Timeline for Easiest proof of computability of homotopy groups of spheres
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 10 at 10:09 | comment | added | Trebor | @TylerLawson Church and Turing had this idea that these two things are equal in power... (assuming enough food, paper, electricity etc) | |
| Feb 11, 2022 at 17:36 | comment | added | Tyler Lawson | I had the impression that a computation could be something carried out by a computor, rather than a computer... | |
| Feb 11, 2022 at 16:31 | comment | added | Peter LeFanu Lumsdaine | @JoeShipman: Complaining about topologists’ use of “computation” is like an American going to Italy and complaining that the “cappucino” isn’t like he’s used to from Starbucks. OK, not quite — that’s an exaggeration — but it’s like complaining that Italian cappucino isn’t like at his nice hipster coffee shop in Brooklyn. The logician’s usage of “computation” is a good and important one — but the topologists’ looser usage is not merely legitimate, but also older than ours, and is essentially what ours was originally derived from. | |
| Feb 11, 2022 at 8:23 | comment | added | Denis Nardin | @JoeShipman Effective computation is just a sufficiently rare concept in most homotopy theory (in fact most of modern math) that it's easier to append "effective" whenever it's needed than to specify every time whether a particular computation is effective. I'm sorry if that causes you discomfort, but different fields will have different priorities. | |
| Feb 11, 2022 at 1:41 | history | became hot network question | |||
| Feb 11, 2022 at 0:59 | comment | added | user164898 | I have never thought that the use of "computation" as a synonym for "calculation" was unique to topologists, but maybe it is, and I never noticed. A web search for "computation of the Chow ring" reveals that some algebraic geometers use the word "computation" this way too, at least. | |
| Feb 11, 2022 at 0:36 | comment | added | Ryan Budney | Homotopy theory has a tendency to call things "computations" when they reduce the computation of something to some relatively standard other object -- but that object itself may be just some mathematically-presented object, i.e. often it is not of a finitary nature. I think the abuse of language, while occasionally frustrating, is understandable, given that specialization has this tendency to put people into bubbles where they are talking with like-minded folks. Whenever you talk to a new person, you have to get used to whatever dialect they are speaking, before your can really converse. | |
| Feb 11, 2022 at 0:27 | comment | added | Carl-Fredrik Nyberg Brodda | @JoeShipman Saying that Wikipedia's use of "computation" for things that are not algorithms is "absolutely insulting crap" seems a little excessive to me... | |
| Feb 11, 2022 at 0:00 | answer | added | Timothy Chow | timeline score: 14 | |
| Feb 10, 2022 at 22:11 | comment | added | Joe Shipman | When I was in grad school I found it very frustrating that topologists would not grasp the concept of “total recursive function” sufficiently well to appreciate that the “computations” discussed in the textbooks were not actually effective. My conclusion: no one has improved on Kan’s original proof that there really was an effective computation in all the years since; even in the special case of spheres that I am asking about. To reiterate: I don’t need a fast algorithm, just an easy proof that there IS an algorithm. | |
| Feb 10, 2022 at 22:03 | comment | added | Joe Shipman | Thanks but neither of those helps—the quote from the Wikipedia article references two papers in French and I need something in English, and the rest of the Wikipedia article is absolutely insulting crap for calling things that aren’t really algorithms “computations”. For the Curtis/Quillen stuff I’d really like a reference. When I was in grad school I found it very frustrating that topologists would not grasp the concept of “total recursive function” sufficiently well to appreciate that the “computations” discussed in the textbooks were not actually effective. | |
| Feb 10, 2022 at 21:18 | comment | added | Fernando Muro | Curtís showed (in the seventies, I think, one of the first papers in the old Topology journal) that the homotopy groups of a connected free simplicial group coincides with those of its projection to nilpotency class $2^n$ up to dimension $n$. One can compute the homology of the Moore complex of a free nilpotent simplicial group in an inefficient but algorithmic way. The same result was later obtained by Quillen and Bousfield-Kan by slightly different techniques. | |
| Feb 10, 2022 at 17:48 | comment | added | მამუკა ჯიბლაძე | Citing from Wikipedia (concerning the method opposite to the Postnikov one), "In principle this gives an effective algorithm for computing all homotopy groups of any finite simply connected simplicial complex, but in practice it is too cumbersome to use for computing anything other than the first few nontrivial homotopy groups as the simplicial complex becomes much more complicated every time one kills a homotopy group." | |
| Feb 10, 2022 at 17:36 | history | asked | Joe Shipman | CC BY-SA 4.0 |