This Week's Finds in Mathematical Physics (Week 147)
John Rickard
: In sci.math Toby Bartels <to...@ugcs.caltech.edu> wrote:
[...]
: : The answer must be something that won't work
: : if 3 is replaced by an infinite cardinal.
:
: Actually, the problem is intended to apply to any sets, not just
: finite sets.
Sorry; I skimmed what Toby Bartels wrote without reading it properly,
and thought he was saying that the answer must be something that
wouldn't work if x and y were allowed to be infinite.
--
John Rickard <John.R...@virata.com>
Toby Bartels
>Toby Bartels <to...@ugcs.caltech.edu> wrote:
>>What about divergence to 0?
>>If pi_i(X) is infinite for some odd i but no even i,
>>can we say |X| is 0?
>Well, we can, but we might regret it later.
I suspected we might.
>>OTOH, G is already a kind of reciprocal of itself.
>>If G is a discrete group, it's a topological space
>>with |G|_homotopy = |G|_set.
>>But G is also a groupoid with 1 object,
>>and |G|_groupoid = 1 / |G|_set.
>>So, |G|_homotopy |G|_groupoid = 1.
>Believe it or not, you are reinventing BG! A groupoid can be
>reinterpreted as a space with vanishing homotopy groups above the
>first, and if you do this to the groupoid G, you get BG.
I thought that might be what I was doing.
But I wasn't sure what BG looked like.
>And the real point is that a 1-object omega-groupoid can be
>reinterpreted as an omega-groupoid by forgetting about the
>object and renaming all the j-morphisms "(j-1)-morphisms".
! 1groupal omega-groupoids!
The homotopy type of a topological group *would* be such a beast.
>>So, are they ever both defined but different?
>I don't recall any examples where they're both finite, but different.
>I know very few cases where they're both finite! How about the point?
>How about the circle? How about the 2-sphere? I leave you to ponder
>these cases.
In the case of the point, both are 1.
In the case of the circle, the homotopy cardinality diverges to 0,
while the Euler characteristic is undivergently 0.
In the case of the sphere, the homotopy cardinality is infinite,
while the Euler characteristic remains a pure 0.
-- Toby
to...@ugcs.caltech.edu
Marc Bellon
}-
}-In article <8g73jc$b...@gap.cco.caltech.edu>,
}-Toby Bartels <to...@ugcs.caltech.edu> wrote:
}-
[A lot of stuff snipped]
}->>But the fundamental group G is infinite! What's going on?
}-
}->This doesn't seem too surprising. 1/(2 - 2g) is also infinite.
}->Just use the geometric series in reverse:
}->1/(2 - 2g) = (1/2) \sum_i g^i, which diverges since g > 1.
}-
}-Well, what I really want is a way of counting elements of the fundamental
}-group of the surface S which gives me a divergent sum that I can cleverly
}-sum up to get 1/(2 - 2g).
}-
}-Here is a clever attempt which I take the liberty of quoting out of
}-email from my friend Dan Christensen, a homotopy theorist:
[snip]
}- For a wedge of n circles, the Euler characteristic is 1-n and the
}-same filtration gives a cardinality of
}-
}- 1 + 2n(1 + (2n-1) + (2n-1)^2) = 1 + 2n/(2-2n) = 1/1-n !
}-
}- However, I haven't been able to get this to work with a surface of
}-genus 2. pi_1 = <a1, b1, a2, b2 | [a1,b1] = 1 = [a2,b2]>. Every word
Here is the problem. The only relation in pi_1 is [a1,b1][a2,b2] =1.
You have far less relations in pi_1 that it is supposed here.
I do not know of any way of giving a canonical shortest form for elements
of pi_1. In particular, it is possible that the shortest form cannot be
obtained without going through forms of greater length than the original
form. A first approximation would be to ignore this relation or the
equivalent one in higher genus, \prod [a_i,b_i] =1.
Then we obtain 1 + 4g \sum ( 4g -1 )^n = 1/(1-2g),
which is asymptotically the searched for result. It is plausible that
the additional relation gives the right correction.
Does anyone know of a specialist of generators and relations in groups
which could be able to find the exact result ? May be GAP is the
proper computer tool to have this calculation done, but I do not have
a working version.
[snipped, false premises, calculation loose interest. I just
keep the result.]
}-A computer calculation shows that this is -1.
}- So we don't get the answer we would hope to get in this case. There
}-are several possible reasons. Maybe this idea has no value whatsoever,
}-and it is just a coincidence that it works for wedges of circles? Or
}-maybe the idea is just slightly wrong in a way that doesn't matter for
}-wedges of circles? Or maybe I got the recursion relations wrong?
}-(Please check---it would be great if I just made a minor mistake.)
}-Or maybe the identification of the sum with the limit is wrong and
}-I should just solve the recursion and figure out the sum more carefully.
}-
}- Let me know if you have any thoughts."
Marc Bellon.
John Baez
Marc Bellon <bel...@lpthe.jussieu.fr> wrote:
>ba...@galaxy.ucr.edu (John Baez) wrote:
>}Dan Christensen wrote:
>}- However, I haven't been able to get this to work with a surface of
>}-genus 2. pi_1 = <a1, b1, a2, b2 | [a1,b1] = 1 = [a2,b2]>.
>Here is the problem. The only relation in pi_1 is [a1,b1][a2,b2] =1.
>You have far less relations in pi_1 that it is supposed here.
James Dolan noticed this mistake too. And he noticed that if you take
the homotopy 1-type whose fundamental group *is* described by Dan's
presentation here, it *does* have Euler characteristic -1, so Dan's result...
>}-A computer calculation shows that this is -1.
... was actually good news rather than bad news!
And it turns out that mathematicians have already studied this stuff
and proved that things work out well for any Riemann surface:
@article {MR97d:20031a,
AUTHOR = {Grigorchuk, R. I.},
TITLE = {Growth functions, rewriting systems and Euler characteristic},
JOURNAL = {Mat. Zametki},
FJOURNAL = {Rossi\u\i skaya Akademiya Nauk. Matematicheskie Zametki},
VOLUME = {58},
YEAR = {1995},
NUMBER = {5},
PAGES = {653--668, 798},
@article {MR88m:22023,
AUTHOR = {Floyd, William J. and Plotnick, Steven P.},
TITLE = {Growth functions on Fuchsian groups and the Euler
characteristic},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {88},
YEAR = {1987},
NUMBER = {1},
PAGES = {1--29},
I thank Laurent Bartholdi and Danny Ruberman for pointing this out.
John Baez
Toby Bartels <to...@ugcs.caltech.edu> wrote:
>In the case of the sphere, the homotopy cardinality is infinite,
>while the Euler characteristic remains a pure 0.
Everything else was fine, but here you slipped:
The Euler characteristic of the 2-sphere is 1 - 0 + 1 = 2.
Its homotopy cardinality may seem infinite because pi_2(S^2) = Z,
but then pi_3(S^2) = Z comes along and undoes that infinity. From
then on, the higher homotopy groups of the 2-sphere are finite.
Unfortunately nobody knows them all, so it's tough to say if there's
any sense in which the alternating product of their cardinalities
"converges" to the desired result. If such a sense exists, it must
be subtle, because I'm pretty sure there are infinitely many nonvanishing
homotopy groups pi_n(S^2)... so we can't get convergence in the standard
sense.
By the way, the fact that pi_3(S^2) = Z is extremely related to
that line bundle Michael Weiss is trying to understand, the complex line
bundle over S^2 with Chern number 1. If you turn this line bundle
into a circle bundle in the obvious way, you get a bundle with total
space S^3, and thus a map S^3 -> S^2. This map, called the Hopf
map, is a generator of pi_3(S^2). Someday I will explain all this
to Michael and help him visualize this stuff - but not today.
Penrose and Rindler have a very nice picture of the Hopf map in his
book on spinors and spacetime. It's profoundly related to spinors.
John Baez
>1groupal omega-groupoids!
>The homotopy type of a topological group *would* be such a beast.
Exactly. Ain't it cool?
In general, whenever we have a gadget in the category of topological
spaces, we get a corresponding gadget in the category of omega-groupoids.
You've just seen that a group object in the category of topological spaces
gives a group object in the category of omega-groupoids, i.e. a groupal
omega-groupoid. But there are lots of other nice applications of this
idea.
For example, lots of famous categories are really topological categories -
for any pair of objects x,y there is a *space* of morphisms hom(x,y),
and composition is a *continuous map* o: hom(x,y) x hom(y,z) -> hom(x,z).
