May 19, 2008

This Week’s Finds in Mathematical Physics (Week 264)

Posted by John Baez

Then learn a wonderful description of the homotopy groups of the 2-sphere in terms of braids, and guess the significance this sequence:

$1, 1, 2, 3, 4, 5, 6, ...$

And if you can, help me with theta functions!

Posted at May 19, 2008 1:54 AM UTC

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Re: This Week’s Finds in Mathematical Physics (Week 264)

Does Berrick, Cohen, Wong and Wu’s theorem bring us anywhere nearer to demonstrating the Baez-Dolan ‘conjecture’ that somehow or other the homotopy cardinality of the 2-sphere, i.e., the alternating product of its homotopy groups, should equal its Euler characteristic, i.e., 2?

Posted by: David Corfield on May 19, 2008 9:40 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

For those without JAMS access, you can get the Berrick et al. paper here.

Posted by: David Corfield on May 19, 2008 5:40 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

Posted by: John Baez on May 19, 2008 6:34 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

It looks like a useful isomorphism in this context is Theorem 1.3:

$H_n(Brun(D^2)) \cong \pi_n(S^2).$

So the homotopy cardinality of $S^2$ could be calculated by an alternating product of homology groups of the chain complex $Brun(D^2)$.

Why is the Euler characteristic defined as an alternating sum? Is it because it’s dealing with the exponents of the nontorsion part of homology, i.e., the ranks? So why does one generally exclude the torsion part?

Posted by: David Corfield on May 19, 2008 6:05 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

David wrote:

Why is the Euler characteristic defined as an alternating sum?

There are many answers to this question, some going back to Euler. But here’s one: the Euler characteristic $\chi$, defined as an alternating sum of ranks of homology groups, has the marvelous property that

$\chi(E) = \chi(F) \chi(B)$

for any fibration

$F \to E \to B$

of connected spaces with finite Euler characteristic.

In other words, it’s multiplicative not just for ordinary products but also for ‘twisted’ products: the total space $E$ is like a ‘twisted product’ of the base space $B$ and the fiber $F$.

(If that doesn’t make sense, imagine a Möbius strip.)

Similarly, the homotopy cardinality $| \cdot |$, defined as the alternating product of cardinalities of homotopy groups, has

$|E| = |F| |B|$

for any fibration

$F \to E \to B$

of connected spaces with finite homotopy cardinality.

The fun part is to see why clever additive cancellations make this fact true for the Euler characteristic, while clever multiplicative cancellations make this fact true for the homotopy cardinality!

Is it because it’s dealing with the exponents of the nontorsion part of homology, i.e., the ranks?

The ranks of finitely generated abelian groups add when we multiply those groups:

$rk(A \times B) = rk(A) + rk(B)$

So, there’s some sort of logarithmic thing going on when we measure the size of an abelian group by taking its rank. But of course the multiplicative-additive distinction is a bit weird here, because $A \times B = A + B$ in the category of abelian groups!

So why does one generally exclude the torsion part?

The traditional reason is that lots of stuff works fine even if we ignore it.

But, that might not be the best approach. Dan Christensen and were thinking about this a bit. You could try working with the cohomology with coefficients in $\mathbb{Z}/p$ and get some sort of ‘mod $p$ Euler characteristic’ for each prime $p$, or maybe each prime power, and then maybe try to fit these together into some sort of ‘adelic’ Euler characteristic that keeps track of all primes at once.

I have a feeling that some people know a lot more about this than I do! But I don’t know who they are.

Posted by: John Baez on May 19, 2008 7:04 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

David wrote:

Does Berrick, Cohen, Wong and Wu’s theorem bring us anywhere nearer to demonstrating the Baez-Dolan ‘conjecture’ that somehow or other the homotopy cardinality of the 2-sphere, i.e., the alternating product of its homotopy groups, should equal its Euler characteristic, i.e., 2?

Great question. As you hint later, the trick would be to understand their simplicial group $Brun(D^2)$ well enough to calculate the alternating sum

$\sum_{n = 0}^\infty (-1)^n k_n$

where $k_n$ is the number of nondegenerate $n$-simplices. Right now I don’t even know if each $k_n$ is finite! Even if it is, the above sum will surely diverge, so we’d need to use some sneaky resummation method, like Abel summation.

When I perfect my machine that spits out grad students made to order, instantly ready to work on whatever I want, life will get a lot better.

