Fundamental Group of Circle Generator Textbook Typo?

Question

I'm confused about what the generator is for the fundamental group of a circle at point $b_0$. That is, what is the generator for $\pi_1(S^1, b_0)$. Is it $e^{2\pi i (t_0 + t)}$ for $t \in [0, 1]$, where $b_0 = e^{2\pi i t_0}$? My book says it is $e^{2\pi i t}$ for $t \in [0, 1]$, but this seems to have no dependence on $b_0$. Just want to see if this is a typo. Thanks in advance!

Answer 1

Looks like a typo to me (or maybe the book said that $b_0 = (1, 0)$ earlier...)

Answer 2

Since $S^1$ is path-connected, the fundamental group is independent of the choice of point, and will be the same at any base point.