With a wave of the above magic wand, these categories become omega-categories
with all j-morphisms for j > 1 being (weakly) invertible.
What this means is that there are a lot more omega-categories floating
around than you might notice at first. Two nice examples are the topological
category of finite-dimensional real or complex vector spaces. These play
an important role in K-theory, and the role they play really takes advantage
of the fact that they become omega-categories using the above trick.
In fact, these examples Vect_R and Vect_C are *symmetric monoidal*
topological categories. In the world of categories "symmetric monoidal"
is the same as "infinitely monoidal", so these examples give, not just
omega-categories, but infinitely monoidal omega-categories. In other
words, you can think of them as Z-categories with only one j-morphism for
all negative j. Thus they tower infinitely tall and infinitely deep, but
they are sort of trivial on the bottom, and the only non-weakly-invertible
j-morphisms are the 0-morphisms (i.e. the objects of Vect_R and Vect_C,
which are not invertible w.r.t. tensor product).
Toby Bartels
Subject: Re: This Week's Finds in Mathematical Physics (Week 147)
References: <8g73jc$b...@gap.cco.caltech.edu> <8g9r88$har$1...@pravda.ucr.edu> <8gaci4$c...@gap.cco.caltech.edu> <8ghrik$3vk$1...@pravda.ucr.edu>
John Baez <ba...@galaxy.ucr.edu> wrote in part:
>Toby Bartels <to...@ugcs.caltech.edu> wrote:
>>In the case of the sphere, the homotopy cardinality is infinite,
>>while the Euler characteristic remains a pure 0.
>Everything else was fine, but here you slipped:
>The Euler characteristic of the 2-sphere is 1 - 0 + 1 = 2.
Duh! they taught me that in elementary school!
I think I was accidentally putting down its genus.
>Its homotopy cardinality may seem infinite because pi_2(S^2) = Z,
>but then pi_3(S^2) = Z comes along and undoes that infinity. From
>then on, the higher homotopy groups of the 2-sphere are finite.
Ah, I stopped when I got to pi_2.
Are we sure that we can just cancel the infinities?
Infinity/infinity = 1 instead of, say, 2.9?
-- Toby
to...@ugcs.caltech.edu
John Baez
Toby Bartels <to...@ugcs.caltech.edu> wrote:
>>Its homotopy cardinality may seem infinite because pi_2(S^2) = Z,
>>but then pi_3(S^2) = Z comes along and undoes that infinity. From
>>then on, the higher homotopy groups of the 2-sphere are finite.
>Ah, I stopped when I got to pi_2.
>Are we sure that we can just cancel the infinities?
>Infinity/infinity = 1 instead of, say, 2.9?
No, I'm not sure at all! We are just fumbling around here trying to
guess what's going on. Perhaps the whole idea is stupid.
Here are some more details about the Euler characteristic versus
the homotopy cardinality of spheres.
The Euler characteristic of an odd-dimensional sphere is 0; the
Euler characteristic of an even-dimensional sphere is 2.
The homotopy cardinality of the n-sphere matches its Euler
characteristic for n < 2, and is given by a divergent product
for n = 2 or higher. But let's be bold (stupid?) and think
about it anyway...
If n is odd, the only infinite homotopy group of S^n is pi_n(S^n),
which is Z. Thus the homotopy cardinality of S^n is a product
of nonzero numbers and 1/infinity, which is zero - matching the
Euler characteristic. Here I'm being glib, since we are really
taking an *infinite* product of nonzero numbers: our responsibility
is presumably to show that this product, which cannot converge,
nonetheless does not approach 0 or infinity in too decided a manner.
(It probably wavers back and forth getting very big and then very
small. Is this bad?)
If n is even, the only infinite homotopy groups of S^n are pi_n(S^n)
and pi_{2n-1}(S^n): the first is Z and the second is Z plus a finite
abelian group. Thus the homotopy cardinality of S^n is a product of
nonzero numbers together with infinity and 1/infinity. We want this
to equal 2 in some sense. Again, it cannot converge to 2, so our
responsibility is to figure out some more general sense in which it
"equals 2".
There are some estimates known and conjectured concerning the homotopy
groups of spheres which might help us here, but unfortunately not
enough is known to really "compute the answer", even after having
made precise the sense in which we wish to do so. However, if we
are clever enough, perhaps we can take this as an opportunity to
make a new conjecture concerning the homotopy groups of spheres.