UNSMOOTHING OF REEB GRAPH

Abstract

Given a manifold X with a function $f$ we can collapse the path-connected components of level sets to a point. Generically this produces a graph G with vertices corresponding to critical points of $f$ and on which we have an induced function $\overline{f}$. This complex $(G,\overline{f})$ is called the Reeb graph of $(X,f)$. In \cite{categorified} the concept of smoothing of a Reeb graph $(G,\overline{f})$ was introduced. Smoothing produces a one-parameter family of smoothed/simplified Reeb graphs $S_{\epsilon}(G,\overline{f})$. In this paper, we study the inverse process, unsmoothing, and show to what extent this can be carried out uniquely or if at all.