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July 27, 2016

Topological Crystals (Part 2)

Posted by John Baez

k4_crystal

We’re building crystals, like diamonds, purely from topology. Last time I said how: you take a graph XX and embed its maximal abelian cover into the vector space H 1(X,)H_1(X,\mathbb{R}).

Now let me back up and say a bit more about the maximal abelian cover. It’s not nearly as famous as the universal cover, but it’s very nice.

The basic idea

By ‘space’ let me mean a connected topological space that’s locally nice. The basic idea is that if XX is some space, its universal cover X˜\widetilde{X} is a covering space of XX that covers all other covering spaces of XX. The maximal abelian cover X¯\overline{X} has a similar universal property — but it’s abelian, and it covers all abelian connected covers. A cover is abelian if its group of deck transformations is abelian.

The cool part is that universal covers are to homotopy theory as maximal abelian covers are to homology theory.

What do I mean by that? For starters, points in X˜\widetilde{X} are just homotopy classes of paths in XX starting at some chosen basepoint. And the points in X¯\overline{X} are just ‘homology classes’ of paths starting at the basepoint.

But people don’t talk so much about ‘homology classes’ of paths. So what do I mean by that? Here a bit of category theory comes in handy. Homotopy classes of paths in XX are morphisms in the fundamental groupoid of XX. Homology classes of paths are morphisms in the abelianized fundamental groupoid!

But wait a minute — what does that mean? Well, we can abelianize any groupoid by imposing the relations

fg=gf f g = g f

whenever it makes sense to do so. It makes sense to do so when you can compose the morphisms f:xyf : x \to y and g:xyg : x' \to y' in either order, and the resulting morphisms fgf g and gfg f have the same source and the same target. And if you work out what that means, you’ll see it means x=y=x=yx = y = x' = y'.

But now let me say it all much more slowly, for people who want a more relaxed treatment. After all, this is a nice little bit of topology that could be in an elementary course!

The details

There are lots of slightly different things called ‘graphs’ in mathematics, but in topological crystallography it’s convenient to work with one that you’ve probably never seen before. This kind of graph has two copies of each edge, one pointing in each direction.

So, we’ll say a graph X=(E,V,s,t,i)X = (E,V,s,t,i) has a set VV of vertices, a set EE of edges, maps s,t:EVs,t : E \to V assigning to each edge its source and target, and a map i:EEi : E \to E sending each edge to its inverse, obeying

s(i(e))=t(e),t(i(e))=s(e),i(i(e))=e s(i(e)) = t(e), \quad t(i(e)) = s(e) , \qquad i(i(e)) = e

and

i(e)e i(e) \ne e

for all eEe \in E.

That inequality at the end will make category theorists gag: definitions should say what’s true, not what’s not true. But category theorists should be able to see what’s really going on here, so I leave that as a puzzle.

For ordinary folks, let me repeat the definition using more words. If s(e)=vs(e) = v and t(e)=wt(e) = w we write e:vwe : v \to w, and draw ee as an interval with an arrow on it pointing from vv to ww. We write i(e)i(e) as e 1e^{-1}, and draw e 1e^{-1} as the same interval as ee, but with its arrow reversed. The equations obeyed by ii say that taking the inverse of e:vwe : v \to w gives an edge e 1:wve^{-1} : w \to v and that (e 1) 1=e(e^{-1})^{-1} = e. No edge can be its own inverse.

A map of graphs, say f:XXf : X \to X', is a pair of functions, one sending vertices to vertices and one sending edges to edges, that preserve the source, target and inverse maps. By abuse of notation we call both of these functions ff.

I started out talking about topology; now I’m treating graphs very combinatorially, but we can bring the topology back in.

From a graph XX we can build a topological space |X||X| called its geometric realization. We do this by taking one point for each vertex and gluing on one copy of [0,1][0,1] for each edge e:vwe : v \to w, gluing the point 00 to vv and the point 11 to ww, and then identifying the interval for each edge ee with the interval for its inverse by means of the map t1tt \mapsto 1 - t. Any map of graphs gives rise to a continuous map between their geometric realizations, and we say a map of graphs is a cover if this continuous map is a covering map. For simplicity we denote the fundamental group of |X||X| by π 1(X)\pi_1(X), and similarly for other topological invariants of |X||X|. However, sometimes I’ll need to distinguish between a graph XX and its geometric realization |X||X|.

