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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.

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    $\begingroup$ 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 $\endgroup$ – Thomas Rot Dec 8 '18 at 18:28

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