Involutory Hopf algebras and 3-manifold invariants

Details Involutory Hopf algebras and 3-manifold invariants Involutory Hopf algebras and 3-manifold invariants

1990-05-19

arxiv.org/abs/math/9201301v1

Internat. J. Math. 2 (1991), no. 1, 41--66

Mathematics

Quantum Algebra

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Scientific paper

We establish a 3-manifold invariant for each finite-dimensional, involutory Hopf algebra. If the Hopf algebra is the group algebra of a group $G$, the invariant counts homomorphisms from the fundamental group of the manifold to $G$. The invariant can be viewed as a state model on a Heegaard diagram or a triangulation of the manifold. The computation of the invariant involves tensor products and contractions of the structure tensors of the algebra. We show that every formal expression involving these tensors corresponds to a unique 3-manifold modulo a well-understood equivalence. This raises the possibility of an algorithm which can determine whether two given 3-manifolds are homeomorphic.

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