Any connected graph XX has a universal cover, meaning a connected cover

p:X˜X p : \widetilde{X} \to X

that covers every other connected cover. The geometric realization of X˜\widetilde{X} is connected and simply connected. The fundamental group π 1(X)\pi_1(X) acts as deck transformations of X˜\widetilde{X}, meaning invertible maps g:X˜X˜g : \widetilde{X} \to \widetilde{X} such that pg=pp \circ g = p. We can take the quotient of X˜\widetilde{X} by the action of any subgroup Gπ 1(X)G \subseteq \pi_1(X) and get a cover q:X˜/GX q : \widetilde{X}/G \to X.

In particular, if we take GG to be the commutator subgroup of π 1(X)\pi_1(X), we call the graph X˜/G\widetilde{X}/G the maximal abelian cover of the graph XX, and denote it by X¯\overline{X}. We obtain a cover

q:X¯X q : \overline{X} \to X

whose group of deck transformations is the abelianization of π 1(X)\pi_1(X). This is just the first homology group H 1(X,)H_1(X,\mathbb{Z}). In particular, if the space corresponding to XX has nn holes, this is a free abelian group on nn generators.

I want a concrete description of the maximal abelian cover! I’ll build it starting with the universal cover, but first we need some preliminaries on paths in graphs.

Given vertices x,yx,y in XX, define a path from xx to yy to be a word of edges γ=e 1e \gamma = e_1 \cdots e_\ell with e i:v i1v ie_i : v_{i-1} \to v_i for some vertices v 0,,v v_0, \dots, v_\ell with v 0=xv_0 = x and v =yv_\ell = y. We allow the word to be empty if and only if x=yx = y; this gives the trivial path from xx to itself. Given a path γ\gamma from xx to yy we write γ:xy\gamma : x \to y, and we write the trivial path from xx to itself as 1 x:xx1_x : x \to x. We define the composite of paths γ:xy \gamma : x \to y and δ:yz \delta : y \to z via concatenation of words, obtaining a path we call γδ:xz\gamma \delta : x \to z. We call a path from a vertex xx to itself a loop based at xx.

We say two paths from xx to yy are homotopic if one can be obtained from the other by repeatedly introducing or deleting subwords of the form e ie i+1e_i e_{i+1} where e i+1=e i 1e_{i+1} = e_i^{-1}. If [γ][\gamma] is a homotopy class of paths from xx to yy, we write [γ]:xy [\gamma] : x \to y. We can compose homotopy classes [γ]:xy [\gamma] : x \to y and [δ]:yz[\delta] : y \to z by setting [γ][δ]=[γδ] [\gamma] [\delta] = [\gamma \delta].

If XX is a connected graph, we can describe the universal cover X˜\widetilde{X} as follows. Fix a vertex x 0x_0 of XX, which we call the basepoint. The vertices of X˜\widetilde{X} are defined to be the homotopy classes of paths [γ]:x 0x [\gamma] : x_0 \to x where xx is arbitrary. The edges in X˜\widetilde{X} from the vertex [γ]:x 0x[\gamma] : x_0 \to x to the vertex [δ]:x 0y[\delta] : x_0 \to y are defined to be the edges eEe \in E with [γe]=[δ][\gamma e] = [\delta]. In fact, there is always at most one such edge. There is an obvious map of graphs

p:X˜X p : \widetilde{X} \to X

sending each vertex [γ]:x 0x [\gamma] : x_0 \to x of X˜\widetilde{X} to the vertex xx of XX. This map is a cover.

Now we are ready to construct the maximal abelian cover X¯\overline{X}. For this, we impose a further equivalence relation on paths, which is designed to make composition commutative whenever possible. However, we need to be careful. If γ:xy \gamma : x \to y and δ:xy \delta : x' \to y' , the composites γδ \gamma \delta and δγ\delta \gamma are both well-defined if and only if x=yx' = y and y=xy' = x. In this case, γδ\gamma \delta and δγ\delta \gamma share the same starting point and share the same ending point if and only if x=xx = x' and y=yy = y'. If all four of these equations hold, both γ\gamma and δ\delta are loops based at xx. So, we shall impose the relation γδ=δγ\gamma \delta = \delta \gamma only in this case.

