Talk:Torsion coefficient (topology)

Actual definition?

Would it be possible to provide an actual definition for what these are? This appears to be a very idiosyncratic term; only one, single modern textbook on (algebraic) topology mentions them (see below). A google search reveals a barren landscape -- looking at the first 8 pages in google, the first 80 links!! I get just this WP article itself, and the Siefert and Threllfall book as the 25th hit(!). That's all! There are a half-dozen hits to students asking confused beginner questions about Abelian groups, with responses mostly being "never heard of it before I think you mean X"; zero hits for any books or articles on algebra or other math topic, defining the term, in any context whatsoever. The ONLY definition I found for Abelian groups is from a non-math reference: free dictionary Its a decent definition (it does not mention topology).

The article, as currently written, suggests that maybe the torsion coefficient is simply the free-dictionary definition of torsion, of $H_{1}$ but I'm left to guess this. Is it $H_{1}(X;Z)$ because it cannot be $H_{1}(X;Q)$??? I can't quite parse the Siefert and Threllfall book - page 66 gives a definition but I can't make it out because page 65 and 64 are blocked. It sounds like a count of the torsion subgroups in $H_{1}$ (as opposed to the order, as defined in the free dictionary definition). But then page 68 says, I quote: there are no n-dimensional torsion coefficients. By contrast, $H_{n}$ have torsion in general so I'm stumped.

This article also makes some bizarre claims: The Betti numbers and torsion coefficients for a manifold are invariants which fully characterise its topology That cannot possibly be right: a classic textbook example: $S^{1}\wedge S^{2}\wedge S^{3}$ and $S^{1}\times S^{2}$ have the same homology but different homotopy. If the torsion coefficients are homology, the claim is false. 67.198.37.16 (talk) 20:10, 10 August 2016 (UTC)

I know where this article comes from because I've worked on homology (mathematics). This article is about the torsion part of H₁ of smooth surfaces. It uses the very classical terminology "torsion coefficient" to mean "the size of the torsion part". Connected surfaces are simple enough that this is all you need. In greater generality, Seifert and Threllfall define the torsion coefficients to be the invariant factors of the homology groups.

I found it difficult to work on the homology article because I prefer to think of homology in modern and algebraic terms; the most elementary approach that I think is interesting is simplicial homology. But such an approach is historically inaccurate. When homology was first introduced it was thought of geometrically, and the first half or so of the homology article shares this preference for geometry because of the preferences of the other editors. I found it very hard to make correct statements without introducing some algebra, and I feel like the article is misleading because it (despite my preference) is mostly about "continuous deformations" of loops, i.e., classes in the fundamental group. I think that, without using the machinery of chain complexes, one can only reliably make statements about surfaces, and it's only possible for surfaces because there are so few homotopy types that nothing can really go wrong for them.

I'm not sure what should be done with this article. Ozob (talk) 01:29, 11 August 2016 (UTC)

Richeson remarks that the Betti numbers and torsion coefficients fall out of algebraic topology in a natural way. As the creator of this article, my motivation was to give the non-orientable aspect of this idea space. I was able only to present the original context of smooth manifolds (which interests me) in the hope of drawing out from other editors a more modern algebraic treatment (which for the most part is way over my head) and which would in turn allow light to be shone on Richeson's remark. Once my material has been worked over to fix any mistakes, etc., I would also hope to add a similar introductory section to the article on Betti numbers. — Cheers, Steelpillow (Talk) 08:00, 11 August 2016 (UTC)

In my opinion, as this is currently written I don't think it should remain on Wikipedia as a standalone article. The topic (computation of the homology of surfaces and observation that it classifies them, and that orientability is equivalent to a trivial torsion subgroup) should rather be a detailed example on other pages. I see that the page on homology contains a (rather better-written at first view) account of the same subject (by the same contributor?). This could also find a place at Surface (topology) which could include a subsection on the homology of surfaces.

