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.
My attemptation:
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 .
Best Answer
For any $n$ there exists a self map of $S^n$ which is homotopic to the identity and has exactly one fixed point. The map is obtained by starting with the homeomorphism $\mathbb{R}^n \rightarrow \mathbb{R}^n$ given by $(x_1,\dots,x_n) \rightarrow (x_1+1,\dots,x_n)$ and applying one point compactification. This evidently has a single fixed point, and it is homotopic to the identity because the original map $\mathbb{R}^n \rightarrow \mathbb{R}^n$ is homotopic to the identity through homeomorphisms.