QUOTIENT SPACES AND TOPOLOGICAL INVARIANTS OF FLOWS

The paper proves completeness of the abstract weak orbit space equipped with the transverse order ≤∂ for (i) regular gradient flows on closed surfaces and (ii) flows generated by structurally stable Hamiltonian vector fields on compact surfaces (Propositions 10.1 and 10.2). The proofs reconstruct the flow from height-stratified blocks (singularities, separatrices, trivial flow boxes or periodic annuli) and show that ≤∂ encodes the necessary gluing information to recover the phase portrait . The candidate solution argues the same classification via an explicit block-by-block lifting/gluing construction using flow boxes and standard local models, which is consistent conceptually but distinct in style. Minor issues: the candidate asserts S/[v] is compact Hausdorff, whereas the paper treats it as a finite poset-stratified space (not Hausdorff in general) ; and the reduction to Reeb graphs is phrased more carefully in the paper via the extended weak orbit space . Overall, both are correct and compatible.