Let a group $G=G_1 \times G_2$, where 

- $G_1$ is a discrete group (can be finite or infinite), 
- $G_2$ be any compact Lie group or finite group.

>Question: Is there some simple result that we can decompose the spin and pin bordism groups
$$\Omega^{Spin}_d(BG),$$
$$\Omega^{Pin^{+/-}}_d(BG),$$
into lower dimensions of
$\Omega^{Spin}_n(BG_1)$ and $\Omega^{Spin}_m(BG_2)$ and their sums or compositions.

For example, I can prove that
$$\Omega^{Spin}_d(B(\mathbb{Z} \times G_2))=
\Omega^{Spin}_d(B(G_2)) \times \Omega^{Spin}_{d-1}(B( G_2)),$$
when $G_2$ is any group. 

How about the decomposition of
>$$\Omega^{Spin}_d(B(G_1 \times G_2)),$$
>$$\Omega^{Pin^{+/-}}_d(B(G_1 \times G_2)),$$
for the criteria of $G=G_1 \times G_2$ given above?
Similar to Künneth formulas/theorem for bordism groups.


Related questions [here](https://mathoverflow.net/q/313521/27004).