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Let $K$ be a finite simplicial, cubical, delta or CW complex. The persistencemap $\mathrm{PH}$ takes a filter $f:K \rightarrow \mathbb{R}$ as input andreturns the barcodes $\mathrm{PH}(f)$ of the associated sublevel set persistenthomology modules. We address the inverse problem: given a target barcode $D$,computing the fiber $\mathrm{PH}^{-1}(D)$. For this, we use the fact that$\mathrm{PH}^{-1}(D)$ decomposes as complex of polyhedra when $K$ is asimplicial complex, and we generalise this result to arbitrary based chaincomplexes. We then design and implement a depth first search algorithm thatrecovers the polyhedra forming the fiber $\mathrm{PH}^{-1}(D)$. As anapplication, we solve a corpus of 120 sample problems, providing a firstinsight into the statistical structure of these fibers, for general CWcomplexes.

Algorithmic Reconstruction of the Fiber of Persistent Homology on Cell Complexes