Notes from 07/ 14/10

Here are some notes I took on the past Wednesday.
Professor Sullivan continued his topic about "the fundamental group". To construct a fundamental group using the universal covering space, we first need following definitions.
[Definition] A metric space is an ordered pair (M,d) where M is a set and d is a metric on M, that is a function

d: M*M -> R

such that for any x, y, z in M
1. non-negativity: d(x,y) >= 0
2. identity of indiscernibles: d(x,y) =0 iff x = y
3. symmetry: d(x,y) = d (y,x)
4. triangle inequality: d(x,z) <= d(x,y) + d(y,z)

[Definition] Given a metric space (X, d), a subset U is called open iff for any element u in U, there exists a set B(u,r) = {vd(u,v)<=r}.

Now the metric space X is also a topological space.

We will assume that X is locally simply connected. Then we can construct a universal covering space and a covering map to study some important consequence of the fundamental group.
Step 1: Let X be a topological space and choose a base point x in X.
Step 2: For each point x' in X, consider all the continuous paths (i.e a continuous map from [0,1]to X) from the base point x to x'. We say that two such paths are equivalent if they can be deformed to one another without moving the start and end points and without breaking. These two paths are the same "rout".

[Definition:] The set of all such routs is the universal covering space X~. The covering map is the map from X~ to X that maps any rout, from x to x', to x'.

[Definition:] If X is a metric space with metric d, then the length the curve g: [a,b] -> X is
length(g)= sup{These sumnation from i=0 to n-1 of d(g(ti),g(t(i+1)): n is a natural number and a=t0 < t1<... < tn=b}.

[Definition:] A metric space is a length metric space if the intrinsic metric (i.e. the infimum of the length of all paths from one point to the other) agrees with the original metric of the space.

Unsurprisingly, the shortest paths are constant paths and they are nowhere differentiable. Also, since we mentioned last time, the covering map is a local isometry. An isometry is a distance-preserving map between two metric spaces so it is easy to understand that the covering map preserves the path lengths.

Next, we came to a theorem which allows us to do much with the covering spaces.

[Theorem (Homotopy Lifting):] Suppose p: Z -> X is a covering map for a space X. Let f : I^n -> X be a map from the unit n-cube to X, and F: I^ (n+1) -> X a homotopy of f to another map f' : I^n -> X. Suppose g : I^n -> Z is a map satisfying p* g =f. Then there exists a unique map L: I^(n+1) ->Z satisfying LI^n = g; p*L = F

It says that if z in Z is a point such that p(z)=x, and g is a path in X starting at x, then there is a unique path g' in Z starting at z such that p of g'=g. We say that a path in X has a unique lift to Z, once the starting point of the lift has been chosen.

[Definition:] The product of the two paths is called action.
More precisely, Let G be a topological group and X any topological set. If there is a continuous function f: G*X -> X, where G*X is given the product topology such that f(1,x) = x as well as f(g1g2,x) = f(g1, f(g2,x)), then the function is called the action of G on X.

[Theorem:]1. The fiber over the base point is a group.
2. This group Gamma_x acts on every fiber Gamma_x'.

Gamma_x * Gamma_x' -> Gamma_x'

3. The action is by left translation in the Gamma_x.
4. The action on each Gamma_x' is without fixed points unless acting with the identity.

After the lecture given by Professor Sullivan, the undergrad reported their progress during the week.

After having a good understanding of the problem, Keren started calculating the geometric length of a given geodesic in a given hyperbolic surface and study its distribution. She explained that in the upper-half of the Poincare disk model, in order to compute the geometric length, we needed to find out the eigenvalues of the transform matrices and then compute its trace. Actually she is now trying very hard to get these eigenvalues.

Ren's project is about exploring surface transformation. Start with three holes in the plane with a circle enclosing the right two. Then switch the points: left over the middle and right over the middle and keep doing those two steps. The numbers of the times that the loops pass under each hole are always the Fibonacci numbers. The sequence of lengths of the iterates of the curve grows exponentially with rate the Golden ratio.

Also, one of the properties of this surface transformation can be represented by the pattern of the curve words which are composed of A,a,B,b,C,c. The pattern is the following: In all the even steps, we change all the A's (a's) to ABa's (Aba's) and change all the B's (b's) to A's (a's). In all the odd steps, we change all the B's (b's) to C's (c's) and change all the C's (c's) to cBC's (cbC's).

Anand continued his project on finding the minimum intersections of all the cyclic reduced words of a given lengths with another fixed surface word. In the beginning of this project, Anand started with some simple surfaces. Next, he was collecting data under each condition step by step and tried to modify his problem with the help of computer program.