Wikipedia:Featured article candidates/Mayer–Vietoris sequence/archive1

The following is an archived discussion of a featured article nomination. Please do not modify it. Subsequent comments should be made on the article's talk page or in Wikipedia talk:Featured article candidates. No further edits should be made to this page.

The article was not promoted by User:SandyGeorgia 00:04, 10 January 2009 [1].

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Mayer–Vietoris sequence

Nominator(s): GeometryGirl (talk)
Informed: WikiProject Mathematics

I'm nominating this article for featured article because it has been peer reviewed and has passed GAN successfully. I'll try my best to address issues. GeometryGirl (talk) 17:53, 1 January 2009 (UTC)

• You may want to inform the folks at Wikipedia talk:WikiProject Mathematics so some content experts can review the article. BuddingJournalist 02:07, 2 January 2009 (UTC)

Done. GeometryGirl (talk) 11:13, 2 January 2009 (UTC)

Image review

• File:Vietoris4343.jpg - Twice now, I've tried to access the source and permissions links for this image (last night and this morning). Both times I've gotten a "The server is temporarily unable to service your request due to maintenance downtime or capacity problems. Please try again later." message. Anyone else having this problem?

Yes, I'm getting the same error message. The whole website is down, and many picture at Commons come from it. GeometryGirl (talk) 13:26, 2 January 2009 (UTC)

I'll try again tomorrow. Awadewit (talk) 13:32, 2 January 2009 (UTC)

You can view a Google cache of the source page here. Copyright information from the MFO is here. "Those images labelled with "Copyright: MFO" can be used on the terms of the Creative Commons License Attribution-Share Alike 2.0 Germany." BuddingJournalist 20:07, 2 January 2009 (UTC)

Excellent - this image is fine. Awadewit (talk) 01:18, 4 January 2009 (UTC)

• All of the diagrams have descriptions and PD licenses. I am incapable of assessing whether or not they require sources. Someone with more mathematical knowledge than myself will have to decide whether or not the information in them requires verification. Awadewit (talk) 12:59, 2 January 2009 (UTC)

In my opinion, the pictures are very basic. There don't really contain any mathematical content, they only help visualisation. GeometryGirl (talk) 13:26, 2 January 2009 (UTC)

I've looked over the images and agree. They are adequately sourced by the associated text and citations therein. Geometry guy 14:38, 4 January 2009 (UTC)

Could you clarify? Do you think the images need to be sourced on their own? Remember, these images are independent files. Since they are not always associated with this article, all necessary sourcing information needs to be on the image description page. Thanks. Awadewit (talk) 16:22, 4 January 2009 (UTC)

Hi Awadewit, I was wondering if you could tell me what type of info in an image needs to be sourced (for FAC) (or tell me where I can read about it). I've never participated in such a review so I have no idea. Do things only need to be sourced if they are contentious? or does everything need to be sourced? For example, here is an image that is on a current featured article (and was on it when the article became a featured article [2]); this picture expresses a mathematical theorem, does it lack a source? if so there are probably a few images on the article under discussion that could use a source. I was also wondering whether you were also talking about the commutative diagrams that had to be rendered as png's in a separate software and uploaded to wiki as images (such as [3]), or just the other images. Cheers. RobHar (talk) 20:24, 5 January 2009 (UTC)

Sorry I see some of this has been addressed on your talk page. RobHar (talk) 20:31, 5 January 2009 (UTC)

I still do not know whether the mathematical diagrams (what I would call "pictures") need a source. I really am just too ignorant of this field to judge. I didn't even realize the "commutative diagrams" were images until now, I'm afraid - I supposed they had been rendered using LaTeX. If these "diagrams" represent formulas or equations, they do not have to have a source. For example, if they are the equivalent of something like "2x = 4 therefore x = 2" (to be basic), there is no reason to have a source. Does this make sense? In the end, the mathematically-informed users will have to decide the source question. Awadewit (talk) 02:16, 6 January 2009 (UTC)

The diagrams are expressions of facts that can be represented in other ways. The facts need a source; once they have a source, then the diagram does not need an additional source (except in unusual situations). Ozob (talk) 04:52, 6 January 2009 (UTC)

I've consulted Shoemaker's Holiday, who knows some topology and is a Commons admin, and he does not think we need sources. Awadewit (talk) 05:05, 6 January 2009 (UTC)

To try and simplify it a little, these are sort of like Venn diagrams, marking out how you'd divide up simple topological shapes into other simple shapes, and where the overlap is. For instance, you could split the 2-torus (two toruses stuck together) back into two toruses by snipping along the region where the red and blue lines cross. While this may seem an imprecise way to define it, that's because topology is geometry without area, volume, or well-defined positions. Shoemaker's Holiday (talk) 05:17, 6 January 2009 (UTC)