Posted by: John Baez on May 19, 2008 8:52 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

Could your general conjecture about Euler characteristic and homotopy cardinality suggest interesting facts?

I mean, imagine we hadn’t calculated the homotopy groups of the spheres beyond $\pi_n(S^n) = \mathbb{Z}$. Wouldn’t we know that because $\chi(S^{2m}) = 2$ that another $\mathbb{Z}$ would have to appear for at least one $(2k + 1)^{th}$ group, $\pi_{2k + 1}(S^{2m})$.

Or is it that the kind of erratic alternating product we’re dealing with can overcome a $\mathbb{Z}$ just using finite groups?

Posted by: David Corfield on May 27, 2008 7:32 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

Or is it that the kind of erratic alternating product we’re dealing with can overcome a $\mathbb{Z}$ just using finite groups?

In theory it could.

For me the problem is that we already know which groups $\pi_{n+k}(S^n)$ are $\mathbb{Z}$. So, it doesn’t strongly confirm the predictive power of somewhat flaky reasoning involving homotopy cardinality to retrodict these. It’d be much cooler to predict something not yet known which we could then go out and test!

Posted by: John Baez on May 27, 2008 8:56 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

Surely our imaginative readers could come up with a suggestion or two!

What does it say about spaces with periodic homotopy groups? I guess if the period is even, then the homotopy cardinality will grow as the power of the product over the period. So we’re not going to end up with anything other than 0, 1 or $\infty$.

On the other hand, with odd period, after two periods we’d be back to 1, so the product would rotate through a finite sequence of values.

Hmm, I guess in the even case cycling could happen too.

Is much know about spaces with periodic homotopy? Searching Google for that phrase yields this.

Might all that $p$-primary $v_n$-periodic homotopy stuff be relevant?

Posted by: David Corfield on May 27, 2008 10:48 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

David wrote:

Is much know about spaces with periodic homotopy?

I don’t know… but there are some famous infinite loop spaces with periodic homotopy groups, thanks to Bott periodicity. $SU(\infty)$ has homotopy groups that go $\mathbb{Z}, 0, \mathbb{Z}, 0, \mathbb{Z}, ...$, repeating with period 2. $SO(\infty)$ repeats with period 8, in a pattern that Peter May teaches his students by singing it to the tune of “Twinkle Twinkle Little Star”:

$\mathbb{Z}_2, \mathbb{Z}_2, 0, \mathbb{Z}, 0, 0, 0, \mathbb{Z}$

In both cases I’m starting with $\pi_0$, which is a group just because our space is a group. In both cases it’s hard to see how to get an interesting homotopy cardinality!

(We also have complex and real $K$-theory spectra which have negative homotopy groups making the periodicity extend in both directions. But, I don’t see how this helps us here. There are lots of other periodic spectra which might be more interesting… but I don’t know much about ‘em!)

I guess I don’t have what it takes this morning to make really interesting conjectures about homotopy cardinality. I’ve been travelling a lot… now it’s time to catch up on work!

Posted by: John Baez on May 27, 2008 5:40 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

I guess that’s at least reassuring that the Poincaré polynomial of $SU(n)$ is

$P_{SU(n)}(t) = (1 + t^3)(1 + t^5)...(1 + t^{2n - 1}),$

so Euler characteristic 0.

And $SU(\infty)$’s homotopy cardinality has a whole lot of $|\mathbb{Z}|$s as denominators.

Posted by: David Corfield on May 27, 2008 7:16 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

David wrote:

I guess that it’s at least reassuring that the Poincaré polynomial of $SU(n)$ is

$P_{SU(n)}(t) = (1 + t^3)(1 + t^5)...(1 + t^{2n - 1}),$

Hey, good point! And this formula is just a squashed, flattened-out, decategorified residue of something easier to understand: $SU(n)$ is a ‘twisted product’ of the odd-dimensional spheres $S^3$, $S^5$, …, on up to $S^{2n-1}$.

It’s an amazing fact that every compact Lie group is a twisted product of spheres, but it’s easy to see in this case.

To see it, note that $SU(n)$ acts on the unit sphere in $\mathbb{C}^n$ with stabilizer subgroup $SU(n-1)$. So, we get a fiber bundle:

$SU(n-1) \to SU(n) \to S^{2n-1}$

and I like to summarize this informally by saying that $SU(n)$ is a ‘twisted product’ of $SU(n-1)$ and $S^{2n-1}$.

