The structure of Morse flows and co-dimension one gradient flows on the sphere with holes

TL;DR

The paper develops complete topological invariants for Morse and codimension-one gradient flows on the sphere with holes, focusing on flows with at most six singular points. It introduces distinguishing graphs and Morse flow codes (A-code and B-code) to encode separatrix diagrams and bifurcations, and provides explicit classifications of typical one-parameter bifurcations across several surfaces: the 2-disk, cylinder, and sphere with holes. The authors enumerate all possible Morse-flow structures up to six points on these surfaces, derive Euler-characteristic constraints, and present concrete examples of codes for different bifurcation types (saddle-node, saddle connection, and their boundary variants). The work extends existing invariants for Morse flows and chord-diagram representations, enabling precise topological classification and potential generalization to other surfaces and higher point counts.

Abstract

We describe all possible topological structures of typical one-parameter bifurcations of gradient flows on the 2-sphere with holes in the case that the number of singular point of flows is at most six. To describe structures, we separatrix diagrams of flows. The saddle-node singularity is specified by selecting a separatrix in the diagram of the flow befor the bifurcation and the saddle connection is specified by a separatrix, which conect two saddles.

Figures (12)

• Figure 1: Separatrix diagram and destiguished graph for Morse flow with code $\{12]0[\}[123]\{3\}.$
• Figure 2: saddle-nod bifurcation $SN_{+}$
• Figure 3: $BSN_{+}$ bifurcation
• Figure 4: $HS_-$ bifurcation
• Figure 5: HN bifurcation

Theorems & Definitions (15)

• Theorem 1
• Example 1
• Theorem 2
• Theorem 3
• Theorem 4
• Example 2
• Example 3
• Example 4
• Example 5
• Theorem 5