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Fact: The spin cobordism is equivalent to "KO" theory in low dimension if we only consider the 2-torsion.

This is related to a question and an answer supports the claim.

Here we denote the $p$-torsion part (mean $\mathbb{Z}_{p^n}$ for some $n$) $$\Omega_d(BG)_p.$$

Question: Then do we have $$\Omega_d^{Pin^{+}}(BG)_p = ko_d^{+}(BG)_p?$$ $$\Omega_d^{Pin^{-}}(BG)_p = ko_d^{-}(BG)_p?$$ for $p=2$ and free part, for $d\le 7$? for certain versions analogous to $ko$ theories? Here $ko_d^{+}$ and $ko_d^{-}$ for $Pin^{+}$ and $Pin^{-}$, just means the some unknown $ko$-like theories, are there such K-theories?

p.s. How about higher $d>7$? If this is a statement about the spectra, not just about stable homotopy groups, and thus within these Pin cobordism and ko theory, do they completely coincide for any dimensions $d$, instead of just $d \leq 7$?

p.s.2. Namely, the 2-torsion and free part of $MPin$ and $KO$ is the same. If there is an odd $p$ torsion, we need to consider localization at odd prime by $MO$ cohomology. Is this correct?

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  • $\begingroup$ Just for a curiosity, what is so special about $MPin_*(BG)$ which isn't present in $MPin_*(X)$ for arbitrary space $X$? $\endgroup$
    – user43326
    Commented Nov 22, 2018 at 16:34
  • $\begingroup$ Thanks for the nice comment -- $BG$ is simply easier for me. But if you know more general space $X$, it is also fine. $\endgroup$
    – wonderich
    Commented Nov 22, 2018 at 17:03

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