# Talk:Mayer–Vietoris sequence

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):

It would be nice to have full-length arrows so we could actually put the names of the maps on top
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)

Can use

  
    
      
        X
        
          
            →
            
              name of map!
            
          
        
        Y
      
    
    {\displaystyle X{\xrightarrow {\text{name of map!}}}Y}
  
.
The code for this is <math> X \xrightarrow{\text{name of map!}} Y </math>. 131.111.8.96 (talk) 11:38, 3 November 2008 (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)

A separate introductory section might be helpful.  The lead, however, should still comply with MOS:LEAD and provide a summary of the article.  I don't see any way to "dumb down" the lead while still retaining all of the information there.  Anyway, just for some perspective, the Mayer-Vietoris sequence is not something that is likely to be of any interest to someone who doesn't know what a topological space is, and the article further should presuppose some familiarity with the notion of a homology group. siℓℓy rabbit (talk) 19:35, 15 December 2008 (UTC)
I guess you could probably say it is a tool for computing certain important invariants of a space by breaking the space up into simpler subspaces. Though I'm unsure about how I feel about inserting such sentences into the lead of every single math article, so my suggestion of this is not a sign of support. Most disciplines other than math have the advantage that they are studying things that people have heard of, like neurons. If the lead of math articles were "dumbed down" as siℓℓy rabbit puts it then pretty much every article in algebra would start with "... is used to study the symmetry of a certain abstract structure", which isn't very informative. That being said, giving an explanation in an intro section makes sense. The Examples section in this article seems to basically serve that role though, in my opinion. Probably more could be added to it though. Cheers. RobHar (talk) 20:26, 15 December 2008 (UTC)
Actually, I think your first sentence is pretty good.  I suggest adding it to the lead in some form or other. siℓℓy rabbit (talk) 21:47, 15 December 2008 (UTC)
I agree with SR and RobHar: the lead is okay, but what is missing is an introductory/background section. At the moment the lead does not summarize the article because (e.g.) the sentence "Singular (co)homology cannot be computed directly from its definition for most topological spaces, the groups of singular chains and singular cycles need to be managed by theoretical tools before becoming understandable." is not elaborated in the article. What are singular chains and cycles? Why can't they be computed directly? Even readers who know what a topological space is need a "Background, Context and Motivation" section (or something similar) to explain the need for a Mayer-Vietoris sequence. {{see also}} can be used to direct them to more details. Since this is a fairly prototypical long exact sequence, some explanation of what a long exact sequence is would not go amiss either. Geometry guy 22:52, 15 December 2008 (UTC)
My issue when reading through the lead section is that it wasn't clear to me how the space is broken up in smaller pieces. Admittedly, this was largely because I was confused by the word "subspace" which subtly made me think about lower-dimensional pieces (the meaning of subspace in linear algebra); I guess that just proves that I rarely do topology. What do you think about adding after "two of its subspaces" in the second sentence the phrase "that together cover the whole space", or something like that? -- Jitse Niesen (talk) 15:54, 16 December 2008 (UTC)
I have attempted to implement some of your suggestions. siℓℓy rabbit (talk) 16:17, 16 December 2008 (UTC)
I like it much better now, thanks. -- Jitse Niesen (talk) 21:29, 16 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)

Actually, I think this article should aim to be comprehensible to a good math major who is willing to look up any concepts they are unfamiliar with (for instance, homology or cohomology group if they didn't take a course on algebraic topology). Hopefully an editor willing to review has now been found. Geometry guy 19:22, 16 December 2008 (UTC)

This review is transcluded from Talk:Mayer–Vietoris sequence/GA1. The edit link for this section can be used to add comments to the review.


### Well-written
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?

I prefer removing all punctuation, I find it obtrusive. Many authors do that (e.g. Hatcher). GeometryGirl (talk)


#### Lede
A wikilink for "singular (co)homology"?
Done GeometryGirl (talk)
Consider removing "so" in "So carefully choosing ..." to avoid the repeated word.
Done GeometryGirl (talk)
The M-V sequence "holds" in ... Can a sequence hold? Or would "is valid in" or "can be used in" be better?
The word 'result' or 'theorem' is implicit: "The M-V sequence result holds..." But maybe changing is better. GeometryGirl (talk)
I think it would be helpful to have a brief sentence (if possible) about what homology and cohomology is and why they are useful to put the M-V sequence into context.
I've made it clear that it was the "invariant" we were interested in. GeometryGirl (talk)
The word "patch" is used informally. But might a reader wonder whether what it is?
There are different ways of patching, each specific to the kind of space studied. The different spaces are listed in the Motivation section. GeometryGirl (talk)