On the other hand the torsion homology for higher-dimensional manifolds (which the article mistakenly states distinguishes them, as noted above but there are many more subtle examples) is a rather active current research topic in low-dimensional topology, with rich relations to geometry and number theory. I am not sure it is yet fit for an encyclopedic treatment and I won't write about it because I worked and am still working on the subject. In case anybody is interested here is a survey on (a subset of) this: https://webusers.imj-prg.fr/~nicolas.bergeron/Travaux_files/bergeron_proc_ems.pdf (disclosure: the author is my former phD advisor) jraimbau (talk) 06:32, 17 August 2016 (UTC)

It seems to me that any approach to merging should also consider the (presumably closely related) article on Betti numbers. — Cheers, Steelpillow (Talk) 09:41, 17 August 2016 (UTC)

I don't agree with this at all. "Betti numbers" has an historical significance that "torsion coefficients" has not, and the former is actually actual used in modern mathematical research and teaching, while I never heard of the latter under this name until now. Looking purely at the mathematical significance of both objects I think it is also fair to say that Betti numbers have been more prominent and versatile then torsion homology. jraimbau (talk) 10:27, 17 August 2016 (UTC)

By "consider" I meant raise the kinds of issue which you have just raised. You have agreed with me in act if not in word! Thank you. From what you say, I am quite prepared to accept that creation of this article may have come too soon. — Cheers, Steelpillow (Talk) 14:08, 17 August 2016 (UTC)

OK, there are multiple conflated issues in the above discussion.

• We still need an accurate definition. I own four books on algebraic topology, and none of them discuss or define "torsion coefficients". They all discuss torsion, and the ext and tor functors, and the universal coefficient theorem for homology and what-not, but none cover the contents of this article.
• Based on the google/arxiv search, this is an obscure dead-end of mathematical research. I believe its worth keeping (and not merging) as an article because maybe its historically interesting. But it should at least be precise and accurate. The errors and non-definitions need to be fixed.
• Re: Ozob's rant about geometry vs algebra in homology -- The problem of naive viewpoints in WP is pervasive in physics and math articles, not just in homology. Many/most articles on general relativity are written by students who seem to be self-taught, and seem never to have taken a modern class in the topic. They're just absolutely turgid, consist plug-n-chug of equations, and completely fail to provide insight into the phenomena. There's a different issue with the quantum articles, which seem to get edited by confused undergrads. If only these were written with some historical approach, because the history of QM is really quite interesting (as it is for GR). Alas. So, for homology, I am not surprised. The intro should state clearly that there are two approaches: one that is historical and geometric, one that is modern and categorial. The JP May book provides a good, modern approach, but it is hard to read unless you've previously studied the subject in a more concrete setting. I believe that many/most WP articles could/should have dual-speed versions: a classical, concrete, low-brow approach, mirroring intro-to textbooks from the 1990's. A second, abstract, modern approach, as actually perceived by current researchers in the field, using modern notation and devices. For homotopy, I'm thinking the difference between fundamental group vs groupoid approaches.
• Re: this article: the primary issue is that the primary source for this article is a book by Siefert, written in 1934!! That's more that's 80 years ago. It predates the invention of category theory, which completely revolutionized the field. Its reporting on research done during the Edwardian era! Before television, for cripes sake! So I think this article could be very interesting, if it was re-written to take on a historical development slant, adding dates and crediting people, and indicating where things went, later in the century, and why this approach seems abandoned.

Interesting. But we still need an accurate definition of what a "torsion coefficient" actually is. 67.198.37.16 (talk) 19:50, 26 August 2016 (UTC)

Ugh. I just looked at the homology (mathematics) article. That article is embarrassingly bad, in a run-and-hide your tail in shame kind of way. It bears just about zero resemblance to what I have learned to be the field of homology. It could be a poster-child for everything that's wrong with WP -- but again, it seems to illustrate the need for two-speed versions of articles. What it writes there is adequate if you were explaining homology at a cocktail party to your serious hi-IQ non-math friends. Except its not even good for a cocktail party, because it doesn't have any punch to it -- e.g. no mention of .. I dunno -- the E8 manifold, or the Hauptvermutung, which would be the giant big "wow you won't believe this" kind of juicy, punchy result that cuts through the Martinis. So its mostly a boring intro-to-undergrads article. WTF. Clearly, we need another article, a different article, or something, that provides an overview to homology in the modern sense. 67.198.37.16 (talk) 20:16, 26 August 2016 (UTC)

Re: the subject of this article I agree with you that if it should stay only if it can be justified by taking a more historical stance. I'm not sure I agree with all your other points but they're moot for the moment.