Then use induction to express $SU(n)$ as a twisted product of odd-dimensional spheres… but be careful about the starting point of the induction! You see, $\mathrm{U}(n)$ is a twisted product of all the odd-dimensional spheres $S^1$, $S^3$, … $S^{2n-1}$, but for $SU(n)$ we must leave out the $S^1$. (Think about it.)

Now, the Poincaré polynomial of $S^k$ is $(1 + t^k)$, and Poincaré polynomials are multiplicative for twisted products, so we get the formula above.

Then, setting $t = -1$, Poincaré polynomials reduce to Euler characteristics.

Or, work directly: the characteristic of an odd-dimensional sphere is $0$, and Euler characteristic is multiplicative for twisted products, so the Euler characteristic of $SU(n)$ is $0$ times itself a bunch of times! So, the Euler characteristic of $SU(\infty)$ is $0^\infty$. It’s really, really zero.

And $SU(\infty)$’s homotopy cardinality has a whole lot of $|\mathbb{Z}|$s as denominators.

Right! So this is $0^\infty$ as well! And not by coincidence, I think.

We’ve stumbled on something interesting here. The homotopy groups of spheres are intractable, so it’s hard to compute homotopy cardinalities of spheres. On the other hand, the homotopy groups of $SU(n)$ are tractable and indeed periodic up to a certain point, which increases as $n \to \infty$. And, $SU(n)$ is a ‘twisted product’ of spheres.

In particular, the homotopy groups of $SU(\infty)$ are completely understood, and it’s a twisted product of all the odd-dimensional spheres.

So, while the homotopy cardinalities of individual spheres are tricky, something nice seems to happen when we take a twisted product of all the odd-dimensional spheres. Of course the homotopy cardinality is just $0$, which is not very thrilling, but it suggests that variants of this game might be more interesting.

Posted by: John Baez on May 27, 2008 7:56 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

There seems to be a lot of this something being easy in one set-up but more difficult in another going on. E.g., homotopy cardinality factors easily with respect to primes, and copies of $\mathbb{Z}$. While over in the realm of Euler characteristic, you said earlier

You could try working with the cohomology with coefficients in $\mathbb{Z}/p$ and get some sort of ‘mod $p$ Euler characteristic’ for each prime $p$, or maybe each prime power, and then maybe try to fit these together into some sort of ‘adelic’ Euler characteristic that keeps track of all primes at once.

Is there a notion of ‘distance’ between spaces, so that two whose homotopy matches for a considerable stretch (or adelic Euler characteristic, if you can make that work) are ‘close’?

Posted by: David Corfield on May 28, 2008 1:57 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

twisted product’ - a direct translation from the Russian for bundle’

so a twisted bundle is a bundle over a bundle!

Posted by: jim stasheff on May 28, 2008 4:31 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

I suppose for the homotopy cardinality of $SO(\infty)$ you’d take the $\mathbb{Z}_2$s to cancel. Then every eight entries there are 2 $|\mathbb{Z}|$s in the denominator. $SU(\infty)$ racks up 0s twice as quickly, corresponding to its being a product of spheres whose dimensions are 3, 5, 7, …, while $SO(\infty)$ is a product of spheres whose dimensions are 3, 7, 11,…

Posted by: David Corfield on May 30, 2008 2:43 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

David wrote:

I suppose for the homotopy cardinality of $SO(\infty)$ you’d take the $\mathbb{Z}_2$s to cancel. Then every eight entries there are 2 $|\mathbb{Z}|$s in the denominator. $SU(\infty)$ racks up 0s twice as quickly, corresponding to its being a product of spheres whose dimensions are 3, 5, 7, …

I like that! I’m not sure what it means, but I like it.

We could make this stuff make more sense by taking $SU(\infty)$ or $SO(\infty)$ and killing off all their homotopy groups past the $n$th. We’d get some topological groups that are not just $SU(n)$’s or $SO(n)$’s: instead, they mimic the $SU(n)$’s and $SO(n)$’s up to a certain dimension, but are then neatly ‘capped off’ instead of having mysterious homotopy groups extending up to arbitrarily high dimensions.

… while $SO(\infty)$ is a product of spheres whose dimensions are 3, 7, 11…

I’m afraid I have to disagree here, though I see why you guessed this! If you adjust my explanation of why $SU(\infty)$ is a twisted product of spheres of dimension 3, 5, 7, etcetera, you get this:

It’s an amazing fact that every compact Lie group is a twisted product of spheres, but it’s easy to see for $SO(n)$.

