The Kuperberg invariant is a topological invariant of closed
–manifolds
based on finite-dimensional Hopf algebras. In this paper, we initiate the program of constructing
–manifold invariants in the
spirit of Kuperberg’s
–manifold
invariant. We utilize a structure called a Hopf triplet, which consists
of three Hopf algebras and a bilinear form on each pair subject to
certain compatibility conditions. In our construction, we present
–manifolds
by their trisection diagrams, a four-dimensional analog of Heegaard diagrams. The
main result is that every Hopf triplet yields a diffeomorphism invariant of closed
–manifolds.
In special cases, our invariant reduces to Crane–Yetter invariants and generalized
dichromatic invariants, and conjecturally Kashaev’s invariant. As a starting
point, we assume that the Hopf algebras involved in the Hopf triplets are
semisimple. We speculate that relaxing semisimplicity will lead to even richer
invariants.
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