Let $S_g$ be a closed orientable surface of genus $g$ and $S_{g}^1$ a closed orientable surface with one boundary component. Let $p$ be in $S_g$ and let's note $\mathrm{Diff}_+(S_g,p)$ the set of orientation preserving diffeomorphisms fixing the point $p$ and $\mathrm{Diff}_+(S_g^1,\partial S)$ the set of orientation preserving diffeomorphisms of $S_g^1$ fixing $\partial S$ pointwise. Seeing $S_g^1$ as a subset of $S_g$ in such a way that $p\in \partial S$ we get an embedding

$$ f : \mathrm{Diff}_+(S_g^1,\partial S) \longrightarrow \mathrm{Diff}_+(S_g,p) $$ by extending any element of $\mathrm{Diff}_+(S_g^1,\partial S)$ requiring it to be the identity on $S_g \setminus S_g^1$.

$f$ induces $$f_* : \pi_0(\mathrm{Diff}_+(S_g^1,\partial S)) \longrightarrow \pi_0(\mathrm{Diff}_+(S_g,p)) $$

My question is : can one describe the rank and the kernel of $f_*$ ?

1) I want to believe that it is onto. It reduces to the following question : can one isotope a diffeomorphism of $S_g$ fixing $p$ to one fixing a neighborhood of $p$ ?

2) For the kernel, I have the impression that something around the point happens, like the number of turns one can make.


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