Notes from 6/2/10

Thought I'd jot down some of the notes I took from last week's meeting:

We started with some more formal definitions.

Definition: A subset U of Rn is called open if for each element x in U, there exists some n-ball centered at x and contained in U. U is an open set.

Definition: A neighborhood of a point x in Rn is a subset X of Rn such that X contains an open set, U, containing x.

Exercise: The collection of open sets in Rn is:

1. Closed under arbitrary union

2. Closed under finite intersection

3. The null set is open, Rn is open

Any neighborhood X of x is also a neighborhood for all y in V for some open set V which also contains x.

4. Hausdorff Property of Rn: If x is a distinct point from y, then there exist disjoint neighborhoods of x and y.

Fact: A function f which maps Rn to Rm is continuous if and only if fro all x in Rn and any neighborhood V of f(x) in Rm there is a neighborhood, U, of x in Rn so that f(U) (the restriction of the function to the subset U) is a subset of the neighborhood V in Rm.

Definition: A Hausdorff topological space X is a set of points provided with a family of subsets called "open sets" which satisfy properties 1,2,3,4 listed above.

We say two spaces X1 and X2 are homeomorphic if there is a bijection carrying the open sets of X1 to the open sets of X2.

Definition: An n-manifold is a Hausdorff topological space so that each point has a neighborhood homeomorphic to a neighborhood of a point in Rn.

There is a deep result that n is well-defined because of invariance of domain.

Definition: A closed manifold is a manifold that can be covered by finitely many finite neighborhoods.

After these series of definitions, we moved into the problem of classification of surfaces and curves by words.

Given a surface with boundary given by a surface word (which is obtained from constructing the surface by gluing a polygon), there exists a bijection between the free homotopy classes of curves of S and the cyclic reduced words in the alphabet of the surface word.

Let us take our surface word to be aAbB. Some curves are a, aB, AB, etc.

We are also interested in reduced words, which do not have a letter and its associated gluing letter next to each other in any cyclic permutation of the word.

For example, abA is not reduced, since it has a and A next to each other in another cyclic permutation of the word (say Aab).

Definition: A primitive word is a word that is not a power of other words.

We see for instance that the word abab is not a primitive word, as it is (ab)^2.

We can then ask questions about minimal self-intersection. For instance, fixing a particular surface, what are the possible self-intersections of a word of length 1? We then created a small chart characterizing some of the possibilities for curves.

We can ask a series of similar questions. For example, instead of considering just self-intersections, we can take a pair of curves and consider how many total intersections occur.

We also consider the idea of multiwords. Multiwords have a length that is the sum of the length of their component words. For example, some multiwords of length 2 might be:

{ab}, {a,b}, {bA}, {a,a}.

As with other words, we can attempt to classify these multiwords too by self-intersections.

Later on we shifted to a discussion on isometries associated with hyperbolic geometry. We discussed two different models, the Poincare disk model and the upper half-plane model.

Please let me know if anyone has any corrections/things I should re-word to these notes.