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Let $\mathcal{C}$ denote some Unitary Braided Modular Fusion Category. It is well known that the Crane-Yetter state-sum, $Z_{CY}(\bullet|\mathcal{C})$ is an oriented-cobordism invariant. In other words, let $N^5$ be a 5-manifold with $\partial N^5=M_1^4 \sqcup \bar{M}_2^4$, which is a cobordism between 4-manifolds $M_1,M_2$. Then $Z_{CY}(M_1|\mathcal{C}) = Z_{CY}(M_2|\mathcal{C})$.

Using the fact that the oriented cobordism group $\Omega_4^{SO}=\mathbb{Z}$ is generated by $CP^2$ and the bordism invariant of $M$ is the signature $\sigma(M)$, we can express $Z_{CY}(M)$ in terms of the signature of $M$ and the value of $Z_{CY}(CP^2|\mathcal{C})$, in particular as $Z_{CY}(CP^2|\mathcal{C})^{\sigma(M)}$. There are well-known expressions for $Z_{CY}(CP^2|\mathcal{C})$ in terms of the Braided Fusion Category data.

Is there a proof of the cobordism invariance using combinatorial/triangulation-based methods? The proofs I've seen all use fairly abstract skein theory arguments - it'd be illuminating if there were a more hands-on approach.

It would also be nice if there were versions of the Pachner move theorem for cobordisms. In particular, are there a set of additional moves to the Pachner moves that can generate cobordisms, as opposed to just PL homeomorphism? Even comments about some generating set of 4-manifold cobordisms (e.g. via connected sums with some set of manifolds) would be helpful, even if they're not explicity translated into the language of triangulations. Recall that the PL-invariance of $Z_{CY}(\bullet|\mathcal{C})$ can be proved via Pachner moves.

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  • $\begingroup$ > It would also be nice if there were versions of the Pachner move theorem for cobordisms. I'm a bit surprised by this. The Pachner theorem comes from looking at specific triangulations of cylinders, so restricting bordisms to those that induce PL homeomorphisms on the boundaries. If you want to remove that restriction, you don't really have Pachner's theory anymore, but simply triangulated bordisms. I don't know what else you'd expect there. $\endgroup$ Sep 23, 2021 at 8:09

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I know very little about this topic, but I think this may be answered in this paper that studies a cubical analogue of Pachner moves:

Karim Adiprasito and Gaku Liu, Normal crossing immersions, cobordisms and flips.

Abstract: We study various analogues of theorems from PL topology for cubical complexes. In particular, we characterize when two PL homeomorphic cubulations are equivalent by Pachner moves by showing the question to be equivalent to the existence of cobordisms between generic immersions of hypersurfaces. This solves a question and conjecture of Habegger, Funar and Thurston.

Instead of working with manifolds broken up into $n$-simplices (triangulations), this result concerns manifolds which are broken up into $n$-cubes (a "cubulation") and defines combinatorial rules for going from one cubulation to another via making local changes (Definition 2.1).

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  • $\begingroup$ This is interesting, but seems like a special case and doesn't directly answer my question. In particular they only talk about PL homeorphisms of manifolds and cobordisms of some special objects. $\endgroup$
    – Joe
    Dec 4, 2020 at 4:45

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