It looks that there are different types of topological K-theories, with similar names but they are totally different outputs for the same input.
The first theory is called the KO theory. There are very limited information I can find on nlab.
The second theory is called the ko theory. Say the ko-Homology studied in Chapter 10 of this book.
My question is that is there a simple way to contrast the twos:
KO theory v.s. ko theory?
For example, we can compare:
$KO_d(BG)_p$
$ko_d(BG)_p$
with the spin bordism group:
- $\Omega_d^{Spin}(BG)_p$
Here we use the subindex to denote the $p$-torsion part (mean $\mathbb{Z}_{p^n}$ for some $n$). We can focus on $p=2$ and free part, for $d\le 7$; since there is a theorem given here, saying that $$ko_d(BG)_2=\Omega_d^{Spin}(BG)_2.$$