# Can one show corbordism invariance of the Crane-Yetter state-sum using simplicial methods / are there 'Pachner-like' moves for cobordisms?

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.

• > 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. Sep 23, 2021 at 8:09

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).