Let $w_1$ be the 1st Sitefel-Whitney classes of the tangent bundle of a 4-manifold $M$ ($M$ is non orientable). My question is

Is $$\exp\left(\frac{i\pi}{2}\int_{M_4} \mathcal{P}(w_1^2)\right)$$ well defined? Here $\mathcal{P}$ is the Pontryagin square.

It is known that for any $\mathbb{Z}_2$ cocycle $u$, the following quantity
$$\exp\left(\frac{i\pi}{2}\int_{M_4} \mathcal{P}(u)\right)$$ is well defined, in the sense that it does not depend on the lifting of $u$ from $\mathbb{Z}_2$ cocycle to $\mathbb{Z}_4$ cocycle. However, when we plug in $u=w_1^2$, it seems to be not well defined and depends on the lift of $w_1$, because $P(w_1^2)= w_1^2 \cup w_1^2+ 4 w_1^2 \cup_1 w_1^3 = w_1^2 \cup w_1^2\mod 4$. So there is a contradiction. How to resolve this contradiction?



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