Timeline for "Why the heck are the homotopy groups of the sphere so damn complicated?"
Current License: CC BY-SA 2.5
15 events
| when toggle format | what | by | license | comment | |
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| May 16, 2017 at 15:03 | comment | added | Dror Speiser | An anthropic prinicple argument for mathematics? | |
| Apr 28, 2010 at 22:52 | comment | added | Sean Tilson | I think the example of refining an invariant that is presented at the beginning of mosher and tangora exemplifies Andrews point, this is essentially the fact that being a module over the steenrod algebra is not as easy as being a graded ring. The more structure something has, the more structure the morphism must preserve and so the invariant is more refined. | |
| Apr 28, 2010 at 12:10 | comment | added | Andrew Stacey | @Pete: I suspect that what we mean by "complicated" and "simple" are slightly different things and that if we could figure out what the other means we'd probably end up broadly in agreement. But that's a conversation to have over a beer someday if we ever end up at the same conference! | |
| Apr 28, 2010 at 11:48 | comment | added | Pete L. Clark | @Andrew: regarding your comment about usefulness and power, I again (respectfully, of course) disagree. Complication certainly need not imply power, but simplicity does imply usefulness. For instance, by any reasonable measure, calculus and linear algebra are among our most powerful tools: the range of problems that they have been used to solve is unparalleled, and almost every research mathematician still uses these things in their work. The history of mathematics is rife with examples of simplifications of theorems and techniques which led to much wider usefulness. | |
| Apr 28, 2010 at 9:45 | comment | added | Lennart Meier | Another important role of stable homotopy groups is that they classify stable spherical fibrations. And like smooth manifolds have a tangent bundle, Poincare complex (i.e. "wannabe manifolds") have an associated canonical stable spherical fibration, the Spivak normal fibration, which is very important to decide whether a given Poincare complex posesses actually the structure of a manifold. | |
| Apr 28, 2010 at 9:44 | comment | added | Lennart Meier | @Pete: As far as I understand it, the homotopy groups of spheres are essential in high-dimensional topology. For example, the classification of exotic spheres (i.e. smooth manifolds homeomorphic to the sphere) would (essentially) be known if the homotopy groups of spheres would be known. This is only the tip of the iceberg - in whole of differentiable surgery theory the role of the stable homotopy groups of spheres (aka framed bordism groups) is essential. | |
| Apr 28, 2010 at 9:35 | comment | added | Andrew Stacey | (continued) so in algebraic topology, we use the simplest cohomology theory with enough power to "see" the structure that we are trying to study. The proof of the Kevaire invariant problem is a great illustration of this: ordinary cohomology theory couldn't see it, complex cobordism could but is very complicated, so they constructed a cohomology theory with enough complexity to see the map, but simple enough that it could be computed. | |
| Apr 28, 2010 at 9:32 | comment | added | Andrew Stacey | @Pete: to answer your second sentence (I'm not qualified to answer the last one): if one wanted to hair-split, one could distinguish between something's usability and its power. The simpler something is, the more usable it is. The more complicated something is, the more powerful. To be truly useful, it has to be usable and powerful so we need to balance the two. Cohomology is a great example: ordinary cohomology is simple, but misses so much as it is very weak. Complex cobordism is complicated, but powerful. In between, we have the Morava K-theories (and others). | |
| Apr 28, 2010 at 8:50 | comment | added | Pete L. Clark | I'm afraid I don't understand this answer at all. In order to be useful, things need to be sufficiently simple, not sufficiently complicated. And indeed the parts of algebraic topology which are most applicable in other areas of mathematics are all much simpler and better understood: e.g. (i) singular co/homology, (ii) fundamental groups, (iii) fiber bundles and characteristic classes, etc. Have the homotopy groups of spheres ever been applied to anything, including in algebraic topology itself? | |
| Apr 28, 2010 at 8:28 | comment | added | Tom Church | The cohomology of Eilenberg-Mac Lane spaces includes in particular all of group cohomology, and it's fair to say there's just as much unknowable blackness there as in the homotopy groups of spheres. There are even parallels: e.g. lots of people would like to know the cohomology of mapping class groups, we only recently know the stable cohomology, and there's so much unstable cohomology it seems plausible that we may never know what all the cohomology is. I'm sure hundreds of mathematicians could say the same about their own favorite groups. Not all EM spaces are simply-connected! =) | |
| Apr 27, 2010 at 14:28 | comment | added | Tyler Lawson | @Jeffrey: On the other hand, Thomas seemed to be making a statement about relative complexity compared to homotopy groups of spheres, which I would say is completely fair. | |
| Apr 27, 2010 at 9:26 | comment | added | Jeffrey Giansiracusa | If you don't think that Eilenberg-MacLane spaces have complicated cohomology then go ahead and compute the integral Steenrod algebra. I dare you. | |
| Apr 27, 2010 at 7:42 | comment | added | Andrew Stacey | What if I replaced "Eilenberg-Mac Lane spaces" with "infinite loop spaces"? I was contrasting homotopy with cohomology. When I hear "cohomology" then I automatically add "generalised" so when I wrote "EM spaces" I meant "as an example of representing a generalised cohomology theory" so was thinking, essentially, of infinite loop spaces. But I can see that it reads as though "cohomology" means "ordinary cohomology". Taken as a whole, the cohomology of infinite loop spaces is pretty complicated, as I'm sure you'll agree! I could edit it to make that clearer - would that be more palatable? | |
| Apr 27, 2010 at 7:29 | comment | added | Thomas Kragh | I really dont think Eilenberg MacLane spaces have complicated cohomology compared to the homotopy groups of spheres. They do play the dual role of spheres for cohomology, but for some reason they are much nicer. Otherwise I fully agree with the answer. | |
| Apr 27, 2010 at 6:59 | history | answered | Andrew Stacey | CC BY-SA 2.5 |