On Reeb spaces of non-proper smooth functions which are $1$-dimensional CW complexes

Abstract

We discuss Reeb spaces of non-proper smooth functions. (Non-)proper functions are maps between topological spaces the preimages of compact sets by which are compact (resp. may not be compact). Their Reeb spaces are the spaces of connected components of their level sets and with the natural quotient topologies. In proper cases, explicit theory are developing since the 1950s. This is closely related to Morse(-Bott) function theory and in such cases we have graphs naturally. Recently, related general topological studies with combinatorial ones are actively developing, mainly due to Gelbukh and Saeki, and they are at most $1$-dimensional in tame cases. We extend a theorem by Saeki in the 2020s: the Reeb space of a smooth function on a closed manifold is naturally a graph if and only if its critical value set is finite. The author has been a pioneer of the non-proper case and previously investigated examples.