I am studying a paper that about finite-order homeomorphisms of a closed oriented floor, say $f: M rightarrow M$, $f^n = id$. I do know that from such an $f$ we get a overlaying $P: M rightarrow M_f$, the place $M_f$ is the orbit area of $M$. (For simplicity, I’m simply assuming that this has no department factors.)
However then the creator frequently refers to “the illustration of $P$“, supposedly a map $rho: pi_1(M_f) rightarrow mathbb Z/nmathbb Z$. I simply don’t see what this map is meant to be, or how it’s induced by $f$.
Right here is an instance I’m making an attempt to explicitly work by. Let $M$ be the genus-$4$ floor within the form of a triangle, and let $f$ is a rotation of $2pi/3$ in regards to the heart (think about a fidget spinner..). Then $M_f$ is the genus-$2$ floor, with elementary group
$$pi_1(M_f) = langle a_1,b_1,a_2,b_2 mid [a_1,b_1][a_2,b_2] = 1rangle.$$
What’s $rho$, a map from this into $mathbb Z/3mathbb Z$, imagined to be?
A search by Google and Hatcher (my reference textual content) didn’t yield any definition. Please let me know if any extra particulars can be useful.