It seems that from [this webpage](http://www.map.mpim-bonn.mpg.de/Spin_bordism#Classification), the spin cobordism is equivalent to KO theory in low dimension.


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


**Question 1**:
Then do we have
$$\Omega\_d^{spin}(BG)\_p = ko\_d(BG)\_p?$$ for $p=2$ and free part, for $d\le 7$? (how about higher $d>7$?)


And
$$
\Omega\_d^{spin}(BG)\_p = \Omega\_d^{SO}(BG)?
$$
for $p \neq 2$ and $p$ is an odd prime?


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


**Question 2**:
If this is a statement about the spectra, not just about stable homotopy groups, and thus within these spin cobordism and ko theory, do they completely coincide for any dimensions $d$, instead of just $d \leq 7$?



Spin cobordism v.s. KO theory in low or in any dimensions
First of all, let me say that the page you are quoting is a little bit 
misleading if not inaccurate on the Anderson-Brown-Peterson splitting. 


$ko\langle 4n(J)\rangle $ should read $\Sigma ^{4n(J)} ko$ and $ko\langle 4n(J)-2\rangle $ should read $\Sigma ^{4n(J)-4} ko\langle 2\rangle $


See, e.g. <https://pdfs.semanticscholar.org/c377/7dd83a6ba0d959c5c62d633ed4109cddb660.pdf> 


With this correction, at 2, $Mspin \wedge BG$ splits whose bottom piece is $ko\wedge BG$, other pieces are at least 7-connected since the "next bottom" piese is $\Sigma ^8ko \wedge BG$ as is pointed out by
ArunDebray, corresponding to the partition $J=(2)$.


Thus the answer to your questions, at the prime 2 is that 


1. We have an isomorphism up to $d\leq 7$
2. The map is always surjective, but the kernel is in general non-trivial for $d\geq 8$.


