Morse hyper-graphs of topological spaces and decompositions

Though the cell complex structures, the Morse decompositions, and Reeb graphs are part of the foundation of topological data analysis, they cannot be applied to numerous structures (e.g., foliations (even on surfaces), line fields, topological spaces, pre-order sets). To analyze various structures, we introduce analogous concepts of recurrence in dynamical systems for topological spaces and decompositions, and construct new topological invariants for topological spaces and decompositions, which generalize and unify the Reeb graphs of Morse functions and the Morse (hyper-)graphs, and refine abstract cell complexes, by using a dynamical systems point of view. The new invariants provide new topological data reduction methods for topological spaces and decompositions, capture the “gradient-like” (i.e., non-recurrent) behaviors, and can apply various phenomena (e.g., nematic liquid crystals) modeled by line fields and computer science via domain theory.