More mathematics
Oct. 25th, 2006 09:08 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
michiexileIt was waaaaaaay too long ago I wrote mathematics here. Most of it ends up at my blog. But I thought I'd share a morsel from the last Oberseminar here: important since it is a comparatively hands-on fact, which however came as news for almost everyone involved - including my advisor.
Homotopy is one of the key tools in my branch of mathematics. The very easiest idea is that of putting rubber bands on thingies. Say you have an object - a coffee cup, or a ball, or something else. This object of yours we shall call a 2-manifold in order to put a name on it. (this immediately implies that we don't care for anything but the actual surface of your thingie)
And you are equipped with funky rubber bands. The rubber bands start out as just bands, and you can connect a band once to form a loop. Once it forms a loop, it sticks at your object as tight as it can go, and stays as tight as it can around whatever it ends up against. Once there, you can start nudging it around, but you are not allowed to break it off again.
For the ball, any way you start, you can always nudge the band until it pops off the ball again, having drawn itself together down to a small clot of rubber. For the coffee cup, the number of times you wind it around the handle decides what you can do with it: no times -> you can pull it off, n times -> you can make it look like anything else that's wound n times around, but not like anything else. And then there are some funky things going on with various ways you could wind it around the handle.
Now, this precise observation is what topologists call homotopy, and algebraists use as a really, really funky tool. The rubber bands get formalized as "maps f from the interval [0,1] to the object, with f(0)=f(1)", nudging between two rubber band positions f and g gets formalized as "a family of maps ft(x) such that f0(x)=f(x), f1(x)=g(x), and the nudging is continuous as a function", and we turn out to be interested in what remains if we put everything as equal that can be nudged into each other.
This all gets far more mangled if you do algebra - so you'll have families of vector spaces, organised into chains of vector space, linear map, vector space, linear map, et.c. Between a pair of such spaces, you can have maps that take all vector spaces in the first chain into corresponding spaces in the second chain, and which don't care about the linear maps in the chains. Two such maps correspond to two rubber bands, and we call the maps f,g between chains C and D with chain-internal maps d and d' homotopic precisely if we can find another map h from the chain C to the chain D so that f-g = d'h+hd.
Now, these things don't really look as if they have much to do with each other, now do they?
What happened at the seminar was that another homotopy for chains was introduced: there is one particular chain thingie called I that corresponds precisely to the [0,1] interval: it looks like
0 -> k -> k2 -> 0
with the single map k -> k2 given as a matrix by
/ 1 \
\ -1 /
and this one gets combined with the chain C to form a new thingie C*I. Then this h is the same thing as a map t from C*I to D such that it takes on the values of f and g for some specified values. This, all of a sudden, looks a lot more like the rubber bands: a function from this C*I to the D is precisely a function in two variables: the t and the x in ft(x), and the conditions we put on the map correspond precisely to that f0(x)=f(x), f1(x)=g(x).
And the funky thing in the seminar was proving that these two homotopies: the h such that f-g=d'h+hd and the map C*I -> D are really the very same thing.
Homotopy is one of the key tools in my branch of mathematics. The very easiest idea is that of putting rubber bands on thingies. Say you have an object - a coffee cup, or a ball, or something else. This object of yours we shall call a 2-manifold in order to put a name on it. (this immediately implies that we don't care for anything but the actual surface of your thingie)
And you are equipped with funky rubber bands. The rubber bands start out as just bands, and you can connect a band once to form a loop. Once it forms a loop, it sticks at your object as tight as it can go, and stays as tight as it can around whatever it ends up against. Once there, you can start nudging it around, but you are not allowed to break it off again.
For the ball, any way you start, you can always nudge the band until it pops off the ball again, having drawn itself together down to a small clot of rubber. For the coffee cup, the number of times you wind it around the handle decides what you can do with it: no times -> you can pull it off, n times -> you can make it look like anything else that's wound n times around, but not like anything else. And then there are some funky things going on with various ways you could wind it around the handle.
Now, this precise observation is what topologists call homotopy, and algebraists use as a really, really funky tool. The rubber bands get formalized as "maps f from the interval [0,1] to the object, with f(0)=f(1)", nudging between two rubber band positions f and g gets formalized as "a family of maps ft(x) such that f0(x)=f(x), f1(x)=g(x), and the nudging is continuous as a function", and we turn out to be interested in what remains if we put everything as equal that can be nudged into each other.
This all gets far more mangled if you do algebra - so you'll have families of vector spaces, organised into chains of vector space, linear map, vector space, linear map, et.c. Between a pair of such spaces, you can have maps that take all vector spaces in the first chain into corresponding spaces in the second chain, and which don't care about the linear maps in the chains. Two such maps correspond to two rubber bands, and we call the maps f,g between chains C and D with chain-internal maps d and d' homotopic precisely if we can find another map h from the chain C to the chain D so that f-g = d'h+hd.
Now, these things don't really look as if they have much to do with each other, now do they?
