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.
