Timeline for Classification problem for non-compact manifolds
Current License: CC BY-SA 2.5
13 events
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| Jan 13, 2010 at 2:14 | comment | added | Jason DeVito - on hiatus | That's amazing, I didn't realize. I also didn't realize how many examples of what I thought were "classification" (according to my definition) were actually examples of "classification" according to yours. Thanks for the enlightenment! | |
| Jan 13, 2010 at 1:24 | comment | added | Ryan Budney | Small follow-up. There is an algorithm to determine if simply-connected manifolds are homeomorphic. Nabutovsky and Weinberger Contemp. Math. 1999 245-250. | |
| Nov 18, 2009 at 4:59 | comment | added | Ryan Budney | Stiefel-Whitney classes are similarly combinatorial, as their original definition is obstruction-theoretic. | |
| Nov 18, 2009 at 4:58 | comment | added | Ryan Budney | Homotopy classification does mean classification in my strict world, too, it's just not stated that way very often nowadays, but that's more a historical anomaly as far as I can tell. Grassmannians have Schubert cell decompositions. So homotopy-classes of maps from a CW-complex to a Grassmanian can be separated via obstruction theory, which boils down to CW-homology issues. So if you restrict to some some reasonably combinatorial class of CW-complex, like PL-CW or such, you can systematically determine when two vector bundles are isomorphic in this manner. | |
| Nov 16, 2009 at 19:59 | comment | added | Jason DeVito - on hiatus | I just wanted to add that I definitely think your definition of "classification" is far superior to mine (in terms of usefulness and information content), but that I think my way (in the sense of reducing to a problem in another field) is, if not MORE commonly used, then at least commonly used. If it is more commonly used, I'd guess it's simply because classification, as I've defined it, is MUCH easier to achieve than as you've defined it. | |
| Nov 16, 2009 at 19:50 | comment | added | Jason DeVito - on hiatus | We seem to disagree on terminology, which is, of course, fine. Is there a polynomial time algorithm for determining whether or not the second Stiefel-Whitney classes of two compact 1-connected 5 manifolds are the same? If we don't know how complicated this problem is, should we wait on calling the "Barden-Smale classification of 5-manifolds" this, and instead call it the "Barden-Smale equivalence of 5-manifolds"? I still feel like "classification" often means "reduce to a problem in another field". See for example, the "classification" of vector bundles in terms of homotopy theory. | |
| Nov 16, 2009 at 18:33 | comment | added | Ryan Budney | There's a big gap between your two examples -- given two triangulated 4-manifolds, determining whether or not their intersection forms are isomorphic, there's a polynomial-time algorithm for that. The isomorphism problem for group presentations isn't so easy. I take a "classification" to mean: you have a collection C with an equivalence relation ~ on it, and you want a list of one representative from each object of C/~ and a decision procedure to determine, given an object c from C, which representative it is isomorphic to. What you call classification I'd call an equivalence. | |
| Nov 16, 2009 at 13:18 | comment | added | Jason DeVito - on hiatus | The point I'm trying to make is that the term "classification" is often used once a (reasonably computable) complete set of invariants has been found. Freedman's classification of 1-connected (smooth) compact 4-manifolds up to homemomorphism depends solely on the intersection form, a readily computable object. This is considered a "classification" regardless of how easy it is to compare 2 intersection forms to see if they really are the same. In short, often "classification" means "reduced to a problem in another field of mathematics", even if this new problem is (known to be) hard. | |
| Nov 16, 2009 at 5:19 | comment | added | Ryan Budney | Rereading your post I think you're blurring the line between "a presentation for a group" and "a group". The former is readily computable for manifolds via Seifert - VanKampen, the latter isn't. | |
| Nov 16, 2009 at 5:01 | comment | added | Ryan Budney | I disagree. The Seifert - VanKampen theorem doesn't compute the fundamental group, it computes one presentation for it from the ingredient presentations and amalgamation maps. I think the more you stare at impossibly ugly presentations the more you'll be swayed by this point of view. I'd rephrase what you've been saying as something like this "given an oracle for the isomorphism problem between finitely presented groups, is there an algorithm to classify compact n-manifolds?" Off the top of my head I don't know the answer to that, but I suspect the answer is no. | |
| Nov 16, 2009 at 0:36 | comment | added | Jason DeVito - on hiatus | I agree (especially that "classification" is somewhat vague), but there is a difference between a computable invariant and comparable invariants. For example, the fundamental group is "reasonably computable" via van Kampen's theorem, but telling when 2 groups are isomorphic is algorithmically hard. Thus, given a manifold M, I can output the fundamental group, but given 2 manifolds, I cannot (algorithmically) distinguish them via their fundamental groups. That said, I think almost anyone would be willing to accept "fundamental group" on any list of "reasonable" invariants. | |
| Nov 15, 2009 at 22:33 | comment | added | Ryan Budney | If classification simply means "a short list of invariants for n-manifolds which [distinguishes] them up to diffeomorphism", then we already have such a list -- the diffeomorphism type of the manifold is the unique invariant you're after. The problem with this "classification" is we don't know how to compute it. So what does "classification" mean? I think for many people (most?) a classification requires computability starting from some kind of reasonable prescription for the type of objects you're interested in -- triangulations or surgury typically for manifolds. | |
| Nov 15, 2009 at 18:40 | history | answered | Jason DeVito - on hiatus | CC BY-SA 2.5 |