What happened at the seminar was that another homotopy for chains was introduced: there is one particular chain thingie called I that corresponds precisely to the [0,1] interval: it looks like
0 -> k -> k2 -> 0
with the single map k -> k2 given as a matrix by
/ 1 \
\ -1 /
and this one gets combined with the chain C to form a new thingie C*I. Then this h is the same thing as a map t from C*I to D such that it takes on the values of f and g for some specified values. This, all of a sudden, looks a lot more like the rubber bands: a function from this C*I to the D is precisely a function in two variables: the t and the x in ft(x), and the conditions we put on the map correspond precisely to that f0(x)=f(x), f1(x)=g(x).
And the funky thing in the seminar was proving that these two homotopies: the h such that f-g=d'h+hd and the map C*I -> D are really the very same thing.
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Hmm ...
Date: 2006-10-25 10:34 pm (UTC)I'll read this again tomorrow and see if it makes more sense to me then ...
Re: Hmm ...
Date: 2006-10-26 07:37 pm (UTC)Re: Hmm ...
Date: 2006-10-27 07:33 am (UTC)Re: Hmm ...
Date: 2006-10-27 07:48 am (UTC)We all know and love the topological homotopy, most naïvely expressed as a continuous function H(x,t):X x I -> Y such that H(x,0)=f(x) and H(x,1)=g(x).
We also all know and love the chain homotopy h, given as a degree -|d| map X -> Y such that f-g = dh+hd.
Now, the funky thing appears when we recognize that the chain complex most naturally associated with the topological construction is, in fact, I=0 -> kc -> ka(+)kb -> 0 with the single non-trivial differential c -> a-b.
So we'll consider X (x) I as a chain complex. This has in a given degree n the composition
X_(n-1) (x) I_1 (+) X_n (x) I_0
Now, the statement that would be neat to have proven (and that got proven in the seminar in question) is that existence of a chain homotopy h: f ~ g is equivalent to existence of a chain map H: X (x) I -> Y such that H(x,a) = f(x), H(x,b) = g(x).
Firstly, lets suppose we have H and want to find a h. Well, consider the function (curry if you wish) h = H(-,c): X -> Y. We'll want to take a look at what happens if we take a close look at dh+hd.
So
dh(x) = dH(x,c)=Hd( x(x)c )=H(dx(x)c - x(x)dc) = H(dx(x)c) - H( x(x)a - x(x)b ) =
= h(dx)-(H(x,a)-H(x,b)) = hd(x) -(f(x)-g(x))
and thus (up to me screwing up signs), dh+hd=f-g.
Suppose now instead we have a h and seek a H. Well, define H by
H(x,a) = f(x)
H(x,b) = g(x)
H(x,c) = h(x)
Now
dH(x,a)=df(x)=fd(x)=H(dx,a)-H(x,da)=Hd(x,a) (where da=0 and db=0 by the differential in I)
dH(x,b)=Hd(x,b) by the same argument and
dH(x,c)=dh(x)=f(x)-g(x)-hd(x)=H(x,a)-H(x,b)-H(dx,c)=H(x,dc)-H(dx,c)=Hd(x,c) (again with fudging the signs that don't bother coming out just right)
So, this juggling shows us that the chain homotopy and topological homotopy, really, really are the same thingie.
Re: Hmm ...
Date: 2006-10-28 03:43 pm (UTC)By the way, here's a similar (but simpler) result: let I be the category with two objects, 0 and 1, and one non-identity arrow 0->1. If C and D are categories, and F,G: C -> D are functors, then a natural transformation F -> G is exactly a functor alpha: CxI -> D for which alpha(_,0) = F and alpha(_,1) = G.
(*) Actually, is that true? I know Top is a model category, but are the weak equivalences given by homotopies?
Re: Hmm ...
Date: 2006-10-28 07:52 pm (UTC)Weak equivalences X -> Y are maps such that the induced maps on the pointed homotopy groups end up being isomorphisms.
Fibrations are Serre fibrations
Cofibrations are what they need to be to fit with these two.
And every homotopy equivalence is a weak equivalence according to this definition. There may, I think, however be more weak equivalences.
***
So, in other words, a natural transformation between two functors really is "just" a homotopy between these two functors, eh?
no subject
Date: 2006-10-26 04:06 am (UTC)no subject
Date: 2006-10-26 07:28 am (UTC)This is all stuff pretty close to the core of what I do for research, and I'm in a way constantly stumped for how to convert it to chitchat: the best I can do seems to be "You see these rubber bands? And we can nudge them and stuff? Now I do a big switcheroo, throw everything you -can- visualize out the window and replace them with odd stuff that you don't understand."
It's embarrassing when I discover that I cannot do much better when talking to the non-algebra professors in our department either.
no subject
Date: 2006-10-26 02:21 pm (UTC)Similar things happen here too sometimes: We'll routinely discover things that other people thought of as basics but we somehow didn't know for years.
no subject
Date: 2006-10-26 02:40 pm (UTC)