  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.
