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Current status: Good article |
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There's a slight issue with the sign conventions on this page: here the explicit maps (i_*, j_*) and k_* - l_* are used, whereas the cited reference (Hatcher) uses (i_*, - j_*) and k_* + l_* (i.e. a change in signs).
The first instance of this is "For unreduced homology, the Mayer–Vietoris sequence states that the following sequence is exact:..."
The source here is [Hatcher 2002, p. 149], but I can't determine what edition/version this actually is. The widely available version has a copyright of 2001 and several reprints, so I can't verify the source or see if this was the actual convention used. It is possible that it is a mistake since the 2015 reprint (which is currently available online) uses the latter convention.
(This becomes an issue in the Klein Bottle example, where the latter convention is used as a critical part of the calculation.)
Dzackgarza ( talk) 23:37, 31 May 2020 (UTC)
Somebody wrote (in a comment near the diagram):
We should replace "\to" with "\buildrel{name}\over{\to}" or with "\mathrel{\mathop{\to}\limits^{name}}". Of course, we can't do either of these with the current incomplete TeX system. I've put in a half-assed version, but I won't be insulted if somebody says «That looks terrible.» and takes it out. (Just be sure to also take out the parenthetical references to the names in the following text.) -- Toby Bartels 23:40, 12 Jun 2004 (UTC)
It might be useful to mention the corresponding result for reduced homology groups with mention of the caveat that A and B have to have a non-empty intersection. -- CSTAR ( talk) 18:49, 5 December 2008 (UTC)
I realize that this article is well-written from the viewpoint of a mathematician, but from the viewpoint of anybody else it is useless. I spent several years in math grad school (before switching to neuroscience), and I can't understand even the first sentence. One of the things I like about neuroscience is that, even though it is becoming a pretty specialized discipline, with rare exceptions any neuroscientist can understand the first sentence of any neuroscience paper. In math, many papers have first sentences that can't be understood by more than a couple of dozen people in the entire world. This article follows in that unfortunate tradition: nobody except an algebraic topologist would be able to get anything out of it. There is no way that an article that makes no assertion of importance that anybody short of an advanced graduate student can understand can possibly be a good Wikipedia article. There is plenty of space available to explain why this topic is worth writing an article about, if there is any explanation that can be given. (Okay, I know this is a bit of a rant, that's why I'm not going to formally review the article…) Looie496 ( talk) 19:23, 15 December 2008 (UTC)
Regardless of my own views, let me point out that this article was nominated for Good Article status over a month ago (see WP:GAN). Since nobody except an algebraic topologist will be able to understand the article, the only hope of getting it reviewed is for some reader of this page to do so. Anybody who has not contributed significantly to the article can review it. Instructions can be found at WP:GAN. Looie496 ( talk) 17:29, 16 December 2008 (UTC)
Extremely well written in general. I have made a few minor adjustments and have some further comments by section. A general note about displayed equations: should they be punctuated properly? For example if they occur at the end of a sentence, should there be a full-stop?
This section seems out of place here. Could it be incorporated into the "background" section higher up?
I find the references perfectly adequate. It is of course great that the major source is freely available! It might be preferable to link the short footnotes to the main references. (I have done this for Hatcher as a start.)
The article provides comprehensive coverage of the topic. I feel however that a little more background is required, both in the lede and in the "background" section. It would be nice if the first paragraph was accessible to even a non-mathematician; at the moment it is not. I know from experience that this is very hard, but could some attempt be made to explain why the M-V sequence is important and useful, without using any kind of technical language such as "algebraic invariants" or "homology"?
Then perhaps in the second paragraph there could be a little more info on homology and cohomology. Interested readers can of course click the link, but there should be just enough in this article to give some context.
Could there be included some details about other related areas of mathematics? For example, what other tools are available for computing the homology groups and how do they compare to the M-V sequence?
I know what I am asking is probably difficult verging on impossible ;)
Fine.
Fine.
This article has some fantastic images and I commend the work that has gone into them. They really help to explain and clarify the subject. The hand-drawn diagram is also excellent and clear. I never knew that you could glue two mobius strips together to make a Klein bottle!
