What is $S^1 \times S^3$ with an open disk removed?

Question

I am trying to calculate the de Rham cohomology of the connected sum of $S^1 \times S^3$ with $\mathbb CP^2$. But I have trouble in figuring out what is $S^1 \times S^3$ with a disk removed. If I have that figured out, I can proceed using Mayer Vietoris, since $\mathbb CP^2$ with a disk removed is simply $\mathbb CP^1$.