Talk:Fundamental group/Archive 1

Archive 1

how to find a space with a given fundamental group

is there a general procedure for finding or constructing a space with a given fundamental group? especially a smooth manifold? -Lethe | Talk 22:48, August 8, 2005 (UTC)

There is such a procedure starting from a presentation for the group - see Corollary 1.28 in Hatcher's on-line book. But there's no way to get a manifold in general, because there aren't enough homeomorphism types of connected manifolds. --Zundark 11:31, 9 August 2005 (UTC)

I can't make out what Zundark is saying. There are uncountably many countable groups, true, but there are uncountably many manifolds, as well. You weren't assuming compactness, were you? I believe that it is correct to say that every finitely presented group can be obtained as the fundamental group of a compact four-manifold. Now, obtaining a general countable group seems a bit harder. But if we give up compactness it should be possible. Best, Sam nead 00:40, 11 October 2005 (UTC)

The original question didn't say anything about countable groups. A connected manifold has cardinality at most $2^{\aleph _{0}}$, so there can't be more than $2^{2^{2^{\aleph _{0}}}}$ different connected manifolds. (In fact, I think it's known that there are only $2^{\aleph _{1}}$ connected manifolds, including non-paracompact ones). But there are an unlimited number of groups, since there is at least one of every non-zero cardinality. (Of course, the original question didn't say anything about connectedness either, but the fundamental group depends only on the path-component, so it suffices to consider connected manifolds.) --Zundark 08:26, 11 October 2005 (UTC)

I have a brief moment and would like to jump in. For finitely generated fundamental groups on CW-complexes the matter is easy. Let G be a finitely generated group on n generators. Now form the complex of n 1-cells attached to a single point, (or the one-point union or wedge of n circles, if you prefer). Label the edges to correspond to the generators. Now use 2-cells to mod out by relations, by stretching a 2-cell around each sequence of edges that correspondes to a word in the presentation. Okay, now we have a CW-complex with G as fundamental group. The next part is a bit cheap and hand-wavy, but convinces me, so I'll share, but may need help with details. We have an embedding theorem for CW-complexes into R^k, for some large k. We should also be able to find an open neighborhood of this embedding that deformation retracts to it. Then we have it! a (flat!) manifold with fundamental group G.

Oh, BTW, we're using only finitely generated groups here. These are countable. If one uses something like a hawiian ring ($\cup _{n}\{(x,y)\in \mathbb {R} ^{2}|(x-1/n)^{2}+y^{2}=1/n^{2}\}$), one can construct CW-complexes for countably generated G, but the deformation retract idea won't work here. I haven't even addressed the case for uncountable groups here, mentioned by Zundark, above, but for finite dimensional manifolds the fundamental groups are always countable. The worst case for a manifold, i believe, is a countable direct sum (not product!!) of countable groups, which is countable. MotherFunctor 02:30, 16 May 2006 (UTC)

The classifying space construction is also relevant, but it won't give you manifolds in general. - Gauge 04:19, 15 December 2005 (UTC)

Same for Eilenberg-MacLane spaces. --Orthografer 04:36, 11 March 2006 (UTC)

Since everyone seems to be in the right mood, what about morphisms? That is, given a topological space X, a group G and a group morphism from \pi_1(X) to G is there a top. space and a continuous function that induce that situation?

hmmm, what happens if you quotient the manifold by a curve representing for each relation defining the kernal. A problem might be that the loop may be forced to intersect itself. Where is this problem from, anything specific? I don't know how to do it off hand. MotherFunctor 06:09, 15 November 2007 (UTC)

Yeah, sure -- just take a classifying space K(G,1), and realize the given homomorphism by a map X --> K(G,1). Turgidson 13:33, 15 November 2007 (UTC)

I think the symbol for the wedge sum in the third paragraph of section "Functorality" is not a wedge sum but a smash product. Sorry I couldn't change it since I didn't find the right character.. --Cheesus 16:16, 10 August 2006 (UTC)

fundamental group is abelian iff its path independent

I added this. easy to show.

128.226.164.120 (talk)jesusonfire —Preceding comment was added at 04:18, 9 December 2007 (UTC)

I have removed it because, as written, "path independent" is undefined and I have no idea what you might mean by this. Please don't use "doesn't" in main space articles. Mathsci (talk) 05:22, 9 December 2007 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Fundamental group/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.

