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Earlier, I was looking at a question here about the abelianization of a certain group $X$. Since $X$ was the fundamental group of a closed surface $\Sigma$, it was easy to compute $X^{ab}$ as $\pi_1(\Sigma)^{ab} = H_1(\Sigma)$, then use the usual machinery to compute $H_1(\Sigma)$. That made me curious about other compelling examples of solving purely (for some definition of 'purely') algebraic questions that are accessible via topology or geometry. The best example I can think of the Nielsen-Schreier theorem, which is certainly provable directly but has a very short proof by recasting the problem in terms of the fundamental group of a wedge product of circles. Continuing this line of reasoning leads to things like graphs of groups, HNN-extensions, and other bits of geometric group theory.

What are some other examples, at any level, of ostensibly purely group-theoretic results that have compelling, shorter topological proofs? The areas are certainly closely connected; I'm looking more for what seem like completely algebraic problems that turn out to have completely topological resolutions.

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    $\begingroup$ "How many index 2 subgroup of $\mathbb{Z*Z}$ exists"...this you can prove using concept of covering space. $\endgroup$ – Anubhav Mukherjee Dec 17 '15 at 17:31
  • $\begingroup$ @Anubhav.K You can also prove that easily directly. See math.stackexchange.com/questions/1542055/… $\endgroup$ – Nex Dec 17 '15 at 18:32
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The Kaplansky Conjecture asserts that the group ring $\Bbb Q G$ contains no non-trivial zero-divisors when $G$ is a torsionfree group.

It is implied by the Atiyah Conjecture, (a version of) which states that for every compact connected CW complex $X$ with $\pi_1(X)=G$, all $\ell^2$-Betti numbers are integers.

The Atiyah Conjecture is now known to be true for a great deal of groups - in particular for groups for which the truth of the Kaplansky Conjecture was unknown before.

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    $\begingroup$ Interesting, thanks. I'm familiar with the Kaplansky conjecture (despite its algebraic nature, it comes up in discussing, for example, Whitehead torsion and surgery theory), but wasn't familiar with the Atiyah conjecture. $\endgroup$ – anomaly Dec 17 '15 at 18:59
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There is an exact sequence, due to Hopf: $$\pi_2(X) \to H_2(X) \to H_2(\pi_1(X)) \to 0,$$ where the first map is the Hurewicz map. In some sense, $H_2(\pi_1(X))$ measures how surjective the (2-dimensional) Hurewicz map is. The purpose of this answer is to sell you on this exact sequence. Here's a small result you can prove with it.

Theorem: if $G$ is the fundamental group of a homology sphere, then $H_1(G) = H_2(G) = 0$. In particular, this is true of the binary icosahedral group (a double cover of $A_5$), which is the fundamental group of the Poincare sphere.

Proof: $H_1(G) = G^{ab} = H_1(X) = 0$. That $H_2(G) = 0$ follows immediately from the above exact sequence, because $H_2(\pi_1(X))$ is a quotient of $H_2(X)$.

Two notes.

1) I don't know an easier or different way at all to prove this fact.

2) This property actually characterizes fundamental groups of homology spheres. It is an old theorem of Kervaire that if $G$ is a finitely presented group with trivial $H_1$ and $H_2$, then for fixed $n > 4$, you can find a homology $n$-sphere with fundamental group $G$. $n=3,4$ are (as usual!) something of a mystery.

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    $\begingroup$ I like the idea that that the notion of groupoid adds a spatial context to the notion of a group, through the set of objects (vertices) of a groupoid. See this exploited in group theory in the book "Categories and Groupoids" by Philip Higgins downloadable from tac.mta.ca/tac/reprints/articles/7/tr7abs.html There is also a strong case for using multipointed spaces in algebraic topology, particularly in connection with van Kampen's theorem. $\endgroup$ – Ronnie Brown Dec 17 '15 at 21:22
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    $\begingroup$ @RonnieBrown: I'm a little bit confused what this has to do with my answer. $\endgroup$ – user98602 Dec 17 '15 at 21:23
  • $\begingroup$ I probably should have put my comment immediately after the question, as a general point. Apologies! $\endgroup$ – Ronnie Brown Dec 17 '15 at 21:27
  • $\begingroup$ Thanks, that's an odd and interesting result. There's a remark in Kervaire's paper about the binary icosahedral group $X$ being the only nontrivial finite group with $H_1 X$ and a balanced presentation. I remember doing some stuff with the latter topic a while ago; I should look into why that's the case. $\endgroup$ – anomaly Dec 18 '15 at 6:10
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Here's a completely algebraic question: what do the subgroups of a free product $G_1 \ast G_2 \ast \dots \ast G_n$ look like?

