I would like to clarify a couple points in the following excerpt from these notes (page 3) discussing Grothendieck's seminal Esquisse d’un programme pointing out the importance to reformalize the fundamentals of topology motivating the necessity to introduce what he called "tame topology":

Regarding his program on tame topology, Grothendieck writes: “I was first and foremost interested by the modular algebraic multiplicities, over the absolute base field $\Bbb Q$, and by a ‘dévissage’ at infinity of their geometric fundamental groups (i.e. of the profinite Teichm¨uller groups) which would be compatible with the natural operations of $\Gamma = \text{Gal}(\overline{\Bbb Q}/ \Bbb Q)$. [...]

Grothendieck recalls that the field of topology at the time he wrote his Esquisse was still dominated by the development, done during the 1930s and 1940s, by analysts, in a way that fits their needs, rather than by geometers. He writes that the problem with such a development is that one has to deal with several pathological situations that have nothing to do with geometry. He declares that the fact that “the foundations of topology are inadequate is manifest from the very beginning, in the form of ‘false problems’ (at least from the point of view of the topological intuition of shape).” These false problems include the existence of wild phenomena (spacefilling curves, etc.) that add complications which are not essential. He states that a new field of topology is needed, one which should be adapted to a theory of “dévissage” (unscrewing) of stratified structures, a device which he was led to use several times in his previous works. Stratifications naturally appear in real or complex analytic geometry [...]

(A): What does Grothendieck mean that the pathologies appearing in former development of topology motivated by analysts have nothing to do with geometry? In which sense? What was for Grothendieck there the nature of a phenomenon in topology which "has nothing to do with geometry"?
More I would like to develop a better feeling for the distinction between phenomena appearing in the "old/classical" topological framework which Grothendieck regarded as "having to do with geometry / are of geometric nature", and those that are not regarded as such. So asking plainly, which distinction Grothendieck made between the notions of "topological" and "geometric"?

(B): What is roughly the heuristic meaning behind the idea of “dévissage” of stratified structures and ‘dévissage’ at infinity?

My rudimentary understanding of dévissage is that roughly it given criteria to decide when say for a fixed space $X$ and $C(X)$ an additive category of "nice" objects naturally attached to $X$ (e.g. is $X$ a scheme, $C(X)$ could be the cat of coherent $O_X$-modules, classes of line bundles, etc.), and $A \subset \text{ob}(C(X))$ subset of the objects of $C(X)$ satisfying (context depending additional properties; archetypal property would be e.g. the $2$-of-$3$-property) and dévissage results are dealing with metastatements when $A$ actually equals $C(X)$.

But how do "stratified spaces" and "at infinity" (...sounds like compactification phenomena like "point or line" at infinite) enter the picture?

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    $\begingroup$ English translation of Esquisse d’un programme is freely accessible. Grothendieck description of ungeometric pathologies is on pp. 258-259. He complains that "it is not really possible" to study the homotopy type of the automorphism group of a space, or of the space of embeddings, or of immersions, etc., while these are "essential invariants" of its "shape". "Topologists elude the difficulty" by switching to smooth or PL category, but "none is “good”, i.e. stable under the most obvious topological operations". $\endgroup$
    – Conifold
    Commented May 22 at 13:19
  • $\begingroup$ Regarding your question (A), even Grothendieck could be wrong. Space filling curves are a very essential way in the theory of hyperbolic 3-manifolds, in the form of the Cannon-Thurston space filling curve. $\endgroup$
    – Lee Mosher
    Commented May 22 at 23:09
  • $\begingroup$ @LeeMosher: Well ... finally it depends on what Grothendieck means by "geometry"/geometric issues (in contrast to topological where he assumes the pathological things "live" and of those "the geometric should be independent"). From my viewpoint theory of hyperbolic 3-manifolds should clearly be "geometric" as it is sensitive for the "local picture" (as every theory involving metrics) and so this construction with space filling curves would contradict to that such "pathologies" are "non geometric", but obviously I'm not Grothendieck so I cannot say precisely what he regarded as "geometric". $\endgroup$
    – user267839
    Commented May 22 at 23:30
  • $\begingroup$ Do you have at least vague conjecture what roughly Grothendieck understood as "geometric" in order to contrast it from "topological"in context of his expose? Maybe that's exactly the key issue of my misunderstanding of point (A). $\endgroup$
    – user267839
    Commented May 22 at 23:39
  • $\begingroup$ There is a historical discussion of Grothendieck's "geometry of shapes" in Morales, The notion of space in Grothendieck, p.11ff, and a variant of formalization of his requirements (which includes so-called triangulability and leads to o-minimal geometry) in Chambert-Loir, Tame topology in number theory and geometry. $\endgroup$
    – Conifold
    Commented May 23 at 6:23


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