Trefor at 04:05, 11 January 2008

2008-01-11T04:05:16Z

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==Class Notes==
<span style="color: red;">The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.</span>

Recall from last class we are going to prove the following lemma:

'''Lemma 1'''

Let <math>e:\mathbb{R}\rightarrow S^1</math> be <math>e(x) = e^{2i\pi x}</math>

Then every path <math>\gamma:[0,1]\rightarrow S^1</math> with <math>\gamma(0) = 1</math> has a unique lift <math>\tilde{\gamma}:[0,1]\rightarrow\mathbb{R}</math> such that

0)<math>\tilde{\gamma}(o) = 0</math>

1) <math>e\circ\tilde{\gamma} = \gamma</math>

'''Definition 1'''

A covering map is a map <math>p:X\rightarrow B</math> such that

0) B is connected and locally connected

1) The map <math>X\rightarrow B</math> is locally a map <math>F\times B \rightarrow B</math> where F the "fiber" is a discrete set

More precisely this condition means that every point in B has a neighborhood U such that <math>p^{-1}(U)</math> is <math>F\times U</math> for some such F

Note 1: As is assumed in this section of the course, our map p is continuous, our spaces have base points <math>p(x_0) = b_0</math>. Furthermore we mean, as always, pathwise connected when we say connected.

Note 2: Recall that connected does NOT imply locally connected. Indeed recall the Cantor Comb from a previous class.

'''Examples of covering maps'''

1) <math>X = F\times B \rightarrow B</math> is trivially a covering map

2) The map from last class, <math>e:\mathbb{R}\rightarrow S^1</math>

3)<math>S^2\rightarrow S^2 /\pm 1 \cong \mathbb{R} P^2</math> where the identification glues antipodal points on the sphere. Here <math>F = \mathbb{Z}/2</math>. I.e., there are two preimages of each point (the two antipodal points)

4) Likewise, <math>S^3\rightarrow\mathbb{R}P^3</math> etc...

5) Consider a bunch of identical floors of "abstract" parking garages all on top of each.

'''Aside 1'''

Recall SO(3) the group of orientation preserving rigid rotations of <math>\mathbb{R}^3</math>.

'''Claim:'''
<math>SO(3)\cong\mathbb{R}P^3</math>

'''Proof:'''

<math>SO(3) = \{v\in\mathbb{R}^3\ |\ ||v||\leq\pi\}/</math> (if <math>||v||=\pi</math> then <math>v\sim -v</math>)

This is because we know that any rotation occurs about an axis. So one can encapsulate the information of a given rotation by a vector where the orientation of the vector denotes the axis and the length of the vector denotes the amount of rotation. However, a rotation by <math>\pi</math> degrees does not matter which direction v or -v we do this in, hence the identification. But the resulting set is precisely <math>\mathbb{R}P^3</math>.

We can now generalize Lemma 1 with our new concept of covering maps opposed to just the map e (which we now know is a specific example of a covering map)

'''Lemma 1 (general):'''

Let <math>p:X\rightarrow B</math> be a covering map. Then every path <math>\gamma:[0,1]\rightarrow B</math> with <math>\gamma(0) = b_0</math> has a unique lift <math>\tilde{\gamma}:[0,1]\rightarrow</math> X such that

0) <math>\tilde{\gamma}(0) = x_0</math>

1) <math>p\circ\tilde{\gamma} = \gamma</math>

'''Proof'''

Cover B with good sets <math>U_{\alpha}</math> (where by "good" we mean the inverse images look like products)

Now <math>\gamma([0,1])</math> is covering by these and so [0,1] is covered by <math>\{\gamma^{-1}(U_{\alpha})\}</math>

The Lebesgue Lemma then implies that there exists N such that [i/N, (i+1)/N] such that <math>\gamma([i/N, (i+1)/N])</math> is in one of the <math>U_{\alpha}</math>

The lemma then follows by an inductive argument which is essentially just bookkeeping. Very loosely it is as follows: In the interval [0,1/N] we know that gamma at 0 is <math>b_0</math> and have to uniquely define <math>\tilde{\gamma}(0) = x_0</math>. We thus get <math>\tilde{\gamma}</math> in this interval recalling that for continuity we have to keep <math>\tilde{\gamma}</math> in one "level". We thus proceed inductively through the other subintervals of [0,1] and it remains to be checked that everything does in fact work out as we expect.

0708-1300/Class notes for Thursday, January 10 - Revision history

Trefor at 04:05, 11 January 2008