My topology teacher (in a lesson about the fundamental group of the circle) says that
$$F: S^1 \times [0, 1] \rightarrow S^1 \\ (z, t) \rightarrow z^{1+t}$$
is not an homotopy between $f(z) = z$ and $g(z) = z^2$ because F is not well-defined ($S^1$ here is seen as subset of $\mathbb C$).
Why? Complex exponentiation is not well-defined?
Intuitively I would do it writing the complex number in polar form $z = e^{i\theta}$ and evaluating $z^{t+1}= e^{i(t+1)\theta}$. I suppose that the problem is that $\theta$ is not unique, but cannot we avoid this by choosing always angles $\in [0, 2\pi)$?
The best approach to define exponentiation with arbitrary exponents is via logarithms, i.e., $a^b:=\exp(b\ln a)$. And in the complex case, we'd define the logarithm best as integral of the reciprocal function $z\mapsto \frac 1z$. As However, the result of this definition of $\ln $ is not well-defined in the sense that it depends on the region $\Omega$ we define it on. That region must be simply connected and not containing $0$ to allow our integration attempt to work. This way, different regions may produce different results. We speak of different branches of the logarithm. It turns out that complex logarithm is best be viewed as defined up to a multiple of $2\pi$ only. The same then applies to general exponentiation.
As said above, we can fix a branch consistently only if we are working on a simply connected region not containing $0$. In the problem, we want to define exponentiation on $S^1$ which is not contained in such a region.