# Decomposition of bordism groups for $BG$ where $G$ is a product of two groups

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.

$$\Omega^{Spin}_d(B(G_1 \times G_2)),$$ $$\Omega^{Pin^{+/-}}_d(B(G_1 \times G_2)),$$ for the criteria of $$G=G_1 \times G_2$$ given above? Similar to Künneth formulas/theorem for bordism groups.
• Of course $B(G_1\times G_2)\cong BG_1\times BG_2$. I never read about this, but there are Künneth formulas for bordism groups, I believe due to Landweber here :jstor.org/stable/1994343?seq=1#page_scan_tab_contents . So you might use this to compute things – Thomas Rot Dec 8 '18 at 18:28