All images are properly licensed and tagged.
In conclusion, congratulations on the editors who have created such a good article. I await some input from others who are more familiar with the subject. Martin 12:21, 21 December 2008 (UTC)
Well it seems I will have to manage without other input. I am satisfied that all the minor points have been settled. In addition there have been some significant improvements to the lede. I still think this could probably be made more accessible, which would be important if this article was ever to make the main page (which I hope it does one day). I am now going to list this as a good article. Martin 22:04, 30 December 2008 (UTC)
I think this may be a good article in an absolute sense, but I'd like to walk you through what happens when somebody like me encounters it. ("Somebody like me" means somebody with a decent knowledge of algebra and topology but no background in algebraic topology.) When I come to the article, the main thing I want to know is what a MVS is and why it is important. Reading the first sentence, I see that it is defined in terms of homology groups. If I knew what a homology group is, I would be fine, but unfortunately I don't, so I follow the wikilink to homology group.
What I find there is totally incomprehensible, but it directs me to homology theory for background. So I go there. In that article, I find that the lead is too indefinite to be useful, and the next section is a "Simple explanation" that starts, At the intuitive level homology is taken to be an equivalence relation, such that chains C and D are homologous on the space X if the chain C − D is a boundary of a chain of one dimension higher.. If I knew what a "chain" was, this might be useful to me. I'll go there in a second, but first I note that the next section, giving an "Example of a torus surface", starts, For example if X is a 2-torus T, a one-dimensional cycle on T is in intuitive terms a linear combination of curves drawn on T…. This is nonsense to me, because you can't have linear combinations without operations of addition and scalar multiplication, and how do you do those things to curves? I can follow the wikilink to curve and hunt through it to find that I really ought to have been directed to algebraic curve. Going there, I find that an algebraic curve is an algebraic variety of dimension 1. So I follow that wikilink, and find that an algebraic variety is basically the set of roots of a polynomial. Hah! Finally something I understand! But how do you add or scalar-multiply such things? It isn't easy to tell.
So I go back to chain (algebraic topology), and find that a simplicial k-chain is a formal linear combination of k-simplices. So I go to simplex, and find a definition that I can understand, but no clue what value a "formal linear combination" of them would have.
So I am basically stuck. After all this hunting around, I have no real idea what a homology group is or why anybody thought this concept was worth inventing.
The bottom line is that the MVS article looks nice structurally, and probably would be very useful to a reader with the right background, but because of the weakness of the articles about underlying concepts, it is currently only useful to somebody with a strong background in algebraic topology. Looie496 ( talk) 20:01, 21 December 2008 (UTC)
One comment: In my view this article is very accessible. I tried reading the lede as if I don't know much mathematics and found out that the reader is likely to understand the lede if he/she understands:
I wouldn't mind helping out here after I finish at ring (mathematics). But in general, I would suggest mentioning that the fundamental group of a space is also an important topological invariant of the space so write something like:
"... like the fundamental group, this is an important topological invariant ..."
Also mention functor for obvious reasons.
Hope this helps.
PST —Preceding unsigned comment added by Point-set topologist ( talk • contribs) 21:18, 23 December 2008 (UTC)
Oh yes. I read the above thread and I agree that this is because the article on homology group is weak. Perhaps when you want to get this to FA, write a brief section on homology groups. Since homology groups are so important in understanding this, I think that this would be appropriate.
PST —Preceding unsigned comment added by Point-set topologist ( talk • contribs) 21:24, 23 December 2008 (UTC)
I just read the article on homology group. I think that it is quite accessible. A word of advice when reading Wikipedia articles: if you don't understand a term, read on until you have finished the section (don't click the link). Then see the links and this should give you a good understanding.
