Invariants of flat connections on 4-manifolds from Hopf group-algebras

TL;DR

The paper develops an invariant of flat $G$-connections on 4-manifolds derived from finite type involutory quasitriangular Hopf $G$-algebras by coloring dotted Kirby diagram components with elements of $G$ and applying a 4D Hennings-type contraction to undotted components. It shows the invariant is well-defined and independent of diagrammatic choices, and proves multiplicativity under connected sum, with an induced manifold invariant obtained by summing over all flat $G$-connections when $G$ is finite. In the trivial connection case, the construction recovers the generalized dichromatic invariant and Crane–Yetter invariant, linking the approach to established 4D TQFT-like structures. The paper provides explicit computations for several 4-manifolds using a cyclic Hopf $G$-algebra, illustrating the interaction between group-theoretic colorings, quantum algebra, and 4-manifold topology, and highlighting the method’s potential for new invariants in 4D topology.$

Abstract

For a given group $G$, we construct an invariant of flat $G$-connections on 4-manifolds from a finite type involutory quasitriangular Hopf $G$-algebra. Hopf $G$-algebras are generalizations of Hopf algebras, equipped with gradings by $G$. In our construction, we color the dotted components of a Kirby diagram with elements of $G$ and employ the Hennings-type procedure. When $G$ is finite, we also define an invariant of 4-manifolds by summing the invariants over all flat $G$-connections.

Figures (31)

• Figure 1: Examples of Kirby diagrams
• Figure 2: A Kirby diagram of $S^1\times S^1\times S^2$. Let $s_1$ and $s_2$ be generators corresponding to the left and right dotted components, respectively. One can then verify that $\pi_1(S^1\times S^1\times S^2)=\langle s_1,s_2\mid s_1s_2s_1^{-1}s_2^{-1}\rangle\cong \mathbb{Z}\times\mathbb{Z}$.
• Figure 6: Contribution at a dotted component
• Figure 7: Contribution at a positive crossing (top) and a negative crossing (bottom)
• Figure 8: Evaluation of each undotted component

Theorems & Definitions (50)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4: radford2012hopf
• Lemma 2.5: radford2012hopf
• Lemma 2.6
• proof
• Definition 2.7
• Definition 2.8
• Remark 2.9