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.