Morse hyper-graphs of topological spaces and decompositions

Abstract

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 for any topological spaces and any invariant decompositions (e.g. compact foliated spaces) exist. Moreover, the abstract weak element spaces generalize the Reeb graphs of Morse functions and the Morse (hyper-)graphs and refine abstract cell complexes.

Morse hyper-graphs and abstract weak orbit spaces of semi-decompositions

“…Generalization of Reeb graphs of Morse functions. Recall that the Reeb graph of a Morse function on a closed manifold is the abstract weak element space of the set of connected components of level sets as abstract multi-graphs [38,Proposition 7.6]. For a semi-decomposition of sublevel sets, we have a similar statement.…”

“…Notice that any decompositions satisfy the conditions ( 2) and ( 3) in the previous definition, and so that any invariant decompositions are invariant as semidecompositions [38, Lemma 7.1 and Lemma 7.2]. Conversely, a decomposition that is invariant with respect to decomposition is invariant with respect to semidecomposition because of [38,Lemma 7.2].…”

“…From [38,Theorem 7.9], notice that the Morse hyper-graph for an invariant decomposition on a topological space is quotient spaces of the abstract element space and the abstract weak element space with respect to decomposition. Therefore we define the Morse hyper-graph of a semi-decomposition of a topological space as follows.…”

“…Notice that the Reeb graph of f is a finite graph as abstract graph (cf. [38,Proposition 7.6]). Denote by Crit(f ) the set of critical points.…”

“…Moreover, using the abstract orbit spaces, the author reconstructed some classes of flows from the time-one maps. In [38], the author generalizes such an abstract orbit structure by using concepts of dynamical systems and defines Morse hyper-graphs for decompositions on topological spaces. In particular, the abstract element spaces of decompositions are quotient spaces of cell complexes, Morse decompositions, and Reeb graphs.…”