Here's a very rough version of a rewrite of this article. What do you think about the level and scope? In many places I've gotten verbose by trying to keep an appeal to intuition going. Should I write out more things in symbols, or just try to sharpen the exposition? Jump in and help out!
A few things which I've been sloppy about and will try to fix up:
- Why we need a basepoint (and I often ignore it later).
- In writing intuition, pay attention to the words path vs. loop vs. class of loops.
- Discussion of SvK theorem is a bit useless for now.
- It would be nice to have an example of a space with finite fundamental group, or at least an element of finite order. is the obvious candidate since there's any easy geometric description of such an element, but that would probably merit a page of its own.
- Need pictures!
--MarkGall 01:21, 4 July 2009 (EDT)
Great work, but I'd like to improve the introductory definition to make it more understandable. Also, the group structure equation is broken towards the end. But your start on this is superb, and a fine place upon which to build.--Andy Schlafly 13:38, 4 July 2009 (EDT)
- I agree about the introduction. I'm trying to think of a better way to motivate it, by talking about distinguishing spaces by looking at holes in them or something along those lines. Whoops, didn't notice the broken equations. The error is that PNG didn't convert right. I'm pretty sure they're correct latex -- anyone know the adjustments that are necessary to get this on-wiki? Do I need to use an array instead of the cases environment? --MarkGall 13:46, 4 July 2009 (EDT)
This is an excellent effort on a really tough problem!
But you need more pictures. Even at the elementary level you are doing, the pace is too fast for the audience. They need to be shown, geometrically, what it means for one path to be homotopic to another. Then you can do shrinking, i.e. being homotopic to zero, and so on.
You need a picture of R^2, (well, a compact region of R^2, or, as mundanes call it, a rectangle :-) with points A and B, and a squiggly line from A to B. And another squiggly line from A to B. And somehow show, geometrically, how one would be continuously deformed into the other. Then do the whole thing again, with a rectangle that has a hole punched in it, and the two paths going on opposite sides of the hole. And argue, geometrically, that they can't be continuously deformed from one to the other because they can't cross the hole.
Only after doing this, do the algebraic presentation, with a path being a map from I^1 that has f(0)=A and f(1)=B, and the deformation (homotopy itself) being a map on I^2, matching one path on one edge of I^2 and the other path on the other edge.
I'm a newbie here, and may be able to help make pictures, but don't know how to upload them. --PatrickD
- I agree about the need for pictures! The one you suggest sounds as good as any. Since I mostly deal with homotopy of based loops I was planning to do a loop in R^2, and then a loop in R^2 - * which is not nulhomotopic. It probably makes sense to do your picture first, then mine. I'm new too -- I don't know how to upload pictures either (can someone enlighten us?). I think the picture should go after the intuitive take on simply connected spaces, and before a homotopy is really defined. What's the best way to create a picture for this? It would be nice to make the diagrams all in one program. A picture for the figure eight might be helpful also. The examples section should also describe the actual paths in the fundamental groups of these spaces -- I'll do that now. It also needs to be slowed down, generally. There are also a few random sentences that are too difficult. --MarkGall 14:28, 4 July 2009 (EDT)
Here's another suggestion:
The fundamental group is a basic construction of algebraic topology. For a given surface, such as a torus or even the (humorously named) dunce cap, an element of the fundamental group is the set of loops in the space that have the following characteristic: each loop can be continuously deformed to every other, a relation known as homotopy.
It's difficult to phrase this elegantly: the fundamental group isn't the set of all loops with the property that all are homotopic. Really, it's a set of equivalence classes: each element of the fundamental group is a bunch of loops, all of which are homotopic (i.e., an element of the fundamental group is an equivalence class of loops with respect to the relation of homotopy). What would you think about not even attempting to give a definition right away, and leaving this for the section on the circle, where it's examined a bit more? This seems likely to scare people off, when it really is a very intuitive thing! --MarkGall 17:16, 4 July 2009 (EDT)
Can we clean up the two equations?
Can we clean up the two equations that are producing the ugly PNG errors? Help is welcome on this!--Andy Schlafly 12:17, 5 July 2009 (EDT)
I'll look at it. Mark is the expert, but I probably know enough about the topic to fix this. However, it appears that it may be that he was using an esoteric feature of LaTeX that isn't really supported in Wikimedia. So the equation may need to be significantly reworked. Stay tuned. PatrickD 12:23, 5 July 2009 (EDT)
- Thanks much. I'm looking forward to the improvements, and learning from them.--Andy Schlafly 12:31, 5 July 2009 (EDT)
Well, he appears to be using the "cases" environment of LaTeX exactly correctly, according to Mittlebach and Goosens. My guess would be that the "vertical formatting environments" of LaTeX don't carry over into web pages, only the horizontal stuff. That doesn't surprise me, actually--Wikimedia is not in the business of making whole pages of pretty math copy. I will look around on Wikipedia to see if any of their math pages (some of which are very hairy) do what Mark is trying to do. If not, we'll have to make his definitions a little less pretty than he wanted. PatrickD 12:40, 5 July 2009 (EDT)
- Patrick, you're a genius! Your fix is fabulous!!!--Andy Schlafly 13:49, 5 July 2009 (EDT)
It turns out that you can't use "textrm" inside a cases environment; you have to use "mbox". Textrm is perfectly legal in real LaTeX, and is my preferred way of doing it, but it is not legal on a wiki. I will create a "hints for math authors" section on the math talk page. PatrickD 14:05, 5 July 2009 (EDT)
- Bravo! That's terrific, and thanks so much for explaining this.--Andy Schlafly 14:16, 5 July 2009 (EDT)
- Ah, that's it! Thanks! Do you have any suggestions for producing good illustrations? I've used Asymptote for similar pictures before and could put something together there, but if there's another preferred way I can do that instead. --MarkGall 14:24, 5 July 2009 (EDT)