A simple model of 4d-TQFT

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

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