To see it, note that $SO(n)$ acts on the unit sphere in $\mathbb{R}^n$ with stabilizer subgroup $SO(n-1)$. So, we get a fiber bundle:

$SO(n-1) \to SO(n) \to S^{n-1}$

and I like to summarize this informally by saying that $SO(n)$ is a ‘twisted product’ of $SO(n-1)$ and $S^{n-1}$.

Then use induction to express $SO(n)$ as a twisted product of spheres… but be careful about the starting point of the induction! You see, $\mathrm{O}(n)$ is a twisted product of all the odd-dimensional spheres $S^0$, $S^1$, $S^2$, … $S^{n-1}$, but for $SO(n)$ we must leave out the $S^0$. (Think about it.)

So, $SO(n)$ is a twisted product of the spheres $S^1, \dots, S^{n-1}$.

Reconciling this with your calculations for $SO(\infty)$ might yield some interesting clues. For example: what do the even-dimensional spheres contribute to the homotopy cardinality of $SO(\infty)$?

(That’s not a puzzle where I already know the answer: it’s a real question!)

Posted by: John Baez on May 30, 2008 5:29 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

So I’m misreading p. 610 of On the Poincaré Polynomials of the Classical Groups by F. D. Murnaghan then?

…when $n$ is odd, the Poincaré polynomial of the $n$-dimensional rotation group is $(z^{2n - 3} + 1)(z^{2n - 7} + 1)...(z^{3} + 1)$ while, when $n$ is even, the Poincaré polynomial of the $n$-dimensional rotation group is $(z^{2n - 5} + 1)(z^{2n - 9} + 1)...(z^{3} + 1)(z^{n -1} + 1)$.

Aha torsion! p. 300 of Hatcher.

Posted by: David Corfield on May 30, 2008 6:10 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

I think this is the problem: I was right about expressing these groups as twisted products of spheres, but I was overoptimistic in claiming that the Poincaré polynomials were always multiplicative for twisted products. For example, it’s really true that $SO(3)$ is an $S^1$ bundle over $S^2$, since

$S^2 = SO(3)/SO(2)$

and $SO(2) = S^1$. But, it ain’t true that the Poincaré polynomial of $SO(3) = \mathbb{R}P^2$ is $(z + 1)(z^2 + 1)$. After all, this 3d manifold is nonorientable, so its rational 3rd cohomology group vanishes.

I guess the problem is that we need to use the Serre spectral sequence to compute cohomology of the total space of a fiber bundle. When the base space is simply connected, (as here), the ‘$E_2$ term’ of this sequence gives the product of the homology of the base tensored with the homology of the fiber. If that were the whole story, the Poincaré polynomial would be multiplicative. But then there are further corrections, in general, which make things more complicated.

Somehow I lucked out in the case of $SU(n)$.

But, more robust reasoning still has me convinced that Euler characteristic is multiplicative for twisted products.

Posted by: John Baez on May 30, 2008 6:59 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

But isn’t that result of Murnaghan that as far as free cohomology groups go it’s as though $SO(\infty)$ is a product $S^3 \times S^7 \times S^{11} \times ...$ quite pleasing to us, in view of the slower rate of homotopy cardinality growth $0^n$?

Posted by: David Corfield on May 30, 2008 7:09 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

SO(3) is RP(3) It is clearly orientable as a lens space of type L(2,1). This 3-mfd is orientable as it is the Thom space of a twisted I bundle over
RP(2). The correspondence is to choose a rotation, it has an axis and an angle, t, between -pi and pi. To get a point in RP(3) then move along the axis a distance t/pi. Antipodal points on the boundary sphere of the 3-ball are identified.

Posted by: Scott Carter on May 30, 2008 9:00 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

For twisted products of spheres (aka sphere bundles over sphere bundles over …)
there’s no need for the steam roller (= heavy machinery). depending on whihc way you want to induct, use the Wang sequence
(preferable) or the Gysin sequence

Posted by: jim stasheff on May 31, 2008 1:30 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

By the way, David — if I can somehow show that the homotopy cardinality of $S^2$ equals its Euler characteristic, you can easily use the information in this Week’s Finds to show the same for $S^3$. See how?