Comments -

• Sources in languages other than English should note that in their reference entry

Otherwise, sources look okay, links checked out with the link checker tool. Ealdgyth - Talk 17:57, 2 January 2009 (UTC)

I added the languages. Thanks. GeometryGirl (talk) 19:19, 2 January 2009 (UTC)

It would also be helpful to note that the link to Mayer's paper requires a subscription. Geometry guy 17:59, 4 January 2009 (UTC)

• Comments. A lot of work has gone into this article, which is now deservedly a GA. However, there is quite a gap from GA to FA, and I don't believe it has been bridged yet. The main challenges are engaging professional prose (1a), comprehensiveness (1b) and more rigorous referencing (1c). The issues can be illustrated by the "Background, motivation and history" section. This has only one citation to a secondary source (Hirzebruch's essay – the papers of Mayer and Vietoris are primary sources). The section combines two distinct but related things: the motivation of the Mayer-Vietoris sequence from a modern perspective, and the history of how and why it arose. The latter is not covered comprehensively, and needs to be well cited. The history does not end with the papers of Mayer and Vietoris, and Eilenberg and Steenrod deserve more than a brief mention at the end of the derivation section. For example, I don't think the concept of an exact sequence existed in 1930 and Mayer and Vietoris did not present or name their results in this way. Who did?

The references could use a good book on the history of (algebraic) topology. The standard one is by Dieudonne: pages 39 and 110 provide some information. Ioan James' recent book may also be helpful.

Concerning the prose, I find the "(co)" appearing everywhere unhelpful. Historically, and primarily, the Mayer-Vietoris sequence is about homology and I suggest concentrating on that. The fact that there is a dual version for cohomology is important too, but it doesn't need to be mentioned continually in the lead and first section (the relative version isn't). An example of the prose is:

• "The Mayer–Vietoris sequence is such an approach, giving partial information about the (co)homology groups of any space by relating it to the (co)homology groups of two of its subspaces and their intersection."

There is a mismatch here between "groups" and "it", and a further confusion when "its" refers to the space rather than the groups; "any space" is also a bit awkward.

Concerning citation, a useful rule of thumb is that Wikipedia has no opinion. Whenever an article contains a statement which is not purely factual, it needs to be cited. The question I ask when I review an article is "according to whom?". This can be asked, for example, of the statement "a theorem such as that of Mayer and Vietoris is potentially of broad and deep applicability." Also, the section uses the word "important" three times. Later on, there is also "As an important special case when G is the group of real numbers...", which could be rephrased. (This segment needs a citation too, but Bott and Tu surely cover it.)

• "He was told about the conjectured result and a way to its solution, and solved the question for the Betti numbers in 1929."

What conjectured result and what question? Is this sourced to Hirzebruch's essay?

There are issues elsewhere. For instance, "holds" (in the lead) is jargon (and confusing here). "Covering subspaces" is confusing too, even with the wikilink. Finally, it would be good to back up Hatcher with another source or two, especially when the citation is to an exercise (refs 14 and 15). J.P. May also has a textbook which is available on line. I hope this and the other references given above help in improving the article. Geometry guy 17:59, 4 January 2009 (UTC)

Comments by Jakob Scholbach Jakob.scholbach (talk) 18:58, 4 January 2009 (UTC):

In general, I think this is a really nice article, generally well-written and with very pretty illustrations. I have a number of relatively trivial issues, and one major concern. My main concern with the present version of the article becoming featured is that it is not comprehensive (a FA criterion). (It is nice that the article is so well-referenced, but also shows, that the content is pretty much taken from one type of book, which increases the potential to miss important points not covered by those references). In my view, to be featured the article should at the very least mention the following topics: MV for multiple open sets/subsets, MV for sheaf cohomology including a mention of cohomology of coherent sheaves, cohomology of sheaves w.r.t. more general topologies (etale cohomology, say). For example, it is an easy, but really useful example of MV that cohomology of coherent sheaves on P^n (projective space of dimension n) vanishes beyond n+1, simply since projective space is covered by n+1 affine spaces, which don't have higher cohomology. (I'm not too much a connoisseur of advanced algebraic topology, but I suspect that there are more advanced applications of MV in this realm, too.) The current article conveys a bit the image as if the parallel statements of MV for singular cohomology and for de Rham cohomology are merely coincidential, but should (IMO) at least tell briefly that both are instances of the more general sheaf cohomology.

• lead section
  • why "attributed to" (sounds a bit like it might be due to somebody else)?
  • the 2nd section of the lead sounds a bit like reduced and relative cohomology are an additional case (in addition to the cases covered by Eilenberg/Steenrod)
  • I suggest moving the sentence "Because the cohomology is not computable directly..." to the first lead section, as a motivational statement.
  • The statement "and a precise relation exists for homology of dimension one" has nothing to do with M/V, and should be removed.
  • The examples are not covered in the lead. They should be, perhaps as part of the motivation.

