
(by my assumption on G1 and G2 both fixing only the 0-sphere in the representation sphere of V, and appealing to tom Dieck splitting).
This construction should given a map from
the G1 x G2-equivariant stable cohomotopy of the point in RO-degree V
to the direct sum Gi-equivariant stable cohomotopy of the point in degree 0, for i =1,2.
My question is: How surjective is this map?
Can one say anything?
That stable cohomotopy in degree 0 is of course given by the Burnside ring. So another way to state the question is:
Can we find the Burnside rings A(G1) and A(G2) as direct summands in the G1 x G2-equivariant cohomotopy of the point, in RO-degree V? Or if not all of them, maybe some part?
Thanks for any comments! Feel free to tell me it's trivial, and that I am an idiot. As long as it's true :-)
 

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