We say two paths are homologous if one can be obtained from another by:

  • repeatedly introducing or deleting subwords e ie i+1e_i e_{i+1} where e i+1=e i 1e_{i+1} = e_i^{-1}, and/or

  • repeatedly replacing subwords of the form e ie je j+1e ke_i \cdots e_j e_{j+1} \cdots e_k by those of the form e j+1e ke ie je_{j+1} \cdots e_k e_i \cdots e_j , where e ie j e_i \cdots e_j and e j+1e ke_{j+1} \cdots e_k are loops based at the same vertex.

My use of the term ‘homologous’ is a bit nonstandard here!

We denote the homology class of a path γ\gamma by [[γ]][[ \gamma ]]. Note that if two paths γ:xy\gamma : x \to y, δ:xy\delta : x' \to y' are homologous then x=xx = x' and y=yy = y'. Thus, the starting and ending points of a homology class of paths are well-defined, and given any path γ:xy \gamma : x \to y we write [[γ]]:xy [[ \gamma ]] : x \to y . The composite of homology classes is also well-defined if we set [[γ]][[δ]]=[[γδ]] [[ \gamma ]] [[ \delta ]] = [[ \gamma \delta ]].

We construct the maximal abelian cover of a connected graph XX just as we constructed its universal cover, but using homology classes rather than homotopy classes of paths. And now I’ll introduce some jargon that should make you start thinking about crystals!

Fix a basepoint x 0x_0 for XX. The vertices of X¯\overline{X}, or atoms, are defined to be the homology classes of paths [[γ]]:x 0x [[\gamma]] : x_0 \to x where xx is arbitrary. Any edge of X¯\overline{X}, or bond, goes from some atom [[γ]]:x 0x[[ \gamma]] : x_0 \to x to the some atom [[δ]]:x 0y[[ \delta ]] : x_0 \to y. The bonds from [[γ]][[ \gamma]] to [[δ]][[ \delta ]] are defined to be the edges eEe \in E with [[γe]]=[[δ]][[ \gamma e ]] = [[ \delta ]]. There is at most one bond between any two atoms. Again we have a covering map

q:X¯X q : \overline{X} \to X

The homotopy classes of loops based at x 0x_0 form a group, with composition as the group operation. This is the fundamental group π 1(X)\pi_1(X) of the graph XX. (It depends on the basepoint x 0x_0, but I’ll leave that out out of the notation just to scandalize my colleagues. It’s so easy to live dangerously when you’re an academic!)

Now, this fundamental group is isomorphic to the usual fundamental group of the space associated to XX. By our construction of the universal cover, π 1(X)\pi_1(X) is also the set of vertices of X˜\widetilde{X} that are mapped to x 0x_0 by pp. Furthermore, any element [γ]π 1(X) [\gamma] \in \pi_1(X) defines a deck transformation of X˜\widetilde{X} that sends each vertex [δ]:x 0x [\delta] : x_0 \to x to the vertex [γ][δ]:x 0x [\gamma] [\delta] : x_0 \to x.

Similarly, the homology classes of loops based at x 0x_0 form a group with composition as the group operation. Since the additional relation used to define homology classes is precisely that needed to make composition of homology classes of loops commutative, this group is the abelianization of π 1(X)\pi_1(X). It is therefore isomorphic to the first homology group H 1(X,)H_1(X,\mathbb{Z}) of the geometric realization of XX!

By our construction of the maximal abelian cover, H 1(X,)H_1(X,\mathbb{Z}) is also the set of vertices of X¯\overline{X} that are mapped to x 0x_0 by qq. Furthermore, any element [[γ]]H 1(X,) [[\gamma]] \in H_1(X,\mathbb{Z}) defines a deck transformation of X¯\overline{X} that sends each vertex [[δ]]:x 0x [[\delta]] : x_0 \to x to the vertex [[γ]][[δ]]:x 0x [[\gamma]] [[\delta]] : x_0 \to x.

