## HOMEWORK: Yes, there will be homework. Not much, but hopefully steady. Not necessarily graded, but probably collected. All sort of flexibility can be worked out, but in order to really make things sink in we should really make a (collective) effort to keep up with it.

 DATE DUE: Jan 28th 1) Exercises 1.1, 1.3 page 21. 2) Prove that a product of two smooth manifolds can be given the structure of a smooth manifold. Describe a possible smooth atlas for the torus. Feb 4th Let f(x,y) and g(x,y) be two polynomials in two variables. By setting z = f(x,y) w = g(x,y) we get a smooth function from the plane (with coordinates x,y) to the plane (with coordinates z,w). What are the conditions on f and g for this function to descend to a smooth function from the projective line to the projective line? Can you generalize this construction to more than two variables and give  an example of a smooth function from P^3 to P^5? Feb 11th Exercises 3-1, page 64. Feb 18th Exercises 3-4, 3-5 page 64. Feb 25th Prove that the following conditions are equivalent: 1)     V is a smooth vector field on X 2)     Every component of every local representation of V is a smooth function. 3)     For every smooth function f: X->IR , the function Vf : X-> IR is smooth. Mar 4th Exercise 4.1, page 90 Mar 11th Exercises 5-1, 5-5, 5-18 pages 126, 128 Apr  8th Exercises 6-4,6-5 page 144 May 6th Compute the de Rham cohomology vecor spaces of: ·        The n-dimensional sphere. ·        Real projective plane. ·        The figure eight. ·        The union at a point of two manifolds. For the last two spaces, what we intend is “some manifold which is homotopy equivalent to…” but you can use the singular space in your constructions.