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December 9, 2009

This Week’s Finds in Mathematical Physics (Week 286)

Posted by John Baez

In week286 of This Week’s Finds, begin learning about rational homotopy theory! See how to build a space whose fundamental group is the rational numbers:

Learn how to ‘rationalize’ a space. And see three descriptions of the rational homotopy category.

Also, guess what this is a picture of:

Posted at December 9, 2009 5:03 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2131

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Re: This Week’s Finds in Mathematical Physics (Week 286)

Dune sea on Mars?

Posted by: John Armstrong on December 9, 2009 5:36 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

A lot of work on rational homotopy theory sidesteps this issue by working only with "1-connected" spaces. These are spaces that are connected and also path-connected. That means the fundamental group is trivial

Don't you mean that they are both path-connected and simply connected … which a lot of people would just call ‘simply connected’? That fits with my understanding of what it means to be nn-connected, and (more low-brow) with my understanding of what makes a fundamental group trivial.

Posted by: Toby Bartels on December 9, 2009 6:01 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

Fixed. Yes, that’s what I meant.

Posted by: John Baez on December 9, 2009 6:13 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

Where you show that 1gG1 \otimes g \in \mathbb{Q} \otimes G vanishes if gg is nn-torsion, there seems to be a factor 1n\frac{1}{n} missing, somewhere. 1g=1nng=1n0=01 \otimes g = \frac{1}{n} \otimes n g = \frac{1}{n} \otimes 0 = 0.

Posted by: Urs Schreiber on December 9, 2009 8:29 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

Is there an adjunction somewhere between rationalization and inclusion of rational spaces?

I’d think about this myself but have a 350 page French thesis to read.

Posted by: David Corfield on December 9, 2009 8:45 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

I will comment a little on rationalizing a non-abelian group as I used that in my thesis (see Chapters 4 and 5). The reason I was going that is because I was trying to find expressions for braid invariants coming from Vassiliev theory, and Chen’s itereated integrals was the way to do that.

Anyway suppose G is a finitely generated group (I don’t know how necessary that is). The idea is to approximate G by a sequence (or tower) of abelian extensions (or better, free abelian extensions) and then rationalize (or in my case realise) these extensions.

I’ll start with something that might be more familiar to some people. You have the lower central series

G=Γ 0Γ 1Γ iG=\Gamma_0 \rhd \Gamma_1 \rhd \dots\rhd\Gamma_i \rhd\dots where Γ i+1[Γ i,G]\Gamma_{i+1} \coloneqq [\Gamma_i ,G]

is the group generated by the commutators of elements of Γ i\Gamma_i with elements of GG. Then the successive quotients Γ iG/Γ i+1G\Gamma_i G/\Gamma_{i+1}G are abelian. We can do better than that in that we can find a sequence which has free abelian quotients. We form the rational closure of the each term of the lower central series:

Δ i{xG|x aΓ iforsomea}.\Delta_i\coloneqq \{x\in G | x^a\in \Gamma_i for some a\in \mathbb{N}\}.

This gives a sequence

G=Δ 0Δ 1Δ iG=\Delta_0 \rhd \Delta_1 \rhd \dots\rhd\Delta_i \rhd\dots

where the successive quotients Δ i/Δ i+1\Delta_i/\Delta_{i+1} are free abelian, so we have a collection of free abelian extensions

Δ i/Δ i+1G/Δ i+1G/Δ i,\Delta_i /\Delta_{i+1}\to G/\Delta_{i+1}\to G/\Delta_{i} ,

and thus a tower of approximations to GG built by free abelian extensions.

G/Δ 3 G/Δ 2 G/Δ 1 {1}\array{ \vdots\\\downarrow\\ G/\Delta_3 \\\downarrow\\ G/\Delta_2 \\ \downarrow \\G/\Delta_1\\ \downarrow\\ \{1\} }

We can now inductively rationalize this tower. As Δ i/Δ i+1\Delta_{i}/\Delta_{i+1} is torsion-free abelian we know what (Δ i/Δ i+1)(\Delta_{i}/\Delta_{i+1})\otimes \mathbb{Q} is, and given (G/Δ i)(G/\Delta_i) \otimes \mathbb{Q} we can form (G/Δ i+1)(G/\Delta_{i+1}) \otimes \mathbb{Q} by using the extension

Δ i/Δ i+1G/Δ i+1G/Δ i,\Delta_i /\Delta_{i+1}\to G/\Delta_{i+1}\to G/\Delta_{i} ,

to form an extension

(Δ i/Δ i+1)(G/Δ i+1)(G/Δ i).(\Delta_i /\Delta_{i+1}) \otimes \mathbb{Q}\to (G/\Delta_{i+1}) \otimes \mathbb{Q}\to (G/\Delta_{i}) \otimes \mathbb{Q}.