So, it all works out! The fundamental group π 1(X)\pi_1(X) acts as deck transformations of the universal cover, while the first homology group H 1(X,)H_1(X,\mathbb{Z}) acts as deck transformations of the maximal abelian cover!

Puzzle for experts: what does this remind you of in Galois theory?

We’ll get back to crystals next time.

Posted at July 27, 2016 10:30 AM UTC

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6 Comments & 0 Trackbacks

Re: Topological Crystals (Part 2)

I guess by Hurewicz that something like this will works only for the first nonzero homotopy group. Do you have a version for other homology groups? Abelianization isn’t going to be obviously helpful any more.

(Stupid, basic question: is ordinary homology representable? I know ordinary cohomology is representable, by Eilenberg-Mac Lane spaces, so I’m sure I should know this.)

Posted by: Allen Knutson on July 27, 2016 12:46 PM | Permalink | Reply to this

Re: Topological Crystals (Part 2)

Homology isn’t representable: it’s covariant but doesn’t preserve products.

Posted by: Omar Antolín Camarena on August 12, 2016 8:47 PM | Permalink | Reply to this

Re: Topological Crystals (Part 2)

Puzzle for experts: what does this remind you of in Galois theory?

Presumably you’re looking for the maximal abelian extension of a number field, something we used to talk about around here.

Posted by: David Corfield on August 10, 2016 8:59 PM | Permalink | Reply to this

Re: Topological Crystals (Part 2)

Right! Just as the Galois group of the algebraic closure k¯\overline{k} over a number field kk is like a fundamental group, and k¯\overline{k} is like a universal cover, its abelianization, namely the Galois group of the maximal abelian extension k abk^{ab} over kk, is like a first homology group — and k abk^{ab} is like a maximal abelian cover.

The abelianized situation can be handled using homology theory (‘class field theory’), so it’s a lot more manageable.

Quoth Wikipedia:

For example, the abelianized absolute Galois group of \mathbb{Q} is (naturally isomorphic to) an infinite product of the group of units of the pp-adic integers taken over all prime numbers p, and the corresponding maximal abelian extension of the rationals is the field generated by all roots of unity. This is known as the Kronecker–Weber theorem, originally conjectured by Leopold Kronecker.

Posted by: John Baez on August 11, 2016 6:47 AM | Permalink | Reply to this

Re: Topological Crystals (Part 2)

That inequality at the end will make category theorists gag: definitions should say what’s true, not what’s not true. But category theorists should be able to see what’s really going on here, so I leave that as a puzzle.

I’d be interested in hearing the answer to this one, which I couldn’t figure out.

Posted by: Omar Antolín Camarena on August 19, 2016 9:49 PM | Permalink | Reply to this

Re: Topological Crystals (Part 2)

Category theorists love the kind of graph that consists of a set VV of vertices, a set EE of edges, and maps s,t:EVs,t: E \to V. These graphs are objects of a topos, you can define them internal to any category, etc.

So let’s call these simply ‘graphs’, and call the category of these GrphGrph. Next we can talk about graphs equipped with a map i:EEi: E \to E that switches source and target:

is=t,it=s,i 2=1 i s = t, \quad i t = s, \quad i^2 = 1

Let’s call these ‘symmetric graphs’, and call the obvious category of these SymGrphSymGrph.

There’s a forgetful functor

U:SymGrphGrph U : SymGrph \to Grph

and it has a left adjoint

F:GrphSymGrph F : Grph \to SymGrph

This left adjoint takes a graph and for each edge ee from vv to ww creates a new edge, its inverse i(e)i(e), from ww to vv. Because it does so freely, we have i(e)ei(e) \ne e.

This is the sort of graph Sunada uses: symmetric graphs in the image of the functor FF. So, they are really ordinary graphs that have been promoted to symmetric graphs!

All the annoying nuances here concern ‘self-loops’: edges with s(e)=t(e)s(e) = t(e). This is the only kind of edge that has the potential of being its own inverse.

Posted by: John Baez on August 20, 2016 4:01 AM | Permalink | Reply to this

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