What is wrong in saying that the map $F:\mathbb{S}^1 \times I \to \mathbb{S}^1$ defined by $F(z,t)=z^{t+1}$ is a homotopy from $f$ to $g$? Let $\mathbb{S}^1$ be the unit circle of complex plane and $f,g:\mathbb{S}^1 \to \mathbb{S}^1$ be two maps defined by $f(z)=z$ and $g(z)=z^2$. What is wrong in saying that the map $F:\mathbb{S}^1 \times I \to \mathbb{S}^1$ defined by $F(z,t)=z^{t+1}$ is a homotopy from $f$ to $g$?

Can someone tell me please what is wrong ?thanks in advance.

Raising a complex number to a non-integer power is more complicated than you're realizing. The "function" $z^{t}$ is really multivalued when $t\notin\mathbb{Z}$, and even after choosing a branch, it won't be continuous.