We can then define GG\otimes \mathbb{Q} to be the limit lim(G/Δ i)\lim (G/\Delta_i) \otimes \mathbb{Q}. This is called the Malcev completion of GG.

Posted by: Simon Willerton on December 9, 2009 10:14 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

Cool! Are the groups G/Δ iG/\Delta_i in this tower of approximations nilpotent?

Posted by: John Baez on December 9, 2009 3:15 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

John exclaimed and then said

Are the groups G/Δ iG/\Delta_i in this tower of approximations nilpotent?

Yes. Firstly, G/Δ 1G/\Delta_1 is free abelian (it’s the abelianization of G modulo torsion, I guess), then you take a central extension of that to get G/Δ 2G/\Delta_2 which is then 22-step nilpotent, etc. etc. If GG is actually nilpotent, then the Δ i\Delta_is eventually stabilize to the torsion part of GG (which is a subgroup when GG is nilpotent, I hope!) so in that case the tower of groups eventually stabilizes at G/torsionG/torsion.

One of the simplest groups to consider is the Heisenberg group of integer upper-triangular matrices with ones on the diagonal. It’s an easy exercise to see what happens in that case.

I must confess I’m very rusty at this as I haven’t thought about it for a very long time. There’s some really nice geometric things going on here as well – you can build a corresponding tower of nilpotent spaces very explicitly. I’ll try to remember this at some point.

Posted by: Simon Willerton on December 9, 2009 11:40 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

Simon wrote: I was trying to find expressions for braid invariants coming from Vassiliev theory, and Chen’s itereated integrals was the way to do that.

did you succeed?

Posted by: jim stasheff on December 9, 2009 3:53 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

Here is a lazy question that I’m embarrassed to be confused about, but I’ve been too lazy to deconfuse myself for years.

Let XX be a topological space and AA the algebra of Sullivan differential forms on XX. Now:

-If XX is 1-connected, then the minimal model MM of AA can be used to compute the higher rational homotopy groups of XX in that M +/(M +) 2=Hom(π *(X),Q),M^+/(M^+)^2 =Hom(\pi_*(X),Q), where M +M^+ is the positively graded portion of MM.

-On the other hand, even if XX is not 1-connected, the 1-minimal model of M(1)M(1) of AA can be used for the Malcev completion of π 1(X)\pi_1(X), in that

M(1) +/(M(1) +) 2=Hom(Lie[π 1(X)Q],Q)M(1)^+/(M(1)^+)^2 =Hom(Lie[\pi_1(X)\otimes Q],Q)

The point I was never able to clarify to myself is:

In general, that is, for XX not 1-connected, does MM give you the QQ-nilpotent completion of the whole π *(X)\pi_*(X)?

One could ask the analogous question for the bar complex B(A)B(A). Is H *(B(A))H^*(B(A)) the QQ-nilpotent completion of π *(X)\pi_*(X)?

Posted by: Minhyong Kim on December 9, 2009 11:17 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

This makes it really easy to spot a mathematician in a roomful of pirates.

Arrr, ye’ve got me!

Posted by: Allen Knutson on December 9, 2009 2:41 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

Heh. I told that joke because when I was looking for a photo of a hand telescope, I typed in ‘pirate telescope’ and discovered that there are plenty of places that will sell you a pirate telescope.

So clearly all pirates, even modern Somali ones, must own pirate telescopes — emblazoned with a skull and crossbones to prove that they’re really pirates. Whereas mathematicians do not!

Posted by: John Baez on December 11, 2009 2:00 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

Something that bothered me for a while about rational homotopy, as an outsider, was this phrase “the homotopy groups are rational vector spaces”. A priori the (higher) homotopy groups are abelian groups. So does this mean that there exists a rational vector space structure? That there exists a unique one? That one is somehow specified?

In fact, these questions are unnecessary, for the following reason. (I think this was explained to me by the James who sometimes comments here.) Fact:

Let AA be an abelian group. Then AA has the structure of a rational vector space in at most one way.

So, despite appearances, being a rational vector space is a property of abelian groups, not extra structure.

The proof is fairly straightforward, I think. If AA admits a rational vector space structure then

for all aAa \in A and all positive integers nn, there exists a unique bAb \in A such that nb=anb = a.

And this condition clearly determines what the scalar multiplication over \mathbb{Q} must be. (In fact, it’s an ‘iff’: an abelian group admits the structure of a rational vector space if and only if it satisfies this condition.)

