Involutory Hopf group-coalgebras and invariants of flat bundles over 4-manifolds
Nicolas Bridges, Shawn Cui
TL;DR
This work develops invariants of pointed flat $G$-bundles over closed 4-manifolds by combining colored trisection diagrams with involutory Hopf $G$-triplets and a tensor-network evaluation. The resulting invariant $Z((\xi,\tilde{x});G;\mathcal{H},e)$ is shown to be diagrammatically well-defined and independent of choices under natural conditions, unifying and extending prior invariants from Chaidez, Cotler, and Cui and Mochida. In special cases, the construction recovers known 4D invariants and reduces to Mochida's invariant for quasitriangular Hopf $G$-algebras, while also connecting to 3-manifold invariants via Hopf pairs/doublets. The paper further demonstrates invariants for flat bundles over 3-manifolds and provides concrete examples distinguishing nontrivial bundles and non-surjective monodromies, highlighting potential new topological distinctions in 4-manifolds.
Abstract
We give invariants of flat bundles over 4-manifolds generalizing a result by Chaidez, Cotler, and Cui (Alg. \& Geo. Topology '22). We utilize a structure called a Hopf $G$-triplet for $G$ a group, which generalizes the notion of a Hopf triplet by Chaidez, Cotler, and Cui. In our construction, we present flat bundles over 4-manifolds using colored trisection diagrams: a direct analogue of colored Heegaard diagrams as described by Virelizier. Our main result is that involutory Hopf $G$-triplets of finite type yield well-defined invariants of $G$-colored trisection diagrams, and that if the monodromy of a flat bundle has image in $G$ we obtain invariants of flat bundles. We also show that a special Hopf $G$-triplet yields the invariant from Hopf $G$-algebras described by Mochida, thus generalizing the construction.