# Self-intersection number of a loop on a surface

Let $$M$$ be an orientable surface$$($$without boundary$$)$$, possibly non-compact. Let $$\alpha\in \pi_1(M)$$ be a primitive element i.e. $$\alpha$$ is not a proper power of some other element. Let $$f:\Bbb S^1\to M$$ be any loop representing $$\alpha$$. Consider the lifting problem

Here, $$M_\alpha$$ be the cover corresponding to the subgroup $$\langle\alpha\rangle$$ of $$\pi_1(M)$$ and $$\widetilde M$$ be the universal cover of $$M$$. Also, all unleveled maps are covering maps.

Note that $$\pi_1(M)$$ acts on $$\widetilde M$$ via covering transformations, and $$M_\alpha$$ be an open annulus$$($$any open connected surface with a finitely generated fundamental group is homeomorphic to the interior of a closed surface$$)$$.

$$\textbf{Problem 1:}$$ Consider two sets $$\mathscr A_f:=\big\{g\in \pi_1(M):\text{ the map } \Bbb R\xrightarrow{g\cdot \ell}\widetilde M\to M_\alpha\text{ runs from one end to other of the open annulus }M_\alpha\big\},$$ $$\mathscr B_f:=\big\{(z,w)\in \Bbb S^1\times\Bbb S^1:z\not= w\text{ and }f(z)=f(w)\big\}$$ Is the cardinality of $$\mathscr B$$ two-times the cardinality of $$\mathscr A$$, i.e. $$|\mathscr A_f|=\frac{1}{2}|\mathscr B_f|$$?

$$\textbf{Problem 2:}$$ Give a Riemannian metric on $$M$$ such that $$\alpha$$ can be represented by a shortest loop $$f^\#:\Bbb S^1\to M$$, i.e. $$f^\#$$ has the minimum length in its free homotopy class. Is $$|\mathscr A_f|\geq |\mathscr A_{f^\#}|$$?

For problem one, the answer is "no" as there may be "bigons". The pair of intersections forming the bigon contribute to $$B_f$$ but not to $$A_f$$.
• How do I relate the words "shortest", "bigons', "$\mathscr A_f$? Jan 30, 2021 at 9:44