Posted by: Tom Leinster on December 9, 2009 4:33 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

Yes. A rational vector space is the same as a divisible torsionfree abelian group. Incidentally, an abelian group is divisible if and only if it is injective in the category of abelian groups, and is torsionfree if and only if it is flat in the category of abelian groups.

Posted by: Todd Trimble on December 9, 2009 4:51 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

Another way to look at this is that the map

\mathbb{Z} \hookrightarrow \mathbb{Q}

is epi in Rings\mathbf{Rings}. A rational structure on GG is a ring map [G,G]\mathbb{Q} \to [G,G], automatically under \mathbb{Z} since \mathbb{Z} is the initial ring; so the epi-ness says that there’s at most one such map.

So in Rings op\mathbf{Rings}^\mathrm{op}, the category of affine schemes (spaces, of a sort), Spec()\Spec(\mathbb{Q}) is a subobject of the terminal object! It’s a truth-value! :-)

…so (sorry, digression), I wish I had a better intuition for categories of schemes, and in particular the rôle played be spectra of fields! I’ve seen just enough to get an intuition of how rich and subtle it can be, but not enough to understand the subtleties much beyond that. For example: Spec()\Spec(\mathbb{Z}) is the terminal object; but it’s in many ways very far from being a point. Conversely, spectra of fields are “points” — certainly in terms of their underlying topological spaces, but they seem to function geometrically as points as well — but in general, they’re not necessarily subterminal! What’s going on? (Besides my parochial expectation from Set\mathbf{Set}-like categories that “terminal object” and “point” should be similar ideas.)

Seriously, I know a full answer to this question would essentially be EGA together with most algebraic geometry since then, but I’d be very interested to hear what others around here (some of whom certainly know much more AG than I do) can suggest in the way of pithy insights.

Posted by: Peter LeFanu Lumsdaine on December 10, 2009 1:06 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

This is something I’d like to understand better as well.

I know that the “points” of SpecSpec \mathbb{Z} (in the locally-ringed-space picture of schemes) are prime numbers, and that localizing or completing at a prime has something to do with looking at fibers over those points. Presumably rationalization has to do with pulling back from SpecSpec \mathbb{Z} to SpecSpec \mathbb{Q}. At least, those are the case for ordinary modules; can the homotopy-theory picture be described that way too, maybe in the (,1)(\infty,1)-topos of \infty-stacks on Rings opRings^{op}?

Of course, the spectra of some fields are subterminal, like \mathbb{Q} as you said, and also each 𝔽 p\mathbb{F}_p. But the spectra of other fields have automorphisms. However, there are other topoi in which there are “point-like things” which have automorphisms. For instance, in a presheaf topos, it makes some sense to think of the representables as “point-like:” they certainly have some point-like properties (indecomposability, projectivity, nonemptiness, and collectively generating the category). And in a topos of GG-sets, it makes sense to consider the orbits G/HG/H as “point-like objects,” which share some of the same properties. Equivariant algebraic topologists are quite used to thinking of GG-spaces as having several different sorts of “points,” which are represented by the orbits G/HG/H: a map GG-map G/HXG/H \to X is a point in the HH-fixed points X HX^H of XX.

That still doesn’t quite tell us what’s special about fields, but I think that this paper may have some insights about that.

Posted by: Mike Shulman on December 10, 2009 9:07 AM | Permalink | PGP Sig | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

Peter wrote

the map \mathbb{Z} \hookrightarrow \mathbb{Q} is epi in rings

thus proving that every abelian group has at most one rational vector space structure.

Ah yes! That’s a very nice way to look at it.

But now I have a question. The prime fields (those with no proper subfields) are precisely /p\mathbb{Z}/p\mathbb{Z}, for primes pp, and \mathbb{Q}. The unique ring homomorphisms /p\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z} and \mathbb{Z} \to \mathbb{Q} are all epi. So, a homomorphism from \mathbb{Z} to a prime field is always epi.

Is there some uniform proof of this?

It’s kind of a silly question because if you split it into two cases (the finite prime fields, and \mathbb{Q}) then it’s dead easy to prove. But I did wonder.

Posted by: Tom Leinster on December 11, 2009 12:53 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

You can look at it this way: Since Z is initial among rings, we’re really asking if any two parallel ring maps out of the prime field K are equal. This is true, since their equalizer is a subfield of K, hence all of K.

Posted by: Owen Biesel on December 11, 2009 2:05 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

Of course! Thanks.

(The only not-totally-trivial step there is seeing that the equalizer of a pair of ring homomorphisms out of a field, a priori only a subring, is in fact a subfield.)

