Refinements of topological invariants of flows

Figure 1.  Reductions of a Hausdorff space into the abstract orbit space

Figure 2.  Left, the vector field $ X_0 $ on a unit closed ball $ M_0 $; middle, the vector field $ Y_1 $ on a handle $ C $; right, the vector field $ X_1 $ on a solid torus $ M_1 $

Figure 3.  Left, the vector field $ Y'_1 $; middle, the projection of the vector field $ Y'_1 $ into the $ x $-$ y $ plane; right, the vector field $ X_2 $ on $ M_2 $

Figure 4.  Left, attaching a copy $ C' $ of the handle $ C $ with the vector field $ Y_1 $; right, attaching a copy $ C' $ of the solid torus $ M_0 \cup C $ with the vector field $ X_1 $

Figure 5.  Left, attaching two copies of $ C $ to a copy of the ball $ \mathbb{D}^3 $ with the vector field $ -X_0 $; right, attaching a copy of $ C $ to the union of a copy of the ball $ \mathbb{D}^3 $ with the vector field $ -X_0 $ and a copy of $ C $ with the vector field $ -Y_1 $

Figure 6.  Left, the vector field $ -X_0 $; middle, the vector field $ -Y_1 $; right, the vector field $ X'_1 $ on $ M'_1 $

Figure 7.  A $ C^\infty $ diffeomorphism on $ \Sigma_2 $

Figure 8.  A schematic picture of the intersection of the stable manifolds of the saddle $ s_1 $ and the limit cycle $ \gamma $ and the unstable manifolds of the three saddles in $ M_2 $

Figure 9.  The list of singular points appeared in quasi-regular flows

Figure 10.  A trivial flow box, an open periodic annulus, and an open transverse annulus

Figure 11.  Self-connected separatrices and non-self-connected separatrices

Figure 12.  Commutative digram consisting of canonical projections among an orbit space, a Reeb graph, an abstract weak orbit space, and an extended weak orbit space of a Hamiltonian flow $ v $ with finitely many singular points on a compact surface $ S $

Figure 13.  Canonical projections and a canonical homeomorphism

Figure 14.  A neighborhood $ U' $ of the transverse $ T $ and its subsets

Figure 15.  The picture on the left is a Hamiltonian flow on a closed disk $ \mathbb{D}_1 $. In the diagram on the right, dotted (resp. up, down) directed lines correspond to $ \leq_\pitchfork $ (resp. $ \leq_\alpha $, $ \leq_\omega $)

Figure 16.  The picture on the left and right are homeomorphisms $ g $ and $ h $ on torus $ \mathbb{T}^2 $ which are time-one map of flows but not topologically equivalent, and whose suspension flows $ v_g $ and $ v_h $ are flows on closed 3-manifolds

Figure 17.  Two flows generated by Morse-Smale vector fields on a torus are not topologically equivalent but the abstract weak orbit spaces are isomorphic

Figure 18.  Two flows $ v $ and $ w $ on a disk $ \mathbb{D} $ that are not topologically equivalent and whose multi-saddle connection diagrams are isomorphic as abstract multi-graphs but not isomorphic as plane graphs. In the diagram, dotted (resp. up, down) directed lines correspond to $ \leq_\pitchfork $ (resp. $ \leq_\alpha $, $ \leq_\omega $). In particular, the extended weak orbit spaces are isomorphic and are pre-ordered sets with the pre-orders $ \leq_{v} $ each of which corresponds to the union of the pre-order $ \leq_v $ and the binary relations $ \leq_\alpha $, $ \leq_\omega $ as a direct product