Thanks!

]]>Actually, in case it saves you trouble, I am currently writing a proof at principal ideal domain that a submodule of a free module over a pid is itself free. This is Theorem 1 in a section on the structure theory of modules over a pid.

]]>Thanks, Todd. Right, I guess I made a mistake here. Let me think…

]]>Sorry, I’m not following the proof. Why is it obvious that $H$ is the abelianization of $p^{-1}(H)$?

]]>I made explicit at *Nielsen-Schreier theorem* statement and proof of the corollary (“Dedekind’s theorem”, I guess) that every subgroup of a free abelian group is itself free abelian. (The proof is below the two proofs of the main theorem. Of course it’s immediate, but I want to record it nevertheless.)

@Urs (30): yes, I think that is a recent-ish Firefox “innovation”. But somehow the “http://” is there invisibly. E.g. in the address box I see now, the address starts with “nforum”. But if I copy and paste the address into a text editor, it begins with “http://nforum”. It’s as if the “http://” really is there in the address bar, but in a typeface too tiny to read.

Although I don’t really like this change, it hasn’t given me any trouble, personally. I copy and paste URLs quite a lot, and the regaining of the http:// seems to work every time for me.

]]>I think I’m done with this. We agree both proofs are about homotopy 1-types. Arguing further about which is ’better’ is not a productive use of time, or not of my time anyway.

]]>The proofs better be the same, that’s the content of the homotopy hypothesis! :-)

What I still don’t see is what its buys us to fatten the combinatorial structure to a topological spayes, in the case at hand. It seems superfluous to me. Of course one can do it. Every invariant statement about groupoids is equivalently a statement about topological spaces that are homotopy 1-types.

Here it seems to me we are naturally talking about combinatorial graphs. Of cours I can say “think of the free groupoid it generates as a topological space”. But I can also not say this. Nothing changes.

]]>Getting back to #25 and #26: I took a quick look at the paper by Cote, and have no idea why she restricted attention to the countable case. Nor am I able to determine where she is or how to contact her to ask.

I hope it is generally agreed by this point that the proofs are very similar indeed. The place where the groupoidal proof ends corresponds to an explicit construction of a covering space projection $E \to X$ between connected graphs, where $E$ has fundamental group $H$. To show that the free groupoid on a connected directed graph is equivalent to a free group, presumably one does something similar to what is indicated in the topological proof (modding out by a maximal tree in the graph to get a bouquet of circles).

]]>I think it is a scanned document and so the pages are images not text.

]]>Thanks, I have moved that information to the entry.

(My pdf search on Higgins’ document doesn’t work, for some reason…)

]]>Can somebody point me to the precise page where in Higgin’s text the groupoid proof is given? Currently the entry claims that there is a proof there.

The Nielsen-Schreier Theorem is Theorem 9, which in Chapter 14, on page number 117 (page 133 of the pdf).

]]>Thanks. (I fixed the link to May’s book, the “http” was missing. Is this something that some recent update of Firefox introduced, to remove the “http” when copy-and-pasting url-s? Because since recently it happens to me all the time, too.)

Can somebody point me to the precise page where in Higgin’s text the groupoid proof is given? Currently the entry claims that there is a proof there.

]]>Thanks.

]]>I edited in several things. I put in a reference to May’s book where full details of the topological proof are given, and I included a brief description of some key technical ingredients in the proof of the topological version, following May. (Partly as brief response to Urs’s #25.) I also included the statement of the index formula in the theorem-statement, and gave a brief description in the topological proof how that is shown.

If desired, I can write out more of the details somewhere, but that might be superfluous since May’s book is freely available online and is linked to in the article.

]]>Should the Schreier index formula go here somewhere? it is a nice result and fits either here or in a separate entry. It is useful when explaining identities among relations.

]]>Well, I don’t think there’s actually any problem here at all; I’d have to look at what Cote wrote to see what the issue is she’s concerned about. (Axiom of choice issues? I think it’s true that the Nielsen-Schreier theorem is equivalent to the axiom of choice.)