Influential in both group theory and topology. Geometry guy 19:58, 10 June 2007 (UTC)

Last edited at 19:58, 10 June 2007 (UTC). Substituted at 02:08, 5 May 2016 (UTC)

Functors, categories and fundamental groups

The introduction seems to say that the fundamental group is functorially associated to a space, which is false:

"Using the concept of a functor in category theory, one can talk about how any topological space is associated to a particular fundamental group and every continuous map between topological spaces is associated to a homomorphism of fundamental groups. The homomorphism associated to a particular continuous map (between topological spaces) is called the homomorphism induced by that map or the induced homomorphism. This is a purely category theoretical way of viewing the fundamental group."

This paragraph is false/not clear: One must consider the category of pointed topological spaces (spaces with a point). --Cbigorgne (talk) 15:51, 19 January 2009 (UTC)

Thanks but I actually noted this in the first paragraph and I thought that it would be redundant to continuously repeat "pointed". Also, not many people will know what a pointed space is (by the way it is not a "space with a point" but rather an ordered pair consisting of (X, p) where X is a space and p is a point). But I included (pointed) in brackets in front of "topological space" in that section. Thanks! --PST 15:55, 19 January 2009 (UTC)

I significantly expanded the lede

Please have a look... --PST 15:43, 19 January 2009 (UTC)

The introduction is now very bad. It is repetitive, both internally and vis-a-vis the article body. I will shortly revert it. best, Sam nead (talk) 21:56, 24 January 2009 (UTC)

Dear PST -- I have severely edited down the introduction. The previous version had the flaws mentioned above. I do feel that this article would greatly benefit from some love and attention -- but erecting a barrier in the introduction will only scare away readers. Sam nead (talk) 22:23, 24 January 2009 (UTC)

Hi Sam,

With most articles, the lede is supposed to be big (see vector space or group (mathematics)). With an article like fundamental group, there are so many important concepts relating to it, that they must be mentioned. See WP:LEAD. Could you please note that when trimming down introductions, that you don't delete material? I would think at least that new material could be moved to different sections. I am reverting your change only because I feel that you are unsure with the lead policies (which say that the lede should include all important concepts relating to the article in question). If you feel differently, we can discuss more, or contact other Wikipedians. --PST 08:08, 28 January 2009 (UTC)

Dear PST - I have read the lead of WP:LEAD and I respectfully (and strongly) disagree that "all" important concepts must be covered. To quote, the lead should summarize the most important points. Nowhere does it say "all". In fact, to follow your advice would prevent us from having a concise lead, which WP:LEAD holds to be even more important.

To address your first point, I was pretty careful about not deleting material -- functoriality is covered later in the article. I would urge you to assume good faith on my part! Frankly, I suspect that, at the heart of the matter, we disagree over the basic importance of category theory as a way of understanding the fundamental group. In fact I hold that the opposite is true... best, Sam nead (talk) 21:03, 31 January 2009 (UTC)

That's fine. The only issue now is WP:NPA (by User:MathSci below). --PST 07:24, 16 February 2009 (UTC)

Point set topologist's lede

What PST wrote in the lede was mathematically misleading, to the extent of being nonsensical and wrong in the 2nd paragraph. This is elementary undergraduate material on extremely well-trodden ground. What might be included in the lede is (a) a mention of covering spaces/deck transformations and (b) the identification of the abelianisation of π₁ with the first homology group. There is no need for commentaries by editors who do not appear to have mastered the subject. Mathsci (talk) 23:25, 28 January 2009 (UTC)

I research homotopy theory (and some parts of algebraic topology) but I guess that is not a relevant background here (could you please be a little more polite? In WP, we want to resolve matters without making unnecessary personal attacks and we also want to make some sort of agreement. Currently, we are fighting from one version to another and nothing in between). The lede I wrote was certainly not up to FA standards; it was merely an intuitive way of understanding the fundamental group. I intended that each paragraph be made more encyclopedic but that for now, the lede was appropriate. Your (b) is a rather basic theorem in homology theory that should be mentioned in a paragraph for "Connection between the fundamental group and homology theory". Your (a) is already mentioned in the fourth paragraph of my lead (when you think a part of something is rubbish, you should not revert the whole thing but rather just that part). Secondly, I find no errors in the second paragraph; if you feel otherwise, could you please explain them? To summarize, I know my lede was not completely formal; but was intuitive. If you see WP:MTAA, it is a requirement, that most articles should be written in a manner that can be understood by as many people as possible. I included an intuitive way to calculate the fundamental group of the circle, as well as noted stuff about category theory, covering spaces and fundamental groups of other objects. I feel, that we want to progress in WP; and by reverting large edits and making personal attacks, we are not going to get anywhere (the article had barely any lead for many years). --PST 08:54, 29 January 2009 (UTC)

