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Everybody in algebraic topology loves homology and cohomology, but sometimes we like homotopy groups also, since they detect different things (think about spheres) .

An interesting and recent application of topology is topological data analysis, when one is given a filtration of topological spaces $\{X_r\}_{r \in [0,1]}$ with a finite number of "critical values". A number $t \in [0,1]$ is called critical if for every $\epsilon >0 $ the map $X_{r-\epsilon} \to X_{r+\epsilon}$ is not a (weak) homotopy equivalence.

There is a cool theorem that classifies the possible filtrations of abelian groups $\{H_k(X_r) \}$ that can arise if $X_r$ have finite dimensional homology in each degree. Morally it is a superposition of "bars": there is a basis in which each generator born at some time and dies at some other time. Correct me if I am wrong in the generality of this theorem.

I was thinking if it is possible to define persistent homotopy groups and to generalize this classification theorem. Persistent homotopy groups are defined exactly the same way. In the case $\pi_k, k\ge 2$, I guess the theorem still holds, but this is a question: cam $\{\pi_k(X_r)\}$ be decomposed in a finite number of bars?

In the $k=1$ case, maybe there is a presentation in terms of resolutions with finite free groups where each generator/relation/relation among relations born at some time and dies at another. Here I always assume finite number of critical values.

Bonus: is there a "persistent homotopy diagram" in which one can nicely represent a $\{\pi_1(x_r) \}$ persistent group?

Bonus 2: if the theory goes through without great changes, why homotopy is not that used? Is it harder to compute?

Edit: to start with something simpler, you can substitute abelian groups with vector spaces over Q, that is to ignore torsion at the moment.

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    $\begingroup$ Yes, homotopy groups are harder to compute. Homology is easy: given a finite point cloud, you can form its Vietoris-Rips complex, then take its simplicial chain complex. This is a finite chain complex of finite-dimensional vector spaces, very tractable for a computer, and you can use standard lin. alg. packages. But to algorithmically calculate the homotopy groups of the VR complex, you have to replace the simplicial set of the complex with a Kan-fibrant model, e.g. the Dwyer-Kan loop group or the Ex^infty construction. Both infinite, and unclear how to encode in a data type, for a computer. $\endgroup$
    – user164898
    Commented May 20, 2021 at 23:06
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    $\begingroup$ Of course there is a theory... but do you want it to have any useful properties? One can talk about the homotopy groups of a filtered complex. This was done before Persistent Homology was an object of study, with success. Your question seems rather vague. $\endgroup$
    – Ryan Budney
    Commented May 20, 2021 at 23:30
  • $\begingroup$ I haven’t read this paper for myself, but you may be interested in ongoing work of Zhou: arxiv.org/abs/1912.12399 . $\endgroup$
    – Eric Peterson
    Commented May 21, 2021 at 7:51
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    $\begingroup$ I think this "barcode" description relies on field coefficients. A sequence of abelian groups like $0\to \mathbb{Z} \to \mathbb{Z}/2\to \ldots$ certainly doesn't decompose into "bars" that just look like a single generator appearing and disappearing again. You can of course still do homological algebra of $[0,1]$-indexed sequences of abelian groups with finitely many critical values, and homotopy groups of a $[0,1]$-filtered space give an example, but it's unclear what exactly you're asking. $\endgroup$
    – Achim Krause
    Commented May 21, 2021 at 8:15
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    $\begingroup$ people do compute homotopy groups on computer, cf e.g. www-fourier.ujf-grenoble.fr/~sergerar/Kenzo $\endgroup$
    – Dima Pasechnik
    Commented May 21, 2021 at 9:11

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