Timeline for A homeomorphism with a prescribed action on the fundamental group - decidable or not?
Current License: CC BY-SA 3.0
18 events
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| Sep 24, 2021 at 7:32 | comment | added | David Roberts♦ | @LeeMosher here's a fixed link for Algorithmic aspects of homeomorphism problems, linked in your comment: arxiv.org/abs/math/9707232 | |
| Aug 9, 2017 at 9:53 | vote | accept | Alex Gavrilov | ||
| Aug 9, 2017 at 9:52 | history | edited | Alex Gavrilov | CC BY-SA 3.0 |
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| Aug 9, 2017 at 9:06 | history | edited | Alex Gavrilov | CC BY-SA 3.0 |
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| Aug 9, 2017 at 0:51 | answer | added | Fedya | timeline score: 7 | |
| Aug 8, 2017 at 20:41 | comment | added | Lee Mosher | To get a readable link in the comment of @AchimKrause I suggest first going to the arxiv page front.math.ucdavis.edu/9707.5232, but not clicking on the .pdf link on that page. Click on the .ps link instead. That's what I needed to do on my machine. | |
| Aug 8, 2017 at 16:19 | comment | added | Tobias Fritz | @AchimKrause: That Theorem 1 proves that for a given simply connected $M$, deciding homeomorphism with $M$ is decidable. That's significantly easier than deciding whether two given simply connected manifolds are homeomorphic! | |
| Aug 8, 2017 at 14:25 | comment | added | Achim Krause | Oh, also, the paragraph before seems to answer OPs question in the negative. They construct two different manifolds with the same $\pi_1$ such that an algorithm checking whether they are homeomorphic would solve the word problem in $\pi_1$. | |
| Aug 8, 2017 at 14:17 | comment | added | Achim Krause | arxiv.org/pdf/math/9707232.pdf this paper claims to prove that the homeomorphism problem for simply connected manifolds of dimension $\geq 5$ is decidable (Theorem 1). | |
| Aug 8, 2017 at 12:49 | history | edited | YCor |
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| Aug 8, 2017 at 12:49 | comment | added | YCor | Such an algorithm would solve the homeomorphism problem between simply connected closed manifolds. I'd be surprised if this exists. | |
| Aug 8, 2017 at 12:46 | comment | added | YCor | @JasonStarr OK indeed I now understand the first interpretation. Actually it meant "decidable" in the sense of set theory, while in the question it's about recursiveness. (Note that the tag "computability-theory" was a hint towards this interpretation) | |
| Aug 8, 2017 at 12:32 | comment | added | Jason Starr | @YCor. I was just trying to get the OP to clarify the question. The answer that was posted (now deleted) seemed to follow the first meaning, whereas the OP has now clarified that the question uses the second meaning. | |
| Aug 8, 2017 at 12:24 | comment | added | Alex Gavrilov | Yes, the input may be a pair of triangulations together with an explicit isomorphism between fundamental groups given by appropriate presentations. I do not think it makes any difference except in some pathological cases. (To avoid them, you may assume manifolds to be smooth, which is fine with me.) | |
| Aug 8, 2017 at 12:03 | comment | added | YCor | Anyway it's a bit vague: by closed manifold you probably mean finite triangulation, and you mean the algorithm to halt (with the correct answer) whenever the space it defines is a topological manifold? So the input would be a pair of triangulations and a pair of homomorphisms between fundamental groups, given by images of generators, which composes to identity on both sides? | |
| Aug 8, 2017 at 12:03 | comment | added | YCor | @JasonStarr What makes you believe the strange first interpretation? Of course it means a Turing machine... | |
| Aug 8, 2017 at 12:00 | comment | added | Jason Starr | What do you mean by, "I am curious if the following topological problem is decidable?" Does the word "decidable" mean "does anybody know if the following is always true?" Or does "decidable" mean "is there a Turing machine ...?" | |
| Aug 8, 2017 at 11:41 | history | asked | Alex Gavrilov | CC BY-SA 3.0 |