PST —Preceding unsigned comment added by Point-set topologist ( talk • contribs) 21:28, 23 December 2008 (UTC)
Any ideas for an image for the lead paragraph? siâ„“â„“y rabbit ( talk) 00:46, 31 December 2008 (UTC)
On the subject of images, MOS:IMAGES suggests that the clear function {{-}} should be used only as a last resort. I have been thinking about the best way to arrange the images so that the text flows without breaks. Perhaps some of the images could be put on the left. And maybe some of them are larger than they need to be. Martin 10:27, 31 December 2008 (UTC)
The image File:Mayer-Vietoris_naturality.png is too wide, and will not fit into many standard-sized windows (to say nothing of potential accessibility issues). Can the horizontal arrows, which take up most of the space, be scaled down 75% or so? siâ„“â„“y rabbit ( talk) 03:10, 5 January 2009 (UTC)
I dug up some references for the mv sequence in more general cohomology theories, and I figured I'd throw them here for now.
Any ideas on how to organise this into the article? RobHar ( talk) 01:56, 8 January 2009 (UTC)
As pointed out in the talk page to sheaf (mathematics), the word equalizer is never actually used in this article, even though the two parallel arrows (i*, j*) of the inclusion maps essentially lead to this!? I thought this was fairly central to the definition!? The confusion is that the sheaf article asks for equalizers when gluing together, notices the sequence is exact, but no other WP article seems to actually delve into this topic... linas ( talk) 22:22, 18 August 2012 (UTC)
The comment(s) below were originally left at Talk:Mayer–Vietoris sequence/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
Now GA. Taking the article further requires more background, history and context, and also more secondary sources and citations (especially for the history). Geometry guy 13:16, 1 January 2009 (UTC) |
Last edited at 13:16, 1 January 2009 (UTC). Substituted at 02:19, 5 May 2016 (UTC)
In the mentioned paragraph, there is the following conclusion.
"Choosing another decomposition x = u′ + v′ does not affect [∂u], since ∂u + ∂v = ∂x = ∂u′ + ∂v′, which implies ∂u − ∂u′ = ∂(v′ − v), and therefore (???) ∂u and ∂u′ lie in the same Homology class".
I don't see how this implication goes. (as Boundary(A∩B)≠Boundary(A)∩Boundary(B))
Another good application of Mayer-Vietoris is the the computation of genus curves as a connected sum of elliptic curves.
![]() | Mayer–Vietoris sequence has been listed as one of the Mathematics good articles under the good article criteria. If you can improve it further, please do so. If it no longer meets these criteria, you can reassess it. | |||||||||||||||
| ||||||||||||||||
Current status: Good article |
![]() | This article is rated GA-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
|
|
There's a slight issue with the sign conventions on this page: here the explicit maps (i_*, j_*) and k_* - l_* are used, whereas the cited reference (Hatcher) uses (i_*, - j_*) and k_* + l_* (i.e. a change in signs).
The first instance of this is "For unreduced homology, the Mayer–Vietoris sequence states that the following sequence is exact:..."
The source here is [Hatcher 2002, p. 149], but I can't determine what edition/version this actually is. The widely available version has a copyright of 2001 and several reprints, so I can't verify the source or see if this was the actual convention used. It is possible that it is a mistake since the 2015 reprint (which is currently available online) uses the latter convention.
(This becomes an issue in the Klein Bottle example, where the latter convention is used as a critical part of the calculation.)
Dzackgarza ( talk) 23:37, 31 May 2020 (UTC)
Somebody wrote (in a comment near the diagram):
We should replace "\to" with "\buildrel{name}\over{\to}" or with "\mathrel{\mathop{\to}\limits^{name}}". Of course, we can't do either of these with the current incomplete TeX system. I've put in a half-assed version, but I won't be insulted if somebody says «That looks terrible.» and takes it out. (Just be sure to also take out the parenthetical references to the names in the following text.) -- Toby Bartels 23:40, 12 Jun 2004 (UTC)
It might be useful to mention the corresponding result for reduced homology groups with mention of the caveat that A and B have to have a non-empty intersection. -- CSTAR ( talk) 18:49, 5 December 2008 (UTC)
I realize that this article is well-written from the viewpoint of a mathematician, but from the viewpoint of anybody else it is useless. I spent several years in math grad school (before switching to neuroscience), and I can't understand even the first sentence. One of the things I like about neuroscience is that, even though it is becoming a pretty specialized discipline, with rare exceptions any neuroscientist can understand the first sentence of any neuroscience paper. In math, many papers have first sentences that can't be understood by more than a couple of dozen people in the entire world. This article follows in that unfortunate tradition: nobody except an algebraic topologist would be able to get anything out of it. There is no way that an article that makes no assertion of importance that anybody short of an advanced graduate student can understand can possibly be a good Wikipedia article. There is plenty of space available to explain why this topic is worth writing an article about, if there is any explanation that can be given. (Okay, I know this is a bit of a rant, that's why I'm not going to formally review the article…) Looie496 ( talk) 19:23, 15 December 2008 (UTC)
Regardless of my own views, let me point out that this article was nominated for Good Article status over a month ago (see WP:GAN). Since nobody except an algebraic topologist will be able to understand the article, the only hope of getting it reviewed is for some reader of this page to do so. Anybody who has not contributed significantly to the article can review it. Instructions can be found at WP:GAN. Looie496 ( talk) 17:29, 16 December 2008 (UTC)
Extremely well written in general. I have made a few minor adjustments and have some further comments by section. A general note about displayed equations: should they be punctuated properly? For example if they occur at the end of a sentence, should there be a full-stop?
