
WHEN: MWF 11 – 11:50 am
WHERE: WEBER 223 // in the oval with good weather
WHAT: A graduate level introduction to the theory of
differentiable manifolds.
TEXTBOOK: Introduction
to Smooth Manifolds (John
M. Lee)


DATE DUE: 

Jan 28^{th} 
1) Exercises 1.1, 1.3 page 21. 
Feb 4^{th} 
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 11^{th} 
Exercises
31, page 64. 
Feb 18^{th} 
Exercises
34, 35 page 64. 
Feb 25^{th} 
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 4^{th} 
Exercise 4.1, page 90 
Mar 11^{th} 
Exercises 51, 55,
518 pages 126, 128 
Apr 8^{th} 
Exercises 64,65 page 144 
May 6^{th} 
Compute the de Rham cohomology vecor spaces of: ·
The
ndimensional 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. 







