It is well-known that if a mapping $f:S^2\to S^2$ is homotopic to identity, then it must have a fixed point. I am wondering that if there exists a map on $S^2$ which is homotopic to identity and has exactly one fixed point.
I try to construct a vector field over $S^2$ which has a single zero. The flow $f_t$ generated by this vector is homotopic to $id$. However, I can't ensure that there exists some $t$, such that $f_t$ has only one fixed point and is exactly where the corresponding vector field vanishes .