This section seems out of place here. Could it be incorporated into the "background" section higher up?
I find the references perfectly adequate. It is of course great that the major source is freely available! It might be preferable to link the short footnotes to the main references. (I have done this for Hatcher as a start.)
The article provides comprehensive coverage of the topic. I feel however that a little more background is required, both in the lede and in the "background" section. It would be nice if the first paragraph was accessible to even a non-mathematician; at the moment it is not. I know from experience that this is very hard, but could some attempt be made to explain why the M-V sequence is important and useful, without using any kind of technical language such as "algebraic invariants" or "homology"?
Then perhaps in the second paragraph there could be a little more info on homology and cohomology. Interested readers can of course click the link, but there should be just enough in this article to give some context.
Could there be included some details about other related areas of mathematics? For example, what other tools are available for computing the homology groups and how do they compare to the M-V sequence?
I know what I am asking is probably difficult verging on impossible ;)
Fine.
Fine.
This article has some fantastic images and I commend the work that has gone into them. They really help to explain and clarify the subject. The hand-drawn diagram is also excellent and clear. I never knew that you could glue two mobius strips together to make a Klein bottle!
All images are properly licensed and tagged.
In conclusion, congratulations on the editors who have created such a good article. I await some input from others who are more familiar with the subject. Martin 12:21, 21 December 2008 (UTC)
Well it seems I will have to manage without other input. I am satisfied that all the minor points have been settled. In addition there have been some significant improvements to the lede. I still think this could probably be made more accessible, which would be important if this article was ever to make the main page (which I hope it does one day). I am now going to list this as a good article. Martin 22:04, 30 December 2008 (UTC)
I think this may be a good article in an absolute sense, but I'd like to walk you through what happens when somebody like me encounters it. ("Somebody like me" means somebody with a decent knowledge of algebra and topology but no background in algebraic topology.) When I come to the article, the main thing I want to know is what a MVS is and why it is important. Reading the first sentence, I see that it is defined in terms of homology groups. If I knew what a homology group is, I would be fine, but unfortunately I don't, so I follow the wikilink to homology group.
What I find there is totally incomprehensible, but it directs me to homology theory for background. So I go there. In that article, I find that the lead is too indefinite to be useful, and the next section is a "Simple explanation" that starts, At the intuitive level homology is taken to be an equivalence relation, such that chains C and D are homologous on the space X if the chain C − D is a boundary of a chain of one dimension higher.. If I knew what a "chain" was, this might be useful to me. I'll go there in a second, but first I note that the next section, giving an "Example of a torus surface", starts, For example if X is a 2-torus T, a one-dimensional cycle on T is in intuitive terms a linear combination of curves drawn on T…. This is nonsense to me, because you can't have linear combinations without operations of addition and scalar multiplication, and how do you do those things to curves? I can follow the wikilink to curve and hunt through it to find that I really ought to have been directed to algebraic curve. Going there, I find that an algebraic curve is an algebraic variety of dimension 1. So I follow that wikilink, and find that an algebraic variety is basically the set of roots of a polynomial. Hah! Finally something I understand! But how do you add or scalar-multiply such things? It isn't easy to tell.