Posted by: John Baez on May 24, 2008 4:33 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

I seem to remember thinking this through before. Isn’t it just that the only difference between the alternating product for $S^3$ and for $S^2$ is the former’s 1 where the latter scores a $|\mathbb{Z}|$? So, the homotopy cardinality of $S^3$ would be $2 / \infty = 0$, matching the Euler characteristic.

Posted by: David Corfield on May 26, 2008 3:50 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

David wrote:

I seem to remember thinking this through before. Isn’t it just that the only difference between the alternating product for $S^3$ and for $S^2$ is the former’s 1 where the latter scores a $|\mathbb{Z}|$? So, the homotopy cardinality of $S^3$ would be $2 / \infty = 0$, matching the Euler characteristic.

Right! So, my challenge was sort of a joke. I asked “if I can show that some divergent product equals 2, can you show that dividing it by $\infty$ gives 0?” And the joke is that I could have made this question much harder, just by turning it around: “if I can how that some divergent product equals 0, can you show that multiplying it by $\infty$ gives 2?”

When you thought this through before, had you known the homotopy groups of $S^3$ mostly matched those of $S^2$? For some reason I was shocked to discover this. It’s like discovering you’ve never noticed that two good friends of yours are twins!

Posted by: John Baez on May 26, 2008 6:30 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

Remember I got into homotopy groups of spheres for that crazy chapter of mine on machines detecting regularities. I couldn’t even get mine to find that $S^2$’s and $S^3$’s groups agreed after $n = 3$.

Posted by: David Corfield on May 26, 2008 11:22 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

Euler characteristic as an alternating sum is related to inclusion-exclusion. I heard Victor Vassiliev make this comment at a lecture at MSRI in the mid 1990s. If you think of an $n$-simplex as the dual picture to $n$-intersecting sets with fullest possible intersections, then the inclusion/exclusion principle is the count of the faces of the simplex with dimensions taken into considerations.

On the stable homotopy groups of spheres (and the infinite projective plane): the third stable homotopy group of $RP^\infty$ is $Z_8$, and it injects into the third stable stem (which is $Z_{24}$) in a very cool way. The generator of the $Z_8$ piece is the Boy’s surface (under the Pontryagin-Thom construction). A generator of the $Z_{24}$ piece can be given by a sphere eversion with one quadruple point and a connected triple point manifold (Koschorke). You cap the sphere eversion off at both ends with balls and the orientations work out exactly as they are supposed to! A sculpture of Boy surface is found at Oberwolfach. In the pictures section of their page are some sketches of my description of it.

In general, for $n \gt 3$ there is an immersed $n$-dimensional manifold (necessarily non-orientable) in $(n+1)$-space with an odd number of $(n + 1)$-tuple points if and only if there is a framed $(n+1)$-manifold of Arf-invariant 1 (Theorem of Eccles). This happens when $n = 2^j-3$, and I think is unknown beyond 61. There is a cheap trick using Boy’s surface and a 3-sphere in 6 space with one double point to get the 5-dimensional example.

Consider the free strict symmetric monoidal category with duals and no pivotal nor rigidity assumed. Objects are non-negative integers, and morphisms are generated in a tensorial sense by cup, cap, and X.

Cup should “create” a particle anti-particle pair, and cap anniliates one. (I read from bottom to top). X comes in the four obvious flavors. The symmetry property XX $\Rightarrow$ || is a 2-isomorphism, and so is the naturality axiom for X. There is also a 2-isomorphism

$\psi: (| \otimes \cap)(X \otimes |) \Rightarrow (\cap \otimes |)(| \otimes X).$

By not allowing a pivotal axiom we are eliminating type I Reidemeister moves (their projections). A sphere eversion is a sequence of 3-morphisms in this appropriately indexed category. The 3-morphisms (or identities among 2-morphisms) are generated by the appropriately restricted class of movie moves. The basic list is the above things are 2-isomophisms, (therefore so is type III), the quadruple point move, and a few moves that involve $\psi$. The axioms can be gleaned from Baez-Langford.

Yet another similarity of form between homotopy theory and HDA.

Posted by: Scott Carter on May 20, 2008 9:02 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

A sphere eversion is a sequence of 3-morphisms in this appropriately indexed category.

That sounds really cool.

Posted by: Bruce Bartlett on May 20, 2008 9:34 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

If you think that’s cool, you should see it in pictures.