#### Background and motivation
Wikilink "exact sequences" (only L.E.S. is linked in the lede)
Done GeometryGirl (talk)
Grammar error? "This does not allow in general to completely compute ..." The passive is needed here.
Could you change it for me please? I don't know English grammar. GeometryGirl (talk)
Done, but then, I'm not a mathematician - might be worth checking to see if that's what you intended to say. Jarry1250 (talk) 17:35, 30 December 2008 (UTC)
Replace "a theorem such as that of Mayer and Vietoris" with "the Mayer–Vietoris sequence"? It is not called a theorem before this point.
This is interesting since the M-V sequence is a theorem. It is not any old sequence, it is exact. That's the theorem. GeometryGirl (talk)


#### Basic versions for singular homology
It wasn't completely obvious what the inclusion maps i,j,k,l were
Added the explicit maps. GeometryGirl (talk)
Link "abelianized" or "abelianization" to abelian?
Done GeometryGirl (talk)


#### Further discussion
Funny grammar: "... for a continuous map f from X1 = A1∪B1 to X2 = A2∪B2 with f(A1) ⊂ A2 and f(B1) ⊂ B2 then f induces a map f* between ..."
Hopefully corrected. GeometryGirl (talk)


#### History
This section seems out of place here. Could it be incorporated into the "background" section higher up?

Done, thanks for the suggestion. GeometryGirl (talk)


### Factually accurate and verifiable
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.)

I've added the links. GeometryGirl (talk)


### Broad in its coverage
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 ;)

I'll think about this. GeometryGirl (talk)


### Neutral
Fine.


### Stable
Fine.


### Images
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!

Yes, this is thanks to RobHar's great job in converting my drawings. The hand-drawn diagram will hopefully also be converted shortly to make things look more professional and consistent. GeometryGirl (talk)
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)

Many thanks Martin! GeometryGirl (talk)


### Follow-up
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)

I agree, but this is due to the weakness of the homology group article rather than this article. Martin 21:28, 21 December 2008 (UTC)
I agree with you, and I agree with Martin even more. I think that from a logical point of view no one should be reading this article before knowing about homology. What I mean is that you do not represent the 'typical reader' of this article. The typical reader will know about homology, and there is no reason a reader without some topological background should land here (unless he clicked "random article", is a reviewer (like you), etc.). Thanks for giving your opinion anyway, it shows work on the background articles is needed. GeometryGirl (talk) 21:50, 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:

Fundamental group (something to relate to)
Direct sum of groups (definitely important here)
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? siℓℓy rabbit (talk) 00:46, 31 December 2008 (UTC)

Yes, I was thinking that it needed one. Martin 08:48, 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?  siℓℓy rabbit (talk) 03:10, 5 January 2009 (UTC)

Good point. Is it better now? GeometryGirl (talk) 11:11, 5 January 2009 (UTC)
Thanks, now that the image size has been reduced it fits.  It originally seemed to me that a better way would to reduce the size of the horizontal arrows so that the font could be kept at the same size, but I don't know how to do this in the amscd package.  I believe the text should also include a plain language description of the image for the benefit of people with screen readers. siℓℓy rabbit (talk) 14:04, 5 January 2009 (UTC)
The text says "f induces a map f_* between Mayer–Vietoris sequences". What more would you add? The vertical arrows are annoying, I'll post a message on the WikiProjectMathsTalk. GeometryGirl (talk) 14:47, 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.

Sheaf cohomology: section 13 of Bredon's Sheaf Theory
Sheaf homology: exercise V.8 of Bredon's Sheaf Theory (Bredon uses a rather general point of view so these need some parsing)
Cohomology on sites on schemes: (such as the étale site) page 110 of Milne's "Étale cohomology". It looks like the subspaces must be Zariski open.
Local cohomology: chapter 3 of Brodmann & Sharp's "Local cohomology". The MV sequence here can be used to relate the cohomological dimension to the number of generators of the ideal that is used in the local cohomology.
Suslin homology: I have no idea what this is, but the MV sequence holds for it according to Theorem 3.5.17 of "Cycles, Transfers, and Motivic Homology Theories" by Voevodsky et al.
Any ideas on how to organise this into the article? RobHar (talk) 01:56, 8 January 2009 (UTC)

Do all of those satisfy the Eilenberg-Steenrod axioms? If not, the ones that don't should be separated out. Algebraist 02:15, 8 January 2009 (UTC)
I think so, don't actually know. The point here is, I think, that a lot of algebraic topology is now done in a category-theoretic setting, and that the lack of coverage of this topic in this article is notable...linas (talk) 22:30, 18 August 2012 (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)

After some digging, it appears that coequalizer article makes it clearest in how it generalizes the notion of a quotient space and a kernel.  It provides the basic building block. The article on exact sequence could/should be extended first, and then finally, the appropriate functorial comments here, viz that there exists a functor into Ab that makes this all work. linas (talk) 23:13, 18 August 2012 (UTC)

The comment(s) below were originally left at Talk:Mayer–Vietoris sequence/CommentsTalk: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.
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 
  
    
      
        g
      
    
    {\displaystyle g}
  
 curves as a connected sum of elliptic curves.