The Kurosh subgroup theorem implies, among other things, that they are all free products of copies of $\mathbb{Z}$ and of subgroups of each of the $G_i$. This generalizes the Nielsen-Schreier theorem, which you get when each of the $G_i$ is $\mathbb{Z}$. The idea of the topological proof is to convert the question into a question about covering spaces of a wedge sum of spaces with fundamental groups $G_1, G_2, \dots G_n$, and it turns out to be possible to describe these covering spaces as generalized graphs in some sense (which can be made precise e.g. using Bass-Serre theory). The copies of $\mathbb{Z}$ show up when these graphs have loops in them.

For example, the subgroups of the modular group $\Gamma \cong PSL_2(\mathbb{Z}) \cong \mathbb{Z}_2 \ast \mathbb{Z}_3$ are all free products of copies of $\mathbb{Z}, \mathbb{Z}_2$, and $\mathbb{Z}_3$. In fact you can even make precise claims about which groups of this form can appear with which indices using the notion of virtual or orbifold Euler characteristic; see, for example, this blog post for some details (and pictures).

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  • $\begingroup$ Niftyy, thanks. I'm familiar with the Kurosh subgroup problem; my background is partially in geometric group theory, so I've at least seen a variety of questions about the structure of free groups, amalgamated free products, HNN-extensions, etc. Very nifty blog post. $\endgroup$ – anomaly Dec 18 '15 at 6:06
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The book Trees, by one of the greatest mathematicians of our time, Jean-Pierre Serre, demonstrates how topology and group theory are well-connected (apologies for the pun). Indeed, Schreier's Theorem, HNN-extensions etc. are viewed here from a topological point of view.

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  • $\begingroup$ Thanks, I've read it; that was the source for my references to Nielsen-Schrier, HNN-extensions, etc. above. $\endgroup$ – anomaly Dec 17 '15 at 21:50
  • $\begingroup$ @Aritra Das - thanks!! $\endgroup$ – Nicky Hekster Dec 21 '15 at 21:21
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Let's say you are given two finitely generated groups $G,H$ and you want to prove that there do not exist isomorphic finite index subgroups $A < G$, $B < H$; the terminology is that you are trying to prove $G,H$ are not commensurable.

It often turns out to be easier to prove a stronger statement, namely that $G,H$ are not quasi-isometric with respect to their word metrics. The reason this can be easier is that there are many interesting and easily applied quasi-isometry invariants, and you simply have to pick the right one.

For just one example, if $G$ is the fundamental group of a closed hyperbolic surface and $H$ is a free group then it comes down to proving that no finite index subgroup of $G$ is free: the Gromov boundary of every finite index subgroup of $G$ is a circle whereas the Gromov boundary of a free group is a Cantor set, and a circle is not homeomorphic to a Cantor set.

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Let $G$ be a finite group. Consider the following question: how many homomorphisms $\pi_1(\Sigma_g) \to G$ are there, where $\Sigma_g$ is the closed oriented surface of genus $g$? Using a presentation of $\pi_1(\Sigma_g)$, this is a purely group-theoretic question. It admits a representation-theoretic answer: we have

$$\frac{|\text{Hom}(\pi_1(\Sigma_g), G)|}{|G|} = \sum_V \left( \frac{\dim V}{|G|} \right)^{\chi(\Sigma_g)}$$

where $\chi(\Sigma_g) = 2 - 2g$ is the Euler characteristic, and the sum runs over all complex irreducible representations of $G$; this is Mednykh's formula.

Classically this result was proven using character theory, but the appearance of the Euler characteristic suggests, correctly, that this result might admit a topological proof, and indeed it does, using the machinery of topological field theory, and more precisely using Dijkgraaf-Witten theory.

The above number turns out to be the value that Dijkgraaf-Witten theory returns when fed the surface $\Sigma_g$, and it can be interpreted as the groupoid cardinality of the groupoid of principal $G$-bundles on $\Sigma_g$. The machinery of TFT provides a way to compute this number by breaking up $\Sigma_g$ into a bunch of surfaces with boundary; the proof is explained (poorly and without references or pictures) here. See also Noah Snyder's take.

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  • $\begingroup$ What a spiffy idea. Thanks, this is exactly the sort of thing I'm looking for, though my knowledge of TQFT consists of one visiting guest lecture when I was in grad school. I should look into that in more detail. $\endgroup$ – anomaly Dec 18 '15 at 6:15
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Symmetries of Things by Conway et al. gives a great example.

A significant portion of the book is devoted to tilings, one goal being to find all groups that arise as the symmetry group of some tiling of some particular surface. Surfaces include

Anyway, the authors use a series of related "Magic Theorems" that essentially use orbifolds to associate costs to features of a fundamental domain. The magic theorems, proven by topological means, assert that each kind of symmetry must have a fixed total cost, allowing us to classify all possible symmetry groups with basic combinatorial reasoning.

