  In the time evolution of fluids, the topologies of fluids can be changed by
the creations and annihilations of singular points and by switching
combinatorial structures of separatrices. In this paper, to describe the
possible generic time evolution of Hamiltonian vector fields on surfaces with
or without constraints, we study the structure of the ``moduli space'' of such
vector fields under the non-existence of creations and annihilations of
singular points. In fact, we describe the relations of bifurcations between
Hamiltonian vector fields to construct foundations of descriptions of the time
evaluations of fluid phenomena. Moreover, we show that the space of
topologically equivalence classes of such vector fields has non-contractible
connected components and is a disjoint union of finite abstract cell complexes
such that the codimension of a cell corresponds to the instability of a
Hamiltonian vector field by using combinatorics and simple homotopy theory. In
particular, there is a connected component of the space that is weakly
homotopic to a three-dimensional sphere.
