Let $S_{g,n}$ be a surface of genus $g$ and with $n$ punctures.
By an essential arc we mean an embeded arc (end points are in punctures) which is:
- Homotopically non-trivial i.e. not homotopic to a point and
- Not homotopic to a puncture.
Suppose $A=\{a_1,a_2.\cdots a_n\}$ is a collection of pairwise non homotopic, pairwise disjoint essential arcs such that each component of $S_{g,n}-A$ is simply connected (i.e. the surfaces obtained by cutting $S$ along $A$ are simply connected).
Let $f$ be an orientation preserving homeomorphism of $S_{g,n}$ which fixes the punctures and also fixes each $a_i$.
Q) Is $f$ is homotopic to identity?