Calculate the De Rham Cohomology of de solid ball with out a solid totus inside.

Question

I am trying to compute the Rham Cohomology groups of $ \overline B_1(0) \; / \; T_\epsilon$: The solid ball of radio 1 in $\mathbb{R}^3$ without a little solid Torus inside.

I guess i should use Mayer Vietoris Theorem. I have been trying with different open covers, but i can not find the proper cover.

Any idea?

Answer 1

Hint: if $B$ is the ball, then consider the covering of $B$ which by the complement of the torus in $B$ and the torus (after enlarging these sets a bit so that they become open)