Posted by: John Armstrong on May 20, 2008 4:42 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

A general position immersed curve (which may have multiple components) represents a 1-morphism from the null object to the null object. Such morphisms are generated by |,X, cup, and cap. Generating 2-morphisms are given by a birth, death, saddle, [| =>zig-zag] and reverse, a psi as above, the projection of a type II move, and the projection of a type III move.

A general position immersed sphere (or any surface) in 3-space can be arranged so that a sequence of horizontal planes chosen to flank all critical (in a general sense) phenomena. The intersection of such a plane with the sphere is an immersed curve or a 1-morphism [ null->null]. The transition from one cross section to another is given by one of the above generating 2-morphisms (tensored and composed with identities on the unaffected regions). So each immersed sphere represents a composition of 2-morphisms

[Null => [null –0–> null] => … => [null –0–> null] => Null]

where Null is the empty 2-morphism from the empty object to itself and [null –0–> null] is the 1-morphism that corresponds to a simple convex closed curve [CIRCLE].

The changes between spheres in the process can also be arranged to involve generic critical points. And these local changes are generated by the movie-moves that don’t involve type I moves (aka branch points). Such a move between immersed spheres, then, can be thought of as a 3-morphism. Let +0 denote a counter clockwise oriented circle, and -0 denote a clockwise oriented circle.
A sphere eversion is the 3-morphism

[Null => +0 => Null] ===> [Null => -0 => Null].

You can cap off one end with the 3-morphisms that start from the NULL 3-morphism and go to an embedded 2-sphere, then cap off the opposite end with the opposite 3-morphism. The result is a generator of the 3rd stable stem.

Posted by: Scott Carter on May 20, 2008 11:03 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

Scott -

Have you seen the discussion on the polyhedral model of Boy surfaces at

http://www.popmath.org.uk/sculpmath/pagesm/brehm.html

(This is more for amusement that for a serious addition to this discussion.)

On a more serious note I remember reading some of the stuff in the 1980s, but dating from 15 years earlier, on immersions using classifying spaces of simplicial monoids. (I mentioned this in another thread a day or two ago.) Perhaps there should be some attempt to quickly check through that stuff for relevant facts since the way of science is to follow the flow of the main ideas as defined by a particular aim and apparent side issues can in fact come to be important later on. I am thinking of the papers by Lashof on Lee’s approach to smoothing. (E.g., if my memory serves me right, the paper: Lee’s immersion theorem and the triangulation of manifolds, R. Lashof, Bull. Amer. Math. Soc. Volume 75, Number 3 (1969), 535-538, was typical of the time).

There were several good survey articles also at that time. The general approach of those papers may still be useful.

Posted by: Tim Porter on May 22, 2008 7:39 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

but didn’t see any polyhedral model
??

Posted by: jim stasheff on May 22, 2008 2:39 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

Towards the bottom of the page, you’ll see the fundamental piece.

Tim: I vaguely remember the stuff of Lashof, but I more or less got out of the immersion business a while ago. The discussion of stable homotopy reminded me of these things.

Posted by: Scott Carter on May 22, 2008 8:23 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

It turns out that some of Coleman’s lecture notes on quantum field theory have been TeXed up by Bryan Chen, and are available here.

Posted by: John Baez on May 22, 2008 3:47 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

has some very relevant information on the sequence

1, 1, 2, 3, 4, 5, 6, …

though it’s phrased not in terms of “sections of line bundles”, but instead in terms of “divisors” (secretly another way of talking about the same thing). Let me quote a portion, just to whet your interest:

We start with a connected compact Riemann surface of genus g, and a fixed point P on it. We may look at functions having a pole only at P. There is an increasing sequence of vector spaces: functions with no poles (i.e., constant functions), functions allowed at most a simple pole at P, functions allowed at most a double pole at P, a triple pole, … These spaces are all finite dimensional. In case g = 0 we can see that the sequence of dimensions starts

1, 2, 3, …

This can be read off from the theory of partial fractions. Conversely if this sequence starts

1, 2, …

then g must be zero (the so-called Riemann sphere).

In the theory of elliptic functions it is shown that for g = 1 this sequence is

1, 1, 2, 3, 4, 5 …

and this characterises the case g = 1. For g > 2 there is no set initial segment; but we can say what the tail of the sequence must be. We can also see why g = 2 is somewhat special.

The reason that the results take the form they do goes back to the formulation of Roch’s part of the Riemann-Roch theorem as a difference of two such dimensions. When one of those can be set to zero, we get an exact formula, which is linear in the genus and the degree (i.e. number of degrees of freedom). Already the examples given allow a reconstruction in the shape

dimension - correction = degree - g + 1.

