>**Fact**: [The spin cobordism is equivalent to "KO" theory in low dimension if we only consider the 2-torsion.](https://mathoverflow.net/q/312050/27004)

[This is related to a question and an answer supports the claim.](https://mathoverflow.net/q/312050/27004)

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?