Morse hyper-graphs of topological spaces and decompositions

Abstract: The cell complex structure is one of the most fundamental structures in topology and combinatorics, the Morse decomposition of a dynamical system analyzes the global gradient behavior, and the Reeb graph of a function is an elementary tool in Morse theory to represent the global connection and is also used to analyze continuous and discrete data in topological data analysis. In this paper, we unify three concepts of cell complexes, Morse decompositions, and Reeb graphs into a concept. In fact, we introduce topological invariants for topological spaces and decompositions, which are analogous to abstract (weak) orbit spaces and Morse graphs for flows. To achieve these, we define analogous concepts of recurrence and "chain-recurrence" for topological spaces and decompositions such that the original and new recurrences correspond to each other for a flow orbits on a locally compact Hausdorff space and the orbit space. We show that the Morse hyper-graphs exist for any topological spaces and any invariant decompositions (e.g. compact foliated spaces). Moreover, the abstract weak element spaces generalize the Reeb graphs of Morse functions and the Morse (hyper-)graphs and refine abstract cell complexes.