I would to like to know if there is an homeomorphism between the unit disk $D^2$ and $S^1\times I$, where $S^1$ is the unit circle. If I prove this homeomorphism I will be able to solve a question related which I'm struggling with.
No they are not: $S^1 \times I$ is a cylinder and is homotopy equivalent to $S^1$. (To see this, stamp on it harshly once with your left foot.)
But $S^1$ is not homotopy equivalent to $D^2$ hence also not homeomorphic to $D^2$. (Every homeomorphism is also a homotopy equivalence.)