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The homotopy groups of $\mathrm{Q}S^0$ are the stable homotopy groups of spheres, of which we know the first 81 exactly, and the first 90 up to some uncertainties. A table is given in Wikipedia: enter image description here

Do we know what are the first few homotopy groups of the smash product $\mathrm{Q}S^0\wedge\mathrm{Q}S^0$?

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    $\begingroup$ My apologies for not being able to formulate this question more precisely; I'm pretty sure it can be done, and I tried my best to do so... $\endgroup$
    – Emily
    Commented Aug 15, 2021 at 2:02
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    $\begingroup$ I think your question is confusing two different tensor products. $S^0$ is the unit for the smash product of pointed sets, which is also the coproduct of commutative monoids therein. The tensor product described by Martin Brandenburg has unit $\mathbb N_+$. These are respectively the pointed analogues of $\oplus$ and $\otimes$ on unpointed commutative monoids. $\endgroup$
    – Marc Hoyois
    Commented Aug 15, 2021 at 8:55
  • $\begingroup$ @MarcHoyois Oh, gosh, you're right! $\endgroup$
    – Emily
    Commented Aug 15, 2021 at 10:29
  • $\begingroup$ @MarcHoyois By the way, is there an analogue of the tensor product $\otimes_0$ in the setting of commutative monoids in pointed spaces, and also of $\mathbf{N}$ and $\mathbf{N}_+$? $\endgroup$
    – Emily
    Commented Aug 15, 2021 at 10:30
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    $\begingroup$ I'm not sure, I don't really understand this pointed tensor product. The analogue of $\mathbb N$ however is the groupoid of finite sets, see mathoverflow.net/questions/401181/… $\endgroup$
    – Marc Hoyois
    Commented Aug 16, 2021 at 6:54

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