Talk:Seifert–Van Kampen theorem

Sufficient conditions[edit]

Section added. —Nils von Barth (nbarth) (talk) 20:15, 26 July 2009 (UTC)[reply]

Time to state at least sufficient conditions for the theorem to hold. Charles Matthews 14:00, 23 Mar 2005 (UTC)

Sample Applications[edit]

It would be nice with a couple of examples of applications. YohanN7 (talk) 18:48, 10 December 2007 (UTC)[reply]

I added a simple application although its formatted very badly, also some pictures of the commutative diagrams would be nice --LkNsngth (talk) 03:54, 24 March 2008 (UTC)[reply]

eh, examples are nice, even better if they are somewhat interesting. that computation of π₁(S^2) is a trivial one; both A and B are disks. maybe we can replace the e.g. Mct mht (talk) 04:08, 24 March 2008 (UTC)[reply]

I added another calculation, the n fold torus, but its formatted quite badly so it would be great of someone better at this could make it more concise and more aesthetically pleasing. I also might have made a few errors so it would be great if someone could help proof read it. LkNsngth (talk) 04:43, 6 April 2008 (UTC)[reply]

I tried to make it more aesthetically pleasing and got completely carried away. I want to say that your argument was correct. Orthografer (talk) 07:02, 6 April 2008 (UTC)[reply]

I think you did a good jobLkNsngth (talk) 17:12, 6 April 2008 (UTC)[reply]

History[edit]

It would be nice if this article had a history section, at least to explain what was done by Seifert and what by van Kampen. Algebraist 23:46, 9 January 2009 (UTC)[reply]

Seconded. Sam nead (talk) 13:58, 13 February 2011 (UTC)[reply]

Name[edit]

I doubt if the name of this article is correct. Egbert van Kampen was Dutch, and according to Dutch orthography (see tussenvoegsel), the "van" in his last name should be capitalized if his first name is not included. Therefore it should be "Seifert–Van Kampen theorem". (I choose to ignore the endash spacing controversy.) I am not confident enough whether the Dutch orthography should be inherited in the English name, so I don't move the article, but I wanted to raise the issue here. --EdgeNavidad (talk) 10:58, 3 February 2010 (UTC)[reply]

Interesting. From the WP:NAME perspective, the key question is which variant of the name constitutes predominant usage by English language sources. Unfortunately, google-searching is not case-sensitive, but it appears that both variants are widely used. WP:USEENGLISH probably provides something of a sway towards using English rather than Dutch orthography in this case. Nsk92 (talk) 13:08, 3 February 2010 (UTC)[reply]

The same problem also applies to van Kampen diagram, also named after Egbert van Kampen. However, curiously, in that case looking through a googlebooks search[1] shows that "van Kampen diagram" is used much more frequently than "Van Kampen diagram". However, for "van Kampen theorem" the results are really split[2]. Nsk92 (talk) 13:23, 3 February 2010 (UTC)[reply]

Google search results for "Seifert–van Kampen -wikipedia": from the first 50 results:

Seifert-Van Kampen:26
Seifert-van Kampen:23
All capitals: 1

Just to show that you are correct about google saying both variants are widely used. To be honest, I don't see exactly how WP:USEENGLISH applies here, because there is no rule about how to capitalize "tussenvoegsels" in the English language. But it is too tricky for me to come up with the full solution, and I'll leave things at this small discussion on the talk page. If this should ever become a featured article, more attention should be given to it.--EdgeNavidad (talk) 17:26, 3 February 2010 (UTC)[reply]

This article is now a mess.[edit]

It seems a number of enthusiasts have added a some repetitive categorical language to the article, along with certain generalizations that tend to add distracting complication. I suggest some of these editors go back and organize the article into sections: an introduction that gives the weaker, noncategorical version that graduate students might come across during quals; then later sections on categorical versions and generalizations. Orthografer (talk) 14:31, 12 February 2010 (UTC)[reply]

I agree. Well, perhaps not totally a mess, but things are kind of inaccessible. Moreover, there are equality signs (=) all over. Those should instead be something indicating isomorphism rather than equality. An inprovement since i last checked in is that there is a nontrivial example. Good.

In the list of equivalent statements of the theorem the (arguably) "simplest" is missing, the one expressing the fundamental group as the free product of the groups coming from U and V modulo the normal closure of injections of elements of the group of W into the free product. This version requires less background on the behalf of the reader than any of the other. I think too that this version is perhaps not "intuitively true", but at least "intuitively plausible". This is especially the case when W is simply connected.

Is the image used in the proof correct? I wouldn't know, but the free product in the middle of it looks suspicious. YohanN7 (talk) 11:04, 14 July 2010 (UTC)[reply]

Proof[edit]

This page should include a proof of the theorem. There is a concise/intuitive proof using descent here: https://www3.nd.edu/~andyp/notes/SeifertVanKampen.pdf — Preceding unsigned comment added by 74.220.45.91 (talk) 04:34, 16 November 2017 (UTC)[reply]

Why do you feel that this article should include the proof? Here's the relevant Manual of Style page. siddharthist (talk) 07:38, 16 November 2017 (UTC)[reply]

Undefined notation/maps[edit]

Under Examples -> 2-Sphere: What is the product $\cdot$ used in $\pi_1(A) \cdot \pi_1(B)$? What is the map $\Phi$?

It would also be extremely helpful to identify why $\ker \Phi$ appears here, since quotients are not mentioned anywhere else in the exposition (except perhaps implicitly as added relations in the amalgamated product). — Preceding unsigned comment added by Dzackgarza (talk • contribs) 22:41, 22 December 2019 (UTC)[reply]