Regarding you remarks on the Homology (mathematics) page: I agree it is very unequally written and lacks a global structure. On the other hand I don't agree at all with your proposed plans for it. In my understanding "Homology in the modern sense" does not make sense. There are numerous versions/theories of homology, and be they humble simplicial/cellular homology in which the subject has its roots or the fancy "abstract nonsense" that you seem to want to reduce the subject to, or the equally fancy but more concrete theories in geometric topology such as Heegard-Floer homology they are all still useful in current "pure" mathematical research. The article needs to take all this into account and not be written only from the reductionist perspective of homological algebra. In my understanding this article is a "front page" for homology, and as such should present an historical account of the subject, an informal and educational part, and a list of the various sub-topics in current and not-so-current research that it includes (this would probably be more at home on the talk page of the article but since it's a response to your comment above I'll leave it here for the time being. Cheers, jraimbau (talk) 21:00, 26 August 2016 (UTC)

Yeah, Sorry, I got into rant mode. As a front-door to homology, it should maybe spend more time reviewing the major results and accomplishments, and yes, this criticism should go on the talk page there. Even if the lead just paraphrased what you just wrote, that would be an improvement. Re: "plans for it", I have none, whatsoever. 67.198.37.16 (talk) 05:28, 27 August 2016 (UTC)

Speaking for myself, it was not my intention to "rant", and I'd like to apologize if I came across that way. I don't want articles to take an uncompromisingly sophisticated viewpoint. I believe there is plenty to say about the history of homology, and I think the geometric perspective on homology of surfaces is quite revealing and is perhaps the right gateway for most beginners. However I also want articles to be correct and to describe modern approaches. I don't believe this article does a good job of this. The term "torsion coefficient" is purely historical, and it doesn't reflect at all the way these phenomena are thought of now: Homological invariants are thought of as groups, not as numbers.

I think the present article would be good if it were only about history. But I don't know if there's enough to say about the history of torsion subgroups in ordinary homology of well-behaved topological spaces to make the article worthwhile. My present feeling is that it should be merged back into the main homology article. Ozob (talk) 23:52, 27 August 2016 (UTC)

Consensus?

It seems that there is a consensus amongst the three editors besides the original author of the page who participated in the conversation, to which the latter seems not to object, on the following points:

• the notability of the notion regarding mathematical research is doubtful;
• the notion may have historical notability but this is not clearly indicated by the article as it stands.
• In consequence the contents should be merged back into the page Homology (mathematics).

If nobody objects in a reasonable delay (or does it themselves) I will make the relevant changes. One point that seems not clear is (in case where this becomes effective) whether the page itself should be deleted or merely changed to a redirect page (I personally favour the latter, if people object I can start an AfD). jraimbau (talk) 10:47, 28 August 2016 (UTC)

As the article creator, I take the point that the term may be obsolete (Richeson's treatment is essentially a historical account). This was something I did not consider at the time. However I wonder whether Homology (mathematics) is the best home. Richeson describes the basic algebra of torsion coefficients and shows how it differs from the equivalent algebra of Betti numbers. I'd suggest that this algebraic discussion needs adding somewhere and its relation (or otherwise) to modern algebras explained. There is a more modern algebraic treatment in the article on orientability, which does not connect with Richeson's that I can discern. It may be more useful to merge with this latter article, thus creating a historical section there. I have no strong opinion either way, but offer the thought. I have a stronger view on redirection: people will still be finding the term in books like Richeson's so I think that a redirect is greatly preferable to simple deletion. As an final thought, I would suggest that the destination article would also benefit from a short history of Betti numbers to balance the content merged from here. — Cheers, Steelpillow (Talk) 12:40, 28 August 2016 (UTC)

I don't have a strong opinion on the merge target, but if material is merged, we need a redirect to preserve attribution history. See WP:FMERGE for details. --Mark viking (talk) 13:57, 28 August 2016 (UTC)

I think that at this point the better choice is to make a redirect to Homology (mathematics) which includes several mentions of the term, and to add a short mention of the fact that orientability of surfaces is detected by the presence of a $\mathbb {Z} /2\mathbb {Z}$-factor in the first homology (represented by the core of an embedded Möbius band. I will do this later today if nobody objects or suggests otherwise in the meantime. jraimbau (talk) 06:55, 1 September 2016 (UTC)