The quote below and much of the rest of the lede is simply bad mathematics: it bears no relations to published texts, has no citations and is unrelated to the main article. It is PSt trying to put together his jumbled and mathematically immature thoughts on wikipedia. PST has not had a university training and this shows in his unhelpful and uninformed commentary. The following sentences hit rock bottom mathematically and could have PST topic-banned from contributing to university level articles on mathematics:

Using the concept of a functor in category theory, one can talk about how any (pointed) topological space is associated to a particular fundamental group and every continuous map between (pointed) topological spaces is associated to a homomorphism of fundamental groups. The homomorphism associated to a particular continuous map (between topological spaces) is called the homomorphism induced by that map or the induced homomorphism. This is a purely category theoretical way of viewing the fundamental group.

PST's edits seem completely clueless in matters mathematical. He has already broken the 3RR rule and is clearly disrupting the encyclopedia; the functor π₁ goes from spaces to groups, not the other way round. To write the opposite is unhelpful and shows a complete lack of expertise in the area. Mathsci (talk) 22:12, 29 January 2009 (UTC)

Dear MathSci - In fact the quoted material is only slightly wrong. You are correct that the first sentence has the functor the wrong way around. But the second gets it right. Indeed the "fundamental group" functor takes pointed maps to fundamental group homomorphisms. Please calm down. I think it is best to assume that PST is editing in good faith, and simply has views that are much stronger than we usually find in a mathematical discussion. Remember, ad hominem remarks are uncalled for. Best, Sam nead (talk) 20:51, 31 January 2009 (UTC)

Dear PST - the functoriality of the fundamental group (that is fun to say quickly!) is covered in the article. The question that I am asking you is "does functoriality belong in the introduction?" It would be nice if the intro was accurate, short, and useful to a person who does not know what the fundamental group is. I personally think that a category theory discussion very much interferes with the second two aspects of a good introduction. all the best, Sam nead (talk) 20:51, 31 January 2009 (UTC)

The lede has been changed to summarise what is in the main text of the article: it agrees with time-hallowed content of undergraduate/graduate courses and can be found in standard textbooks.

PST is in no way helping this discussion by asserting that he does "research in homotopy theory". We know he has not yet been to university, based on his previous incarnation and the way in which he edits. By a researcher in homotopy theory I understand colleagues like Michael J. Hopkins from MIT or Ib Madsen from Aarhus. Could PST please stop misrepresenting himself?

Functoriality applies to almost every branch of pure mathematics and does not need to be spelled out incorrectly and irrelevantly in the lede or in the main text of the article. BTW so far the article has no discussion of classifying spaces or K(Π,1)'s, which could be added. Mathsci (talk) 10:03, 1 February 2009 (UTC)

In order to prevent further harrassment, I have contacted User:Joeldl. Just to note, if one paragraph of the lede is conceived to be incorrect, this is no way implies that the whole lead is incorrect. The paragraph mentioned was perhaps badly worded on my part, for I intended to write that any group is associated to a pointed topological space (known as the fundamental group in this particular context). So really, the misunderstanding was due to my misuse of "associated". I should have said that "any pointed topological space is "mapped" onto a group" rather than "associated". I used "associated" the wrong way around. --PST 07:46, 16 February 2009 (UTC)

Severe problems with style

I am sorry, I slapped the style template on this article. I am trying to improve it myself, but I can only do so much. Please help. This article is currently too chatty and informal. All talking to the reader ("consider" etc) has to go. The use of "we" is a strict no no on wikipedia. Please be concise, and structured. Don't expect the reader to read everything, and keep in mind we are not addressing a single reader of one background here, we are addressing both non-experts and experts, so information needs to be better structured.--345Kai (talk) 12:00, 22 June 2009 (UTC)

Most of the style problems seem to be in the first section "Intuition and definition". There is nothing wrong with giving an informal definition, as long as the formal definition is also there. Mathsci (talk) 12:14, 22 June 2009 (UTC)

Base point dependence

Hi! It is stated that we have (up to isomorphism) independence of the base point provided the space in question is path-connected. I believe this to be true, that's not the problem. The question is if somebody knows why Stephen Willard in his General Topology writes that arc-connectedness is needed for the independence of base point. Willard is usually pretty picky when it comes to making assumptions in theorems. (Willard is cited as reference in several articles...)YohanN7 (talk) 22:09, 15 July 2009 (UTC)