For g = 1 the correction is 1 for degree 0; and otherwise 0. The full theorem explains the correction as the dimension associated to a further, ‘complementary’ space of functions.

Posted by: John Baez on May 24, 2008 3:56 AM | Permalink | Reply to this

Molien concidence? Re: This Week’s Finds in Mathematical Physics (Week 264)

A028310 Expansion of (1-x+x^2)/(1-x)^2.

1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

COMMENT

Molien series for ring of Hamming weight enumerators of self-dual codes (with respect to Euclidean inner product) of length n over GF(4).

Engel expansion of e (see A006784 for definition) [when offset by 1] - Henry Bottomley (se16(AT)btinternet.com), Dec 18 2000

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).

Index entries for Molien series

Index entries for sequences related to Engel expansions

FORMULA

Binomial transform is A005183. - Paul Barry (pbarry(AT)wit.ie), Jul 21 2003

G.f.: (1-x+x^2)/(1-x)^2 = (1-x^6)/((1-x)(1-x^2)(1-x^3)) = (1+x^3)/((1-x)*(1-x^2)).

Euler transform of length 6 sequence [ 1, 1, 1, 0, 0, -1]. - Michael Somos Jul 30 2006

PROGRAM

(PARI) a(n)=(n==0)+max(n, 0)

CROSSREFS

Apart from the extra initial 1, same as A000027.

KEYWORD

nonn,easy,mult

AUTHOR

njas

Posted by: Jonathan Vos Post on May 24, 2008 9:09 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

Right — this is the Sloane’s entry that I mentioned in this Week’s Finds.

Posted by: John Baez on May 26, 2008 6:21 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

Wow! The HiRISE satellite orbiting Mars, which took the amazing closeup of Phobos gracing this Week’s Finds, managed to catch a picture of Phoenix as it parachuted to Mars yesterday!

Posted by: John Baez on May 26, 2008 8:04 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

Who will complete the square?

$\array{ Groupoid cardinality & \rightarrow & Euler-Leinster characteristic\\ \downarrow & & \downarrow \\ Homotopy cardinality & \rightarrow & ??? }$

Posted by: David Corfield on May 28, 2008 8:55 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

Tom Leinster came close to completing this square in his original paper. He described a concept of cardinality for strict $n$-categories at the bottom of page 15.

On the one hand this concept agrees with the Euler-Leinster characteristic of a 1-category whenever we have a strict $n$-category with only identity $j$-morphisms for $j \gt 1$ (and probably any $n$-category equivalent to one of these, too… right, Tom?)

On the other hand, I bet this concept agrees with the homotopy cardinality when our strict $n$-category is a strict $n$-groupoid. (Does it, Tom?)

So, the really interesting step would be to generalize this concept of cardinality to Batanin’s weak $n$-categories.

And then, of course, use it for something.

Posted by: John Baez on May 29, 2008 8:39 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

Why do you mention Batanin’s in particular?

Posted by: David Corfield on May 29, 2008 9:09 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

As you know, a category enriched over some category $K$ is one where instead of hom-sets $hom(x,y)$, we have hom-objects $hom(x,y) \in K$. Tom defines a concept of cardinality enriched over $K$ when $K$ is a category whose objects are already equipped with a concept of cardinality. Since strict $(n+1)$-categories are categories enriched over $n Cat$, and everyone knows the cardinality of an $0$-category — since that’s just a set — by induction we get a concept of cardinality for strict $n$-categories!

Batanin’s weak $n$-categories are close to the usual strict ones in the following sense: for any pair of objects $x,y$ you get an $(n-1)$-category $hom(x,y)$. This is similar to the concept of enrichment — though it’s not so simple, since the usual category axioms hold only ‘weakly’.

So, I can vaguely imagine doing an inductive definition of cardinality for Batanin’s weak $n$-categories that mimics what Tom did.

However, it was just a vague thought. It could be wrong: it could be that Trimble’s definition makes things easier, since it’s more closely based on the concept of ‘iterated enrichment’.

Posted by: John Baez on May 29, 2008 9:28 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 264)

As for uses, perhaps we could employ it profitably on those fundamental $n$-categories of stratified spaces we looked at once.

Posted by: David Corfield on May 29, 2008 9:47 PM | Permalink | Reply to this

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