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One of the most computationally convenient properties of singular cohomology $X \mapsto H^\bullet(X;\mathbb{Z})$ is the fact that one can extract it from a good cover $\{U_i\}$ of $X$ via Cech cohomol …

Since this area is developing rather quickly, there is a dearth of canonical references that would satisfy basic pedagogical requirements. If I were teaching a course on this material right now, I wou …

As mentioned in Carlsson and Zomorodian's paper (to which you have linked), the problem of computing persistence barcodes with coefficients in a ring $R$ relies essentially on classifying graded modul …

One way to make things "minimal" (given the lack of any further information) is to construct a simplicial complex whose cup products are all trivial, so the (co)homology generators don't interact with …

Your question is precisely the subject of Justin Curry's recent preprint. Bottom line: if you agree to identify functions $f,g:[0,1] \to \mathbb{R}$ whenever they have the same merge-tree, then ther …

Thanks to Cosheaf Overlord Justin Curry for bringing this question to my attention. I'm only going to address the first question here, and I think with some computations (whose complexity depends on y …

Fix a ring $R$ and let $C_\bullet$, $D_\bullet$ be (possibly unbounded) chain complexes of $R$-modules. Assume that $f_\bullet:C_\bullet \to D_\bullet$ is a quasi-isomorphism: that is to say, $f$ is a …

Let $R$ be a reasonable ring (maybe I mean a PID, or $\mathbb{Z}$, and when sufficiently desperate, a field). Now consider fixed sequences $C_n$ and $H_n$ of $R$-modules, which are tame in every possi …

The answer to your question is no, nobody has used persistence to improve the algorithmic efficiency of computing differentials, although of course the relationship between persistence intervals of a …

Edit: Modified in accordance with Tom Leinster's entirely reasonable objections. Sorry to exhume this question from 5+ years ago. In case someone is still looking for an answer, note that a very spec …

Let $X$ be a reasonable topological space (say one that has the homotopy type of a finite CW complex) and consider a subset $\Gamma$ of $X \times X$ so that the projection $p:\Gamma \to X$ onto the fi …

Here is a cool new (and very readable) preprint which uses the second barycentric subdivision (as discussed in Zhen Lin, Fernando Muro and Peter May's comments) to construct a cofibrantly generated mo …

Let $S$ be any regular CW decomposition of (a space homeomorphic to) the $n$-sphere, and consider a cell $\sigma$ of dimension $d \in \{0,\ldots,n\}$. Let $S'$ be the regular CW complex which remains …

At least in the degree-theoretic world, the index notation appears to be dominant. When we write the Lefschetz-Hopf theorem $$L(f) = \sum_{x \in \text{Fix}(f)} \text{stuff}_x(f),$$ the $\text{stuff} …

Let $(X,\leq)$ be a poset with distinguished element $p$, and let $P'$ be the poset of "finite chains which weakly descend to $p$" given by all $\sigma = (x_0 \geq x_1 \geq \cdots \geq x_k \geq p)$ or …