Posted by: Tom Leinster on December 11, 2009 2:21 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

Something I am curious about is how “motivic” such constructions are, if mutually compatible realization functors exist which transfer structures between them, and how “anabelian” they are, how much such constructions say about rational points.

Posted by: Thomas on December 10, 2009 1:42 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

They are very motivic if you start from a variety, but in the usual weak sense of having a compatible collection of realizations corresponding to each cohomology theory. This is discussed at some length for the projective line minus three points in Deligne’s famous paper. You’ll see in the introduction there that he admits that rational higher homotopy or Malcev completions are very far from anabelian, because they’re so close to homology. For applications to Diophantine problems, you can look for example, at this paper.

Posted by: Minhyong Kim on December 10, 2009 9:29 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

Tom wrote:

Something that bothered me for a while about rational homotopy, as an outsider, was this phrase “the homotopy groups are rational vector spaces”. A priori the (higher) homotopy groups are abelian groups. So does this mean that there exists a rational vector space structure? That there exists a unique one? That one is somehow specified?

I think it’s because you’re a category theorist. Lately I’ve been explaining this stuff to various people at UCR, and whenever James Dolan overhears me say “the homotopy groups are rational vector spaces” he chimes in and says “and that’s just a property, right?”

Somehow I just took for granted that it was a property, even before I knew this property was “being a divisible torsion-free abelian group”, as Todd pointed out here.

Psychoanalyzing myself, I think it’s because I could easily see that any abelian group homomorphism between rational vector spaces is also a vector space map. So, the forgetful functor from rational vector spaces to abelian groups is full. And since it’s obviously faithful, this functor only forgets a property — not structure or stuff.

At least I think that’s why I was so nonchalant about saying that some abelian group ‘is’ a rational vector space.

Since James had sensitized me to this issue, I considered saying something like this at the start of week286:

First, XX' is a “rational space”: a 1-connected pointed space whose homotopy groups are rational vector spaces. (And in case you’re wondering, this is just a property of an abelian group.)

But I decided this would puzzle more people than it would de-puzzle!

But now I can add your remark to my ‘Addenda’. It is indeed important.

Posted by: John Baez on December 11, 2009 1:37 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

It's always seemed obviously a property to me, because if you can divide by 2, then there's only one way to do it. (There are more details to check, but that's the idea right there.)

Posted by: Toby Bartels on December 11, 2009 6:20 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

if you can divide by 2, then there’s only one way to do it.

Do you mean in an abelian group or in a rational vector space? If the former, that’s false. If the latter, how does it help?

Posted by: Tom Leinster on December 16, 2009 12:40 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

If the former, that’s false.

So it is! I guess that it's not that obvious then.

Posted by: Toby Bartels on December 16, 2009 12:56 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

In case n-cat cafe readers haven’t seen it. Ken Baker’s illustrations are totally awesome!

Posted by: Scott Carter on December 10, 2009 4:01 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

It’s funny how his initial mistake drawing them matched my initial mistake in describing them… which luckily was fixed by the time anyone saw what I wrote!

Posted by: John Baez on December 10, 2009 8:46 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

Kenneth Baker has allowed me to use some of his figures. Here’s an early stage of the construction of the rational circle:

The right edge of the red band is our original circle, drawn in a tricky way to make the whole picture more manageable. The left edge of the red band is homotopic to 2 times the loop traced out by this original circle. The left of the yellow band is homotopic to 6 times it, and the left edge of the green band is homotopic to 24 times it!

If we remove the red band we see how the yellow one wraps around it 3 times:

and if we remove the yellow band we see how the green one wraps around it 4 times:

Here’s a kind of cross-section that reveals more about what’s going on:

You’re probably curious about how Kenneth Baker drew these pictures. Here’s how:

These pictures are done using Rhino 3D. Actually I’m using the beta version of their port to OS X. There’s a function (called Flow) that lets you map a “spine” of an object to another curve to tell it how to deform the object. This is how I went from the chopped open version to the round one. It’s also how I managed to make the orange wrap around the green and the red wrap around the orange.

Posted by: John Baez on December 11, 2009 1:17 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

Wow, those Mandelbulb pictures are truly mindblowing!

Posted by: Bruce Bartlett on December 14, 2009 2:41 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

Wouldn’t that make a delicious cake? Or maybe it’s a speck of pollen?

Posted by: some guy on the street on December 14, 2009 10:48 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 286)

Or maybe it’s a speck of pollen?

That's exactly what I thought. Such good pictures, I'm lucky that I didn't sneeze! (^_^)

Posted by: Toby Bartels on December 15, 2009 12:06 AM | Permalink | Reply to this

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