This is a follow up more advanced question following Twisted spin bordism invariants in 5 dimensions. We follow the definitions in the earlier post.
I had discussed my computation of
$$ \Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{8}}= \mathbb Z_{32} \times \mathbb Z_{2}, $$
My specific question is how to pin down the $\mathbb Z_{32}$ generator as a topological invariant? (like characteristic class or eta invariant).
My current proposal is that the $\mathbb Z_{32}$ whether 1/2 of the $\mathbb Z_{32}$ is just the Postnikov square of $b \in H^2(B \mathbb Z_8, \mathbb Z_8)$.
$$\text{Postnikov square from $b \in H^2(B \mathbb Z_8, \mathbb Z_8)$ to $H^5(B \mathbb Z_8, \mathbb Z_{16})$}$$
Can this be proved or disproved, so falsified?
Let $X$ is any $\mathbb Z_{32}$ topological term in the dimension $d=5$,
then consider $X'$ be the pullback of $X$ from $\Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{8}}$ to $\Omega_5^{Spin \times_{} \mathbb Z_{8}}$,
we find:
$$X' \mod 2 =(a \mod 2) (b \mod 2)^2= Sq^2\big((a \mod 2) (b \mod 2)\big)$$ $$=(w_2(TM)+w_1(TM)^2)\big((a \mod 2) (b \mod 2)\big) =0,$$ the last equality is based on the Wu formula on a 5d spin manifold.
My question is $$\frac{X'}{2}=\text{Postnikov square of } b?$$ where $$a \in H^1(B \mathbb Z_8,\mathbb Z_8),\quad b \in H^2(B \mathbb Z_8, \mathbb Z_8).$$
PS. Recall, we find the Adams $\mathcal A$ module structure for $\Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{4}} =\mathbb Z_{16}$ and $ \Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{8}}= \mathbb Z_{32} \times \mathbb Z_{2}$, are provided in my earlier post Twisted spin bordism invariants in 5 dimensions .
