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? Recall that the PL-invariance of $Z_{CY}(\bullet|\mathcal{C})$ can be proved via Pachner moves.

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.

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. For example, are there versions of the Pachner move theorem for cobordisms? (Recall that the PL-invariance of $Z_{CY}(\bullet|\mathcal{C})$ can be proved via Pachner moves.)

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? Recall that the PL-invariance of $Z_{CY}(\bullet|\mathcal{C})$ can be proved via Pachner moves.