Talk:Fundamental group

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. <math>\mathbb R \mathbb P^2</math> 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