I'm not sure it's any shorter or more efficient than classifying these groups by purely algebraic means, as it's the only proof I've seen of these classifications! It's definitely pretty slick.

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Intuition. For me one of the greatest benefits of studying topology for group theory is that you get a lot of intuition and visualizations of how things work in a group. I know this is not an exact answer, as intuition cannot be formulated as a theorem, but a lot of theorems follow by this intuition.

So there are multiple objects which are the major keywords for intuition: graphs of groups, Cayley graphs, and fundamental groups of nice spaces. Graphs of groups allow us to quickly realize groups as fundamental groups of topological spaces. And as a fundamental group you can see the elements of the group behaving under the group multiplication. Another way to see the elements operating is by studying the Cayley graph. Different topological (or combinatorial) properties will give you the meaning of group theoretic terms and behaviour of group multiplication. Also the fundamental groups of nice spaces can come in many forms (very often as graphs of groups) for example in (short) exact sequences (fibrations (in particular covering spaces), maps with sections...). This semester I am tutoring an Algebra course and I see myself wanting to talk about topology all the time. For example how would you give nice intuitive examples on semi-direct products? (I know, there are some nice purely algebraic examples too).

To make this answer an actual answer I want to mention Stallings theorem on the ends of a finitely generated group $G$: namely $ends(G)>1 \Leftrightarrow G$ is obtained as a non-trivial graph of groups with one edge. By non-trivial I mean that all the vertex groups are non-trivial and the edge group can be chosen finite. This theorem has some nice group theoretic applications such as finite extensions of free groups $F_{s{<\infty}}$ are HNN extensions (Karrass, Pietrowski, Solitar). There are plenty of more results and applications of this theorem for example on the wikipedia page.

Other keywords which come to my mind which I can basically only (or much better) grasp topologically: Group cohomology, Whitehead groups (I am not sure how important they are just algebraically without surgery), and (automorphisms) of free groups. A large part of what we know about the latter is known by using topological methods.

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  • $\begingroup$ I'm only familiar with Whitehead groups in the context of surgery theory (though there are a lot of interesting and purely algebraic questions one can ask about $\mathbb{Z}[X]$ and $\mathbb{Z}[X]/\{\pm X\}$ for reasonable $X$). In particular, I've seen the results but not the proof of the computation of $\text{Wh}(X)$ for $X$ finitely generated and abelian, and I don't know whether they're algebraic (via, say, algebraic $K$-theory) or topological (via surgery) in nature. $\endgroup$ – anomaly Dec 18 '15 at 15:43
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One influence of geometry and topology on group theory has been to extend group theoretic methods. Philip Higgins described to me how he first thought of using groupoids by reading about covering spaces in the book on Homology Theory by Hilton and Wylie, and realising that the account of covering spaces was all about groupoids. You can read about the applications he found in the 1971 book Categories and Groupoids (downloadable); this gives groupoid proofs of subgroup theorems mentioned on this page. Note that groupoids allow notions of fibration and covering morphisms, modelling topological notions.

An application of higher groupoids to homotopy theory which Loday and I published in 1984 led to a nonabelian tensor product of groups which act on each other in a "compatible" way. A simple example is two normal subgroups $M,N$ of a group $P$. The commutator map $[\,, \,]:M \times N \to P, (m,n) \mapsto mnm^{-1}n^{-1}$ has properties of $[mm',n], [m,nn']$, which make it, not bimultiplicative, but a biderivation. The universal biderivation is written $M \times N \to M \otimes N$, so the commutator map factors through a morphism of groups $\kappa: M \otimes N \to P$. Graham Ellis proved that $M \otimes N $ is finite if $M,N$ are finite.

There are applications of this tensor product to the homology of groups. For example if $1 \to R \to F \to P\to 1$ is an exact sequence with $F$ free, then $$H_3(P) \cong \text{Ker} (R \wedge F \to P)$$ where $R \wedge F$ is the quotient of $R \otimes F$ by the relations $r \otimes r =1$ for $r \in R$; and to algebraic topology: for example $$\pi_3S(K(P,1))\cong \text{Ker} (\kappa: P \otimes P \to P),$$ where $S$ is suspension. A number of group theorists have taken up the ideas, particularly on calculating $P \otimes P$ for classes of groups, see this bibliography, which has over 170 items, including a description dating from 1952 of $H_2(P)$ essentially as the kernel of $P \wedge P \to P$.

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  • $\begingroup$ That's a great remark about nonabelian tensor products, which I hadn't run across before. Thanks! $\endgroup$ – anomaly Dec 18 '15 at 19:25

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