The Heisenberg double of involutory Hopf algebras and invariants of closed 3–manifolds

Mihalache, Serban Matei¹ ; Suzuki, Sakie² ; Terashima, Yuji¹

¹ Department of Mathematics, Tohoku University, Sendai, Japan
² Department of Mathematical and Computing Science, Tokyo Institute of Technology, Tokyo, Japan

Algebraic and Geometric Topology, Tome 24 (2024) no. 7, pp. 3669-3691

Résumé

We construct an invariant of closed oriented 3–manifolds using a finite-dimensional involutory unimodular and counimodular Hopf algebra H. We use the framework of normal o–graphs introduced by R Benedetti and C Petronio, in which one can represent a branched ideal triangulation via an oriented virtual knot diagram. We assign a copy of the canonical element of the Heisenberg double ℋ(H) of H to each real crossing, which represents a branched ideal tetrahedron. The invariant takes values in the cyclic quotient ℋ(H)∕[ℋ(H),ℋ(H)], which is isomorphic to the base field. In the construction we use only the canonical element and structure constants of H, and not any representations of H. This, together with the finiteness and locality conditions of the moves for normal o–graphs, makes the calculation of our invariant rather simple and easy to understand. When H is the group algebra of a finite group, the invariant counts the number of group homomorphisms from the fundamental group of the 3–manifold to the group.

Détail

DOI : 10.2140/agt.2024.24.3669

Keywords: invariants of $3$–manifolds, Hopf algebras, Heisenberg doubles