Refresher on singularities of mappings into line and plane
If one works with smooth manifolds and mapping, it makes sense to focus on generic situation, – functions that form an open dense subset in the space of all functions. This often leads to significant simplification of the description of the functions.
For example, For a smooth mapping \(F:M\to\Real^d\) of a compact manifold into Euclidean spaces, the singular set – consists of critical points, where the rank of the Jacobi operator is less than maximal. Generically, this is a set of dimension \((d-1)\).
In the case of \(\Real\)-valued function, generic functions are Morse, i.e. have non-degenerate (thus isolated) critical points, with different critical values. For the maps into the plane, Whitney theory implies that generically such mappings have a smooth curve of critical (“fold”) points, which contains isolated pleat points.
Reeb graphs: given a function (from a manifold, or more generally, a path connected topological space), identify path connected components of the level sets.
Mapper is a software data analysis tool that reconstructs Reeb graph.
More generally, Reeb spaces can be formed using maps into plane or other, higher dimensional spaces, and identifying path-connected components: resulting CW complexes are increasingly harder to construct and interpret.
Merge trees: keep track not of level, but of sublevel sets. Essentially, this is the 0-dimensional persistence homology.
For univariate functions, merge tree completely identifies the underlying function, up to a reparametrization of the underlying space (real line). Is there an analogue of this result in higher dimensional setting? Indeed, there is.
Morse theory and persistence
For Morse functions, there is a natural way to decompose the underlying manifold, and a corresponding way to construct persistent homologies from the Morse-Smale complex. This was done by Barannikov and used to derive spectral asymptotics for Witten Laplacian.
One can also see the importance of (at least, 0-dimensional) persistent homologies in the classical result on gradient dynamics perturbed by white noise: the escape rates from wells are governed by the bar lengths corresponding to those wells.