• background section
  • In general I think that's well done. However statements like "the cocycle groups are often too big to handle directly" would probably be good with a reference.
  • "a theorem such as that of M and V" is vague, just "the MV sequence" would be better.

• basic versions for singular homology
  • OK, if you call this basic, what is the non-basic stuff?
  • It would be good to mention that A and B can simply be open subsets, this is a case often needed.
  • Generally, * should be replaced by ∗ (∗).
  • "In particular, H_1 is the abelianization" is misleading. This is a general statement, unrelated to MV. I suggest rewording it to "since, quite generally, H_1 is the abelianization of pi_1, this is precisely the ..."

• Basic applications
  • "the homology groups for A and B are trivial" should probably read "the higher homology groups" or "the reduced homology"
  • in the Klein bottle, putting some parentheses around Z + Z_2 would be good (in the 2nd term in the last eqn)

• Further discussion
  • "are the usual ones" might deserve a brief explanation
  • "Consider the ..." could be reworded (things like "note that" etc., directly calling for a reader, should be avoided by MOS, where possible)
  • Is "A + B" standard notation for the union? Jakob.scholbach (talk) 18:58, 4 January 2009 (UTC)

Thank you! I am very pleasantly surprised by the feedbacks of Geometry guy and Jakob. There is enough material above to keep me working a bit. Thank you for your criticism and suggestions. GeometryGirl (talk) 20:37, 4 January 2009 (UTC)

I would have to disagree with including a lot of material on MV with three or more open sets. That takes you in to Čech cohomology: The reason why the higher cohomology of a coherent sheaf F on Pⁿ vanishes is because F is quasi-isomorphic to its Čech resolution, which terminates after n+1 steps. (The Čech resolution doesn't appear on the Čech cohomology page, but it's the same idea as the Koszul complex.) That's really a different topic than the present article. I agree that it deserves a mention, but not much. (The generalization of the MV long exact sequence is the Čech-to-derived functor spectral sequence.) Going out into maximal generality, we end up with Verdier's hypercoverings and Deligne's theory of cohomological descent. Here's a set of notes on them: [4]. But these are all well outside the scope of the article, and they deserve a sentence at most.

I agree that it would be good to include statements of MV for other cohomology theories (such as sheaf cohomology). Does anyone know if it holds for extraordinary cohomology theories? (Weibel's K-theory book says there's a version for algebraic K₀ and K₁, [5], pp. 14–15) I think the right general statement for this might be that some cohomology theories preserve homotopy colimits? But I don't know much about these things. Ozob (talk) 21:27, 4 January 2009 (UTC)

Well, call it Cech or call it MV for 3, I see it as really closely related. But simply writing "The MV sequence can be seen as a first step toward (or special case of) Cech, since..." is probably also sufficient. Hypercoverings are certainly a bit far off. And yes, there is a MV sequence for algebraic K-theory (this should be in Quillen's original paper (Quillen, Daniel (1972), "Higher Algebraic K-theory. I", Lecture Notes in Mathematics, 341: 85–147), which is mirrored by the MV sequence for (mixed) motives, and every other cohomology theory in algebraic geometry I know of. I would go as far as saying that the existence of a MV sequence is a first test for the reasonability of a cohomology theory. Jakob.scholbach (talk) 21:41, 4 January 2009 (UTC)

It looks like Ozob has found the way to phrase the Cech vs MV issue over at "Čech-to-derived functor spectral sequence", i.e. the MV sequence is the long exact sequence associated to the spectral sequence coming from a cover by two sets. So something to that effect mentioned in the MV sequence article should be enough (and in fact pretty cool, imho). As for the more general cohomology theories, SGA4.V.3 shows the existence of the Čech-to-derived functor spectral sequence (they call it the Cartan-Leray ss) in an arbitrary topos (as far as I can tell), so the MV sequence holds for sheaf cohomology in an arbitrary topos. Furthermore, Section 2.1 (page 25) of Kōno and Tamaki's "Generalized cohomology" shows that the existence of the MV sequence doesn't depend on the "dimension axiom", and so it exists for extraordinary cohomology theories as well (such as topological K-theory and cobordism). Does this cover everything? And if it does, would the following address your major concern, Jakob?

1. Mentioning the MV sequence is the long exact sequence associated to the Cech spectral sequence;
2. Saying that the MV sequence exists in sheaf cohomology in an arbitrary topos;
3. Saying that the MV sequence exists in both ordinary an extraordinary cohomology theories (in the sense of Eilenberg-Steenrod).

