
 Topology, Math 5520, Winter 2001 
  Differential topology studies manifolds, spaces on which it is possible to say not only that something is continuous, but that it is smooth (i.e., differentiable). 
  It is a very odd topology at first encounter, in comparison with the topologies we are familiar with in the line, the plane, etc. It has proven to be quite a powerful tool, however, playing an important role in the proof of Fermat's last theorem, for example. 
  For example, the proof using algebraic topology of the Brouwer Fixed Point Theorem, that any continuous function from an ndimensional ball to itself must have a fixed point, reduces to the simple algebraic fact that the identity function from the integers to the integers is not a constant function. 
 www.math.wayne.edu /~rrb/classes/m5520w01/Day1.html (1643 words) 
