Combinatorial structures of the space of Hamiltonian vector fields on compact surfaces

TL;DR

This paper explores the topological and combinatorial structures of Hamiltonian vector fields on surfaces, revealing complex moduli spaces with non-trivial topology, which are relevant for understanding fluid dynamics and bifurcations.

Contribution

It characterizes the moduli space of Hamiltonian vector fields on surfaces, showing its decomposition into finite cell complexes and identifying topological properties like non-contractibility.

Findings

The space of equivalence classes has non-contractible components.

The moduli space decomposes into finite cell complexes.

One component is weakly homotopic to a 3-sphere.

Abstract

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.