If it's not path connected, then the fundamental group may depend on the base point. For example: if the space is the disjoint union of a line and a circle (not path connected), then what is the fundamental group? If the base point is in the line, it's trivial. If the base point is in the circle, it's the integers. Staecker (talk) 12:02, 16 July 2009 (UTC)

The point of the question was the distinction between path-connected and arc-connected (which is stronger). Hans Adler 14:45, 16 July 2009 (UTC)

Hans, you caught me there too... The last paragraph of Arc-connected#Path_connectedness has a good discussion. Thinking everything is Hausdorff is a hard habit to break. Orthografer (talk) 04:09, 19 July 2009 (UTC)

I also missed it at first. In fact, I wasn't aware there is such a notion as arc-connected. If I had learned about it it would have been in German, and I still don't know a German word for it. And I still haven't figured out why you would want to assume it in this context. – Unless he also defines the fundamental group in terms of arc-connectedness? Hans Adler 07:21, 19 July 2009 (UTC)

Definition messed up

I see that since September people have tried to simplify the definition section. Unfortunately it now contains errors. I fixed the definition of the homotopy, but the definition of addition of loops is just too messed up for a minor fix. Adding two loops with different endpoints can be done, but you then have to choose a consistent set of paths between every two points in the space (taking care to make the compositions commute up to homotopy), and then use this set for all additions. Choosing a different path between the endpoints for an addition can give different answers! Also this is not exactly a simplification. Much easier to define the fundamental group at a point, and then maybe show using paths how groups at different points are isomorphic. --Ørjan (talk) 19:28, 5 February 2010 (UTC)

Actually the Intuition section seems much better, and afaict that's mostly the original one. --Ørjan (talk) 19:48, 5 February 2010 (UTC)

I deleted the first section since it was seriously confusing - using additive notation for concatenation of paths is very bad style. However, there is a kernel of a good idea there. The author was trying to get at the basepoint independence of the fundamental group of a path-connected space. This is a good thing to point out -- it would be nice to have a reference. Sam nead (talk) 14:22, 6 February 2010 (UTC)

I added a sentence on basepoint independence to the functoriality section. It could probably be improved. Best, Sam nead (talk) 14:28, 6 February 2010 (UTC)

Could you explain why you have replaced "path-connected" by "arc-connected"? It's a long time since I did any proper topology, but I seem to remember that path-connected is enough and simply can't see what you would use the stronger definition for in this context. If arc-connected is really necessary, then this article should have a path-connected counterexample. But it seems to me that the obvious, trivial proof works just fine in the path-connected case. Hans Adler 23:40, 6 February 2010 (UTC)

I was following the discussion in the section "Base point dependence", immediately above this section. Best, Sam nead (talk) 19:01, 7 February 2010 (UTC)

I don't think that section reached a conclusion. I just tried looking it up in my Massey's A Basic Course in Algebraic Topology (1991), but I find he is using arc as a synonym for path... which means his theorem 3.5:

"If X is arcwise connected, the groups π(X, x) and π(X, y) are isomorphic for any two points x, y ∈ X."

is really about what we here call paths. He does not seem to assume that spaces in this chapter are Hausdorff. --Ørjan (talk) 01:54, 8 February 2010 (UTC)

So far as I was concerned, the discussion had a conclusion, it was just not explicit in the discussion itself. If someone (me) can learn algebraic topology for several years without ever encountering the notion arc-connectedness in a memorable way, that's a strong indication that it's not needed in this context.

The fundamental group is obviously defined via paths, not arcs. (Otherwise the space consisting of a single point would have an empty "fundamental group".) A path p from A to B can always be reversed; let's call the result q. Let s be a loop at A. We get the corresponding loop at B as qsp: the map that is q (scaled by 1/3) on the first third of the unit interval, s on the second, and p on the third. If t is another loop at A, then it's easy to see that (qsp)(qtp) is homotopic to q(st)p. (The homotopy happens entirely on the side of the unit interval, so non-Hausdorff is not an issue at all. Basically this is the same proof that pq is zero-homotopic, which also doesn't need any assumptions on the space.)

In addition to Massey, we have Munkres, Topology (2nd edition), Theorem 52.1. It confirms what I just wrote: A path from A to B is enough to get an isomorphism. In light of this I am restoring "path-connected". Hans Adler 08:01, 8 February 2010 (UTC)