"Why can’t you have a retract of the disk onto its boundary circle?"

This is the question we will attempt to answer in this post.  Along the way, we will climb treacherous mountains and dive deep into dark waters — most frightening, perhaps: we will find things that we didn’t even know we were looking for.

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Sometimes, we can read a whole bunch of math and not get it until a simple picture is drawn.  For example, the notion of upper semi-continuous was not clear to me (at all) until a picture was drawn.  The "idea" of the mean value theorem is similar — drawing the tangent and parallel secant really shows what the theorem is trying to say. 

This post is to say in words, and then show in pictures, why we should have the fundamental group about a basepoint x_{0} of {\mathbb R}^{n} be trivial.  In other words, we’d like to prove \pi_{1}({\mathbb R}^{n}, x_{0}) = 0

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Just a quick note here; I wanted to mention a kind of “trick” when doing some algebraic topology problems.  A few problems from Hatcher reminded me of this, and it’s kind of strange that I didn’t bring it up before.

cut cut cut.

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I’m guilty of continuously mixing up homeomorphisms and homotopy equivalences.  It’s terrible, really.

So, in honor of this, let’s show that the circle and the figure 8 both viewed as subsets of {\mathbb R}^2 are not homeomorphic.  This is the same sort of deal as the plane verses line example in my last post, and there’s a really common trick when considering homeomorphisms: take out a point.

So, let’s take out a point of the circle and a point of the figure 8.  There’s actually a few cases to consider, since the figure 8 has two “kinds” of points on it.  But when we take a point out of a circle, what do we get?  We get a connected open line segment — or, at least, we get something homeomorphic to an open line segment.

Say we take out the “center point” of the figure 8.  What do we get?  We get two “cut” circles which are not attached to each other, and these are homeomorphic to two disjoint open line segments.  Since \pi_0 (the number of components) differs in these, they must not be homeomorphic (see the end of the last post if you don’t understand why).

But what if we take out a point that’s not a center point of the figure 8?  We get a circle with horns — something that kind of looks like the Taurus astrology sign.  So, we’ve reduced this to proving that the circle and the line are not homeomorphic.  This is an exercise for the reader.

What if someone came up to you and asked you if there were any way to take the real line and keep bending it around until we got the real plane.  I mean, it doesn’t seem all that far-fetched: maybe we could just keep bending it in half until we got something that looked like the plane!  Can we show that this is impossible?  After the cut.

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Last post I talked about fundamental groups and some of the stuff that we can do with them.  For example, we have that a disk is different from a circle, since the disk has a trivial fundamental group and the circle’s fundamental group is the integers.

\pi_{1}(D^1) = 0

\pi_{1}(S^1) = {\mathbb Z}

But is this enough to say anything about how truly different the circle and the disk are?  Yes, the circle has a hole, and the disk does not have a hole, but is there some kind of continuous deformation that let’s a circle become a disk?  Or one that let’s a disk become a circle?  Maybe we could pull at the disk and spin it around a lot until it becomes a circle, continuously!  We just don’t have the machinery at this point to be able to prove that the circle and the disk are not similar in the sense that there is no continuous deformation between them.  Upsetting, but true.

Before we can build up this machinery, and can truly think about spaces and fundamental groups as truly related species, we need to build up something called Category theory.

After the jump.

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Prereqs: You might want some point-set topology, and a basic introduction to algebraic topology.  This post is not good to learn from (for that sort of thing, I recommend the wonderful Hatcher text, which essentially starts where point-set leaves off, or the lectures by john baez which are wonderful but contain a little bit more category theory than is perhaps necessary to a beginner.) but is just a review for people who have seen algebraic topology but perhaps forget a little of what went on.

After the jump.

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