Homework for June 16

What is the number of  surface words  on an alphabet of 2n letters  x1, x2.. xn, X1, X2.. Xn?

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.

Notes from First Meeting

Here are some of the things I jotted down from last week's meeting: (Feel free to add/correct things)

Topology

What are some of the motivations for the study of topology?
There are some problems that do not depend upon lengths, angles, etc. but on more basic properties (such as connectedness etc.)
Example: Seven Bridges of Konigsberg problem.

Notion of a "surface"

Division into two broad categories:
I.) Configurations of closed curves and geodesics on surfaces
II.) Self-transformations of surfaces and their effects on curves

Surfaces we study are characterized by whether they are orientable (do not contain Moebius strips), the number of boundary components, and the Euler characteristic.

We wish to find invariants that determine the structure of surfaces.

A minimal surface is one where the mean curvature is zero (the radii of curvature for positive and negative sections sums to zero)

An example in real-life is soap bubbles.

For topology, we consider two surfaces the same if we can construct a continuous bijection from one to the other. This continuous bijection between surfaces is called a homemorphism; the two surfaces are said to be homeomorphic.

Manifolds are surfaces which locally take the appearance of Euclidean space. (Tangency of planes at every point of the manifold.)

We can consider curves on these manifolds. While there are an infinite number of curves up to deformation, we are interested in minimal complexity curves, which minimize intersection points.

We call the number of handles in a surface the genus of the surface. For example, a torus (a donut) has genus 1.

We can consider the behavior of curves upon cutting part of our surface.

So broadly, we will study:
surfaces, closed curves of surfaces, classes of closed curves of surfaces, and minimal intersection/self-intersection

We can transition into geometry by endowing our surfaces with further structure, such as a metric (notion of distance).
By adding a metric, we can discuss geodesics, which are the shortest paths between two points on a manifold. Unlike their counterparts in the plane (lines), geodesics can self-intersect.

We can now consider configurations that minimize self-intersection and the distances based on our metric.

We can use geometric arguments to demonstrate that certain configurations are not realizable as geodesics for a given metric, while others can be realized for the metric. These arguments make use of geometric properties though (such as the curvature of the surface).
We studied a particular example (the pair of pants) that was a hyperbolic surface (negative curvature).

Finally, we discussed a method of constructing bounded surfaces from polygons. We take an n-alphabet and consider a 2n-gon. Each letter of the alphabet must appear once. Furthermore, each letter has an associated term. So the letter, a, also has an associated term, a'. To construct the surface, we glue together each letter with its associated term.
For example, we can have our alphabet be {a,b}. (So n=2.) Depending on how we label the sides of our polygon, we can obtain different shapes. So we could have different "surface words" that yield different surfaces. Some surface words are aba'b' and aa'bb'.

We can also consider a different type of word. Given a surface, characterized by some surface word), we can consider different curves on the surface. We can characterize these curves based upon what sides of the polygon (after it has been glued together) they pass through in what order. We can reduce these words by noting that no letter can appear next to its associated letter: a should never appear next to a' in these reduced words.

Generally, we obtain the following statement:

Every free homotopy class of orientable closed curves on a surface with boundary can be labeled by a unique (up to cyclic permutation) reduced word in the alphabet of the surface word.

Readings

For hyperbolic geometry, you can start by reading  some  of 1. and then some of 2. (and repeat 1,2,1,2...)


For topology in dimension three (which depends very much on hyperbolic geometry), start reading 5.  Afterward (possibly much later depending on your knowledge of topology) read some of 3. then some of 4.; and again (3,4,3,4..) back and forth.
 

1.  Hyperbolic geometry, the first 150 years , by John Milnor
2. Hyperbolic manifolds according to Thurston and Jorgensen by Michael Gromov.
3. Towards the Poincaré Conjecture and the Classification of 3-Manifolds,  by John Milnor
4.  Recent progress on the Poincare conjecture and classification of 3-manifolds.by John Morgan
5. "The Poincare conjecture" by John Milnor, and the "abstract". 
6. A lecture about the Poincare conjecture by Curt McMullen

 

euler characteristic

algebraic topology attaches algebraic structures and invariants of these to spaces. this is usually done by dividing the space into chambers which are themselves simple like rooms.one has besides the chambers, the walls , the corner lines and the corner points. One calls these respectively in reverse order. 0 cells. 1 cells 2 cells 3 cells etc. the goal is to find structures that are essentially unchanged when a space divided into cells is subdivided into smaller cells.

the earliest and most famous such invariant is the alternating sum of the number of k cells:

#of zero cells - #of 1 cells + #of 2 cells ... which sum terminates for a space divided into finitely many cells.

there is exactly one more independent topological invariant of this simple form. namely it only depends on the number of cells of each type and not how they hook together...

problem: notice it and prove this claim.

Undergraduate and Graduate Worshop on Topology and Geometry, Summer 2010

This is the blog of the workshop. Every participant is invited to contribute.