The fiber–base Betti-number conjecture for persistent homology

Let $K$ be an arbitrary connected simplicial complex, and let $D_K$ be the barcode with one infinite bar $(0,+\infty)$ in degree $0$, no finite bars, and infinite bars $(1,+\infty)$ of multiplicity $\beta_p(K)$ in each degree $p\geq 1$. The fiber $\mathrm{PH}^{-1}(D_K)$ is the space of filters on $K$ realizing $D_K$.

Fiber–base Betti-number conjecture. The fiber $\mathrm{PH}^{-1}(D_K)$ and $K$ have the same Betti numbers.

The conjecture is motivated by all examples computed by the authors with $\mathbb{Z}_2$ coefficients, where the fiber and the base complex have the same Betti numbers. It is not stated to be proved, so its general validity remains open.