Cheers. RobHar (talk) 20:13, 10 January 2009 (UTC)

• Comments. A couple of quick comments. First, I think it'd be good to mention the dates (1930/31) when the basic result for homology was proved in the lede section, and not just in the main body of the article, to give the reader a better idea of its age. Second, does anyone here know around what time the actual term "Mayer-Vietoris sequence" was introduced (and maybe even by whom)? It is often difficult to pinpoint such things precisely but something approximate should be known. Nsk92 (talk) 03:05, 5 January 2009 (UTC)

Also, I agree with the above comments that the "history" portion of the "Background, motivation and history" section needs to be expanded. E.g. who and when stated the result in its modern form? (Presumably Mayer and Vietoris used different language.) Also, when and by whom was the cohomology version obtained? (I assume that Vietoris stated the theorem for homology only, right?) Same for the relative version. Nsk92 (talk) 03:17, 5 January 2009 (UTC)

Oppose for the moment - I have an amateur interest in topology, but even I find this a bit jargon-dense. Now, obviously, this is an esoteric subject, and noone expects you to simplify the entire thing, but I do think that at least the first paragraph in the lead should attempt to give a brief explanation that a layman could understand. Shoemaker's Holiday (talk) 16:01, 6 January 2009 (UTC)

One could write "In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is a tool to help compute certain invariants of topological spaces. The result is due to two Austrian mathematicians, Walther Mayer and Leopold Vietoris. The method consists of splitting a space into pieces for which these invariants may be easier to compute. The sequence relates the invariants of the space to the invariants of the pieces via a mathematical object called a long exact sequence." Is this preferable? All I've done is remove all the jargon. It is my opinion that the introduction as it stands explains the MV sequence, but perhaps this would be clearer if scary jargon were removed. Or do you think some attempt should be made to explain more of the context such as what are (co)homology groups, what is meant by invariants (and maybe the term algebraic), and maybe even what a long exact sequence is. All this is layman's terms. Is this the sort of explanation you think is lacking? Cheers. RobHar (talk) 22:33, 6 January 2009 (UTC)

Invariant is still jargon. So I don't think this satisfies Shoemaker Holiday's objection. Ozob (talk) 20:17, 7 January 2009 (UTC)

On the other hand, Shoemaker Holiday said it was "jargon-dense", not that he was opposed to the presence of any jargon, but rather the high concentration of it. The appearance of one jargon term may be acceptable to him. Or if there's a way of saying "invariant" in layman terms then that could be substituted in. Let's see what Shoemaker Holiday thinks. RobHar (talk) 18:31, 10 January 2009 (UTC)

Oppose - I have already commented on this at the GA stage. I think it would be unfortunate to have on the main page of Wikipedia an article of which only a small subset of Ph.D. level mathematicians and graduate students can understand even the first sentence. For anybody else, the article doesn't even serve the purpose of provoking curiosity, because the background articles that it links to are, excuse me, crappy. The article may be very useful to a tiny group of specialists, but I don't think we should be advertising it to people who won't have any chance of getting anything out of it. Looie496 (talk) 04:57, 9 January 2009 (UTC)

While I also have my concerns (see above), I want to make it a point that your objection is not based on FA criteria and seems to be largely irrelevant: The argument with the main page is irrelevant, because we don't discuss that here. Also, that background articles are crappy is not the fault of this article. It is true that the article has narrow importance for the general audience, but this does not preclude it from being featured. Also, I (myself a Ph.D. student) think the article is well-written and understandable for an undergrad student willing to delve into the matter (or having had a course on basic topology), which is exactly the right audience for the article. Jakob.scholbach (talk) 08:08, 9 January 2009 (UTC)

Well, as I said in my GA comments, I think of myself as a test case here: I came close to getting a Ph.D. in math before switching to neuroscience, and have a decent understanding of algebra, analysis, logic, linear algebra, differential geometry, dynamical systems theory, and point set topology -- more than the majority of undergraduates would know. But I never delved into algebraic topology, and don't know what a homology group is, and consequently I can't get anything whatsoever out of this article. Of course I could go to the library and check out a book that deals with algebraic topology, spend a day learning the fundamentals, then come back to this article. But it doesn't seem to me that a Wikipedia FA should have such stringent requirements. Looie496 (talk) 18:48, 9 January 2009 (UTC)

Since there seems to be some question about this, could you please list which FA criteria you don't think this article passes? Thanks. RobHar (talk) 18:24, 10 January 2009 (UTC)

I don't believe that the article is consistent with WP:MSM#Article introduction, which is implicitly part of the FA criteria. For what it's worth, a statement at the beginning that "you have to know homology theory to understand any of this article, and you won't be able to learn it using Wikipedia" would resolve some of my issues, as it would prevent people from getting frustrated. Looie496 (talk) 20:26, 10 January 2009 (UTC)

The above discussion is preserved as an archive. Please do not modify it. No further edits should be made to this page.