Re: de Rham cohomology of R^2 minus two points



Lukas Horosiewicz wrote:
An exercise in Bott and Tu reads "Compute H^*(R^2 - P - Q) where P and
Q are two points in R^2. Find the closed forms that represent the
cohomology classes. Could someone provide helpful guidance?


Consider the simpler case of R^2 \ (0,0).

Think of the complex plane C, and recall that the log function
is given by the integral:

log(w) = \integral_p( dz / z )

where the path p is *any* path in C \ 0, connecting 1 to w.

Next, recall that log(z) has an indeterminacy of 2 n pi i, where
n can be changed at will by introducing loops around 0.

Alternatively, recall the Cauchy index theorem.

Once you've done all that reminiscing, try to find a single
closed 1-form that generates the 1-dimensional deRham cohomology
of R^2 \ (0,0). If done correctly, you will almost surely
recognize this 1-form.

Finally, notice that R^2 \ {P,Q} maps into both R^2 \ {P}
and R^2 \ {Q} naturally. Think "pullback".

That's as much hint as I can provide without writing down
the solution for you.

Dale.
.