Talk:Branched surface

An ok start, but there are still a lot of problems with this page. First, the description of branched surface is wrong. The given description is that of a simple branched surface. In general you need to allow immersed curves to be branch locii. Next, no mention is made of smoothness at the branching points, or for train tracks, at the switches. The branches should be C^1 fitting together in a certain way at the switches. Also, train tracks usually, in most contexts, have conditions on their complementary domains. In other words, "train track" does not just refer to the graph, but a certain kind of embedding into a surface. On tori, one allows digon regions, but on higher genus surfaces, one doesn't, for example. The comment that weights for train tracks are usually natural numbers is strange. They usually aren't natural numbers, since train tracks are used most heavily with laminations. Finally, it is wrong to say a train track or branched surface carries a foliation since a foliation fills up the whole manifold. Such branched thing carry laminations. --C S 22:12, Dec 14, 2004 (UTC)


 * Whew! Okay. It needs work. I didn't include a bunch of stuff ($$C^1$$ at the branch points, no digons) either because I forgot to (e.g., no digons) or because I didn't bother (e.g., smoothness). But I did strive for accuracy in what was included, so I apologize for my inaccuracies. perhaps you can help polish the article. As to the natural-number thing, I didn't say they're "usually" natural numbers, but that they're "often but not always" natural numbers. My intent was to emphasize the "not always", but I guess that didn't come across well. (I'm changing that.) And as to carrying a foliation, I'm not sure you're right: Consider the foln of R^3 by vertical planes. Would one vertical plane (a branched surface) not carry the foln? (I'm honestly not sure.)  &mdash;msh210 22:49, 14 Dec 2004 (UTC)


 * That's why it's important to at some point, include an actual precise definition of "carry". (Of course, it's easier to criticize than fix the problem :-) but now that I have more time I will pitch in; I've been very busy the last month or so). The precise definition of carry is to take a regular neighborhood of the branched surface that looks locally like the second picture on Oertel's webpage.  This has a foliation by vertical segments.  A surface is carried by the branched surface if it is contained in a neighborhood of this type and is transverse to every vertical segment (it may avoid some of them).  If it also intersects every such segment, the branched surface fully carries the surface.  The definition for train tracks carrying curves is similar.  You make each branch a long rectangle (with vertical segments transverse to the direction of the branch) and at the switches, instead of a cusp, you have a rectangular corner (hard to describe, but if you look on Google for fat train track or train track neighborhood, you may find some pictures...).  So to answer your question, no, the one plane would not carry the foliation.  The idea is that a branched surface/train track carries something if it runs "close" along the branched surface/train track.  That is what is meant by a lamination looking like a train track/branched surface to a myopic person.  --C S 23:31, Dec 14, 2004 (UTC)

To see a nice local picture of a branched surface, see Oertel's webpage. Maybe we can use his pictures. --C S 22:20, Dec 14, 2004 (UTC)


 * Or just re-draw the same thing. It's a standard enough picture that I don't think it'd be a copyright issue. (Ianal.)   &mdash;msh210 22:49, 14 Dec 2004 (UTC)


 * Hehe; yes, it's definitely not a copyright issue! I was just saying if we were lazy, we could get permission to use his graphics.  To describe "carry" we will definitely need a picture like Oertel's second one.  Anyway, what are you using to draw your pictures?  I recommend using Adobe Illustrator; the pictures come out pretty crisp with it.  --C S 23:31, Dec 14, 2004 (UTC)

Weights
The weights cannot be negative! I've added that to the article, but now, unfortunately, the picture of a weighted train track is wrong. The ActiveDiscuss tag is a good idea; at least nobody will write their dissertation using this article while we get it into shape :-) --C S 23:36, Dec 14, 2004 (UTC)

Branched manifold?
I'm not sure if there is such a general concept that has a precise, standard definition. As I pointed out above, a branched surface isn't merely a trivial extension of a train track: its branch locii can be immersed circles with double points. I'm familiar with train tracks and branch surfaces from 3-manifold theory, but I don't know about higher dimensions. Even these two cases make clear that the general definition for higher definitions would be more complicated. I'm not sure what such a definition would be, or if it's even useful. --C S 23:50, Dec 14, 2004 (UTC)

train track is singular
I just noticed that the article is written in a way (especially with the comment on the redirect from 'train track') that suggests that a 'train track' does not exist, and is always referred to in the plural form 'train tracks'. This is incorrect. When talking about one train track, well, it's just 'track' not 'tracks'. That is how the math terminology is. One only says 'train tracks' when refer to them in the plural. I've changed the article accordingly. This may be a more important comment if train track gets its own page. --C S 09:31, Dec 15, 2004 (UTC)