- The $SO$ bordism group of Eilenberg–MacLane space $K(\mathbb{Z}_2,2)$ is
$\Omega_4^{SO}(K(\mathbb{Z}_2,2))=\mathbb{Z}_4$.
- The cohomology group of $K(\mathbb{Z}_2,2)$ with $\mathbb{R}/\mathbb{Z}=U(1)$ coefficent, I suppose is also $$ H^4(K(\mathbb{Z}_2,2),U(1))=\mathbb{Z}_4. $$
Questions:
- What is the explicit group cohomology cocycle $\omega_4$?
where $$\omega_4\in H^4(K(\mathbb{Z}_2,2),U(1))=\mathbb{Z}_4.$$ Can we find all the $\mathbb{Z}_4$-class of $\omega_4$ written on the 4-simplex where the group elements $\mathbb{Z}_2$ is assigned to the 2-simplex (the 3-cell or the 3-triangle). What is this $\omega_4$ explicitly?
(p.s. I think the $\omega_4$ corresponds to the $\mathbb{Z}_2$ subgroup within $\mathbb{Z}_4$ is actually easier to write down, which looks like $$ \omega_4 =(-1)^{X \cup X} (?),$$ where $X \in H^2(K(\mathbb{Z}_2,2),U(1))$ or $H^2(K(\mathbb{Z}_2,2),\mathbb{Z}_2)$, where $X$ is assigned to the 2-simplex of the 4-simplex.) Moreover, how about the $\omega_4$ in the $\mathbb{Z}_4$ class? I think it is related to the Pontryagin square, but how to write in terms of group cocycle?
- What is the precise relation between $\Omega_4^{SO}(K(\mathbb{Z}_2,2))$ and $H^4(K(\mathbb{Z}_2,2),U(1))$?