Jtdylktuy

A surface is an oriented connected sum of $ggeq 0$ tori, with $b geq 0$ open disks removed, and $n geq 0$ punctures in its interior.

Let Aut$^+(S,partial S)$ denote the group (under composition) of orientation preserving homeomorphisms from $S$ onto itself which restrict to the identity map on the boundary $partial S$. This is endowed with the compact open topology. Let Aut$_0(S,partial S)$ denote the connected component of $mathrm{id}:Sto S$.

In A Primer On Mapping Class Groups, the mapping class group $mathrm{Mod}(S)$ of a surface $S$ is defined in the following two ways:

1. $mathrm{Mod}(S) = pi_0(mathrm{Aut}^+(S,partial S), mathrm{id})$


2. 
   $mathrm{Mod}(S) = mathrm{Aut}^+(S,partial S) / mathrm{Aut}_0(S,partial S)$.

It is easy to show that the first definition corresponds to the set of boundary-fixing isotopy classes of maps in Aut$^+(S,partial S)$, for which the group operation is simply $[f][g] = [fcirc g]$. However, I'm struggling to show that the second definition makes sense, let alone that it is equivalent to the first.

My problem is this: for the definitions to be equivalent, quotienting by $mathrm{Aut}_0(S,partial S)$ must correspond to quotienting by isotopy. Thus $mathrm{Aut}_0(S,partial S)$ must be the isotopy class of the identity. It is easy to see that this is just the path component of the identity. But why should I expect this to be the same as the connected component of the identity? I've tried to show that Aut$^+(S,partial S)$ is locally path connected (which I believe is true, since the definition of a "surface" doesn't allow for much pathology). However I haven't been able to make any progress.