What I might do later is copy out or paraphrase the topological proof of NS from A Concise Course in Algebraic Topology (pp. 34-35), which also has the more refined statement regarding ranks of the free groups, to remove any lingering concerns. If all this gets sorted out, we’ll have two nice proofs! :-)

]]>I need to think about where the countability assumption enters. How do you circumvent it in the topological proof?

]]>The text points to the proof of that statements at free groupoid – fundamental group!

Thanks. I see that the text points to a reference (which I haven’t looked at yet) which advertises a proof for the case of *countable* directed graphs.

That apparent lacuna needs to be addressed before calling the groupoidal proof a complete proof. Do you agree?

]]>17 confused me because your picture has three unlabeled vertices and three edges labeled

Sorry for the picture being confusing, I had tried to say in words right below how to to read it. The picture is the way it is because that’s how the covering space of $*\sslash F(S)$ comes out: this covering space is the path space $(*\sslash F(S))^{I}$ restricted to one endpoint of paths held fixed, hence the pullback

$\mathbf{E}F(S) = * \times_{(*\sslash F(S)) }(*\sslash F(S))^I \,.$That pullback has as morphisms the triangles in $*\sslash F(S)$ as indicated.

I want to hear your explanation.

The text points to the proof of that statements at free groupoid – fundamental group!

]]>Urs: #17 confused me because your picture has three unlabeled vertices and three edges labeled $g_1$, $g_2$, $s$ – instead of two vertices labeled by $g_1$, $g_2$ and one edge labeled $s$. (This isn’t nitpicking – this is honest confusion.)

Anyway, thanks – I think we’re on the same page now. So $F(S) \sslash S$ is the Cayley graph of the canonical presentation of $F(S)$ by generators $S$ and no relations. And now that you’ve explained $(H \backslash F(S)) \sslash S$, I think I’m clear on that too. (But all of this notation should be explained somewhere in the Lab.)

I think all I need now is your response to my question

On the other hand: what are you using to justify, “It is then sufficient to observe that this quotient is still a free groupoid on a directed graph to conclude that H is a free group.”?

To me, the idea is that the graph $(H \backslash F(S)) \sslash S$ is “homotopy equivalent” to a graph with one vertex and a bunch of loops – but I want to hear *your* explanation.

The word “action graph” I made up, it is meant to be the immediate restriction of the notion of action groupoid along a graph inside a groupoid.

In #17 I wrote it out. $F(S)\sslash S$ is supposed to denote the graph whose vertices are the elements of $F(S)$ and which has edges $g \stackrel{s}{\to} g \cdot s$ for all $g \in F(S)$ and $s \in S \hookrightarrow F(S)$.

Similarly $(H \backslash F(S)) \sslash S$ is supposed to denote the graph whose vertices are the elements $[g]$ in the left coset with edges $[g] \stackrel{s}{\to} [g \cdot s]$ for all $[g] \in H \backslash F(S)$ and $s \in S \hookrightarrow F(S)$.

]]>Right?

Well, I don’t know! :-)

Maybe my problem is that I’m still not sure I know what you mean by “the action graph” (this is not defined anywhere in the nLab, and I even have trouble finding a clear explanation anywhere on the Internet). I took a guess that the directed graph $F(S) \sslash S$ is the infinite $S$-ary tree, but that might be totally wrong. Would you mind giving a formal definition? I gather it’s a directed graph. The picture you drew in your last comment makes me afraid I’d gotten it wrong.

The expression $(H \backslash F(S)) \sslash S$ looks like a set of cosets, modulo a set. Off the bat, I wouldn’t know what that means, so I was guessing what you meant. This illustrates what I was saying in #4 about having to digest certain notation.

]]>I’m not sure you answered my question about placement of parentheses, involving particularly the subgroup H.

Maybe I misunderstand. Let’s see: the difference in the placement of the parentheses is that in one case we first take the quotient on the left by $H$ and then the weak quotient on the right by $S$, or the other way round.

Right?

So I suggested that both versions are equally good. But maybe we are talking past each other. Or I am mixed up. I admit of being involved in something else at the moment, so I might not be concentrating enough here. Let me know.

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