So I go back to chain (algebraic topology), and find that a simplicial k-chain is a formal linear combination of k-simplices. So I go to simplex, and find a definition that I can understand, but no clue what value a "formal linear combination" of them would have.
So I am basically stuck. After all this hunting around, I have no real idea what a homology group is or why anybody thought this concept was worth inventing.
The bottom line is that the MVS article looks nice structurally, and probably would be very useful to a reader with the right background, but because of the weakness of the articles about underlying concepts, it is currently only useful to somebody with a strong background in algebraic topology. Looie496 ( talk) 20:01, 21 December 2008 (UTC)
One comment: In my view this article is very accessible. I tried reading the lede as if I don't know much mathematics and found out that the reader is likely to understand the lede if he/she understands:
I wouldn't mind helping out here after I finish at ring (mathematics). But in general, I would suggest mentioning that the fundamental group of a space is also an important topological invariant of the space so write something like:
"... like the fundamental group, this is an important topological invariant ..."
Also mention functor for obvious reasons.
Hope this helps.
PST —Preceding unsigned comment added by Point-set topologist ( talk • contribs) 21:18, 23 December 2008 (UTC)
Oh yes. I read the above thread and I agree that this is because the article on homology group is weak. Perhaps when you want to get this to FA, write a brief section on homology groups. Since homology groups are so important in understanding this, I think that this would be appropriate.
PST —Preceding unsigned comment added by Point-set topologist ( talk • contribs) 21:24, 23 December 2008 (UTC)
I just read the article on homology group. I think that it is quite accessible. A word of advice when reading Wikipedia articles: if you don't understand a term, read on until you have finished the section (don't click the link). Then see the links and this should give you a good understanding.
PST —Preceding unsigned comment added by Point-set topologist ( talk • contribs) 21:28, 23 December 2008 (UTC)
Any ideas for an image for the lead paragraph? siâ„“â„“y rabbit ( talk) 00:46, 31 December 2008 (UTC)
On the subject of images, MOS:IMAGES suggests that the clear function {{-}} should be used only as a last resort. I have been thinking about the best way to arrange the images so that the text flows without breaks. Perhaps some of the images could be put on the left. And maybe some of them are larger than they need to be. Martin 10:27, 31 December 2008 (UTC)
The image File:Mayer-Vietoris_naturality.png is too wide, and will not fit into many standard-sized windows (to say nothing of potential accessibility issues). Can the horizontal arrows, which take up most of the space, be scaled down 75% or so? siâ„“â„“y rabbit ( talk) 03:10, 5 January 2009 (UTC)
I dug up some references for the mv sequence in more general cohomology theories, and I figured I'd throw them here for now.
Any ideas on how to organise this into the article? RobHar ( talk) 01:56, 8 January 2009 (UTC)
As pointed out in the talk page to sheaf (mathematics), the word equalizer is never actually used in this article, even though the two parallel arrows (i*, j*) of the inclusion maps essentially lead to this!? I thought this was fairly central to the definition!? The confusion is that the sheaf article asks for equalizers when gluing together, notices the sequence is exact, but no other WP article seems to actually delve into this topic... linas ( talk) 22:22, 18 August 2012 (UTC)
The comment(s) below were originally left at Talk:Mayer–Vietoris sequence/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
Now GA. Taking the article further requires more background, history and context, and also more secondary sources and citations (especially for the history). Geometry guy 13:16, 1 January 2009 (UTC) |
Last edited at 13:16, 1 January 2009 (UTC). Substituted at 02:19, 5 May 2016 (UTC)
In the mentioned paragraph, there is the following conclusion.
"Choosing another decomposition x = u′ + v′ does not affect [∂u], since ∂u + ∂v = ∂x = ∂u′ + ∂v′, which implies ∂u − ∂u′ = ∂(v′ − v), and therefore (???) ∂u and ∂u′ lie in the same Homology class".
I don't see how this implication goes. (as Boundary(A∩B)≠Boundary(A)∩Boundary(B))
Another good application of Mayer-Vietoris is the the computation of genus curves as a connected sum of elliptic curves.