Fiber of persistent homology on Morse functions (Q6102904)

Let \(M\) be an arbitrary smooth, \(d\)-dimensional compact manifold with boundary \(\partial M\) and let \(f : M \rightarrow \mathbb{R}\) be a Morse function, which means that \(f\) has isolated critical points none of which belongs to \(\partial M\), as well as, \(f\) is constant on each boundary component of \(\partial M\). Consider the group of diffeomorphisms that are isotopic to the identity map, denoted by \(D_{Id}(M)\). This group acts on the set of Morse functions on \(M\) by pre-composition. In other words, for any \(\phi\in D_{Id}(M)\) and Morse function \(f\), we obtain a new Morse function \(\phi\mapsto f\circ\phi\). Hence, one gets the orbit of \(f\) under this action, denoted by \(\mathcal{O}_{Id}(f)\), which resides within the space of Morse functions. Given a Morse function \(f\), we can consider a nested sequence of sublevel sets \(f^{-1}((-\infty, x])\). By applying homology in degree \(0 \leq k \leq d\) with coefficients in a field, we construct the persistent homology module of \(f\). This module consists of a sequence of vector spaces \(H_k(f^{-1}((-\infty,x]))\) with linear maps between them induced by inclusions between them. We denote this module as \(D := PH(f)\), the associated barcode. The barcode of \(f\) in degree \(k\) is denoted by \(PH_k(f)\). The authors introduce a persistence map defined on Morse functions: \[ PH:f \mapsto [PH_{0}(f),\ldots, PH_{d}(f)] , \] which returns a sequence of \(d+1\) barcodes. The main contribution of the paper is a theorem that establishes the equality between the orbit \(\mathcal{O}_{Id} ( f )\) of a Morse function \(f\) and the path connected component \(PH^{-1}_f(D)\) containing \(f\) within the fiber of \(PH\) over \(D\). In other words: \(PH^{-1}_f(D)=\mathcal{O}_{Id}(f)\). The authors also show that, in a continuous \(1\)-dimensional setting where \(M\) is the interval or the circle and \(f\) is continuous, each component in the fiber is contractible or circular, respectively.