A simple model of 4d-TQFT

We show that, associated with any complex root of unity $\omega$, there exists a particularly simple 4d-TQFT model $M_\omega$ defined on the cobordism category of Delta complexes. For an oriented closed 4-manifold $X$ of Euler characteristic $\chi(X)$, it is conjectured that the quantity $N^{3\chi(X)/2}M_\omega(X)$, where $N$ is the order of $\omega$, takes only finitely many values as a function of $\omega$. In particular, it is equal to 1 for $S^4$, $\left(3+(-1)^{N}\right)/2$ for $S^2\times S^2$, and $N^{-1/2}\sum_{k=1}^N\omega^{k^2}$ for $\mathbb{C} P^2$.