Homotopy Type of Wedge Sum
I wish to show that the solid torus $\overline{B^2} \times S^1$ without a point of its interior is homotopy equivalent to $S^2 \vee S^1$.
- Intuitively, these spaces are even homeomorphic - is this true?
My idea is that $\overline{B^2}$ is homeomorphic to $S^2$ without the point that we "remove" with the wedge sum, and the $S^1$ in the torus can be identified with the $S^1$ in the wedge sum.
- If it is true, is it common to give an explicit formula for the homeomorphism between the two spaces?
Thank you already!
Remark: We have not covered Kampen's theorem yet.
I sketched the process as follows:
