  The vacuum amplitude of the closed membrane theory is investigated using the
fact that any three-dimensional manifold has the corresponding Heegaard diagram
(splitting) although it is not unique. We concentrate on the topological aspect
with the geometry considered only perturbatively. In the simplest case where
the action describes the free fields we find that the genus one amplitudes
(lens space) are obtained from the S3 amplitude by merely renormalizing the
membrane tension. The amplitudes corresponding to the Heegaard diagram of genus
two or higher can be calculated as the Coulomb amplitudes with arbitrary charge
distributed on a knot or a link which corresponds to the set of branch points
of a given regular or an irregular covering space. We also discuss the case of
membrane instanton.
