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A surface is an oriented connected sum of $g\geq 0$ 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 $\mathrm{id}:S\to S$.

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 $[f][g] = [f\circ g]$. 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.

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I forgot all about the fact that I asked this question here, but I answered it myself a while back.

Definitions:

A surface $S$ is an oriented connected sum of $g \geq 0$ tori with $b \geq 0$ open disks removed, and $n \geq 0$ punctures in its interior. Homeomorphism classes of surfaces are in bijective correspondence with $\{(g,b,n): g,b,n \geq 0\}$.

Given a surface $S$, the group (under composition) of orientation preserving homeomorphisms that fix the boundary $\partial S$ is denoted by Aut$^+(S,\partial S)$. This is endowed with the compact-open topology. The path component of Aut$^+(S,\partial S)$ containing id$_S$ is denoted Aut$_0(S,\partial S)$.

Theorem:

Given a surface $S$, the following are three equivalent definitions of its mapping class group, Mod$(S)$:

  1. Mod$(S)_1 := \pi_0 (\mathrm{Aut}^+(S,\partial S), \mathrm{id}_S)$.
  2. $\mathrm{Mod} (S)_2$ is the group of boundary-fixing isotopy classes of maps in $\mathrm{Aut}^+(S,\partial S)$.
  3. $\mathrm{Mod}(S)_3 := \mathrm{Aut}^+(S,\partial S)/ \mathrm{Aut}_0(S,\partial S)$.

We first show that $1$ and $2$ are naturally isomorphic as sets. Explicitly, $\pi_0 (\mathrm{Aut}^+(S,\partial S), \mathrm{id}_S)$ is the collection of homotopy classes of continuous maps $\sigma: \boldsymbol 2 \to \mathrm{Aut}^+(S,\partial S)$ such that $\sigma (0) = \mathrm{id}_S$. Note that $\boldsymbol 2$ denotes the discrete space $\{0,1\}$. Suppose $\sigma, \tau : \boldsymbol 2 \to \mathrm{Aut}^+(S,\partial S)$ belong to the same homotopy class. Then there is a continuous map $$H: \boldsymbol 2 \times [0,1] \to \mathrm{Aut}^+(S,\partial S)$$ such that $H(1,0) = \sigma(1)$, $H(1,1) = \tau(1)$, and $H(0,t) = \mathrm{id}_S$ for each $t \in [0,1]$. Consider the map $$F: S \times [0,1] \to S$$ defined by $F(s,t) := H(1,t)(s)$ for all $s \in S, t \in [0,1]$. Then $F(s,0) = \sigma(1)(s)$ and $F(s,1) = \tau(1)(s)$ for each $s \in S$. Given any $t \in [0,1]$, $F(-,t) \in \mathrm{Aut}^+(S,\partial S)$. Therefore $F$ is a boundary-fixing isotopy from $\sigma(1)$ to $\tau(1)$. In summary, given any homotopy class $[\sigma] \in \pi_0 (\mathrm{Aut}^+(S,\partial S), \mathrm{id}_S)$, evaluation at $1$ gives a boundary-fixing isotopy class of maps in $\mathrm{Aut}^+(S,\partial S)$.


Conversely, suppose $f,g \in \mathrm{Aut}^+(S,\partial S)$ are isotopic. One can similarly construct a homotopy between the maps $\sigma,\tau: \boldsymbol 2 \to \mathrm{Aut}^+(S,\partial S)$ defined by $\sigma(0)=\tau(0)= \mathrm{id}_S$, $\sigma(1) = f$, $\tau(1) = g$.

We now define the group structure on $\mathrm{Mod} (S)_2$. I claim that $$[f][g] = [f\circ g]$$ for each $[f],[g]\in \mathrm{Mod}(S)_2$ defines a group structure on $\mathrm{Mod} (S)_2$. Let $f_0,f_1,g_0,g_1 \in \mathrm{Aut}^+(S,\partial S)$ such that $[f_0] = [f_1]$ and $[g_0] = [g_1]$. Then there are boundary-fixing isotopies $F^f, F^g: S \times [0,1] \to S$ such that $F^f(s,i) = f_i(s)$ and $F^g(s,i) = g_i(s)$ for all $s \in S$. The map $$F: S \times [0,1] \to S$$ defined by $F(s,t) = F^f(F^g(s,t),t)$ for all $s\in S, t\in [0,1]$ is then a boundary-fixing isotopy from $f_0 \circ g_0$ to $f_1 \circ g_1$. Thus $[f][g] = [f\circ g]$ is well defined.

It is now routine to see that this binary operation is a group operation, where $$[\mathrm{id}_S] = e,\ \text{and}\ [f]^{-1} = [f^{-1}].$$ This also induces a group structure on $\mathrm{Mod}(S)_1$: For any $[\sigma],[\tau] \in \pi_0(\mathrm{Aut}^+(S,\partial S), \mathrm{id})$, $$[\sigma][\tau] = [\gamma],\ \text{where}\ \gamma(1) = \sigma(1) \circ \tau(1).$$


Finally, we determine how definitions $1$ and $2$ of $\mathrm{Mod}(S)$ correspond to $\mathrm{Mod}(S)_3$. $\mathrm{Aut}^+(S,\partial S)$ is naturally equipped with a group structure under composition. We first show that $\mathrm{Aut}_0(S,\partial S)$ is a normal subgroup of $\mathrm{Aut}^+(S,\partial S)$, and then that the quotient group $\mathrm{Aut}^+(S,\partial S)/\mathrm{Aut}_0(S,\partial S)$ is isomorphic to $\mathrm{Mod}(S)_2$.

Observe that there is a one-to-one correspondence between isotopies $F: S \times [0,1] \to S$ and paths $\gamma : [0,1] \to \mathrm{Aut}^+(S,\partial S)$. Given an isotopy $F$, simply define the path by $$t \mapsto (s \mapsto F(s,t)).$$ Thus $\mathrm{Aut}_0(S,\partial S)$ is the isotopy class of $\mathrm{id}_S$. From the above discussion about the group operation on $\mathrm{Mod}(S)_2$, it is now clear that $\mathrm{Aut}_0(S,\partial S)$ is a normal subgroup of $\mathrm{Aut}^+(S,\partial S)$. Thus the quotient is truly a group.

I now claim that $\varphi: \mathrm{Mod}(S)_3 \to \mathrm{Mod}(S)_2$ defined by $f + [\mathrm{id}] \mapsto [f]$ is a group isomorphism. Suppose $f,g \in \mathrm{Aut}^+(S,\partial S)$ such that $f+[\mathrm{id}] = g+[\mathrm{id}]$. Then there exists $h \in \mathrm{Aut}_0(S,\partial S)$ such that $f = g \circ h$. Then $[f] = [g\circ h] = [g][h] = [g][\mathrm{id}] = [g]$, so $\varphi$ is well defined. Clearly the map is surjective and injective. Finally, given any $f+[\mathrm{id}], g+[\mathrm{id}]$, $$\varphi(f+[\mathrm{id}])\varphi(g+[\mathrm{id}]) = [f][g] = [f\circ g] = \varphi (f\circ g + [\mathrm{id}]) = \varphi((f + [\mathrm{id}])(g + [\mathrm{id}])).$$ Therefore $\varphi$ is an isomorphism as required. This proves the equivalence of the three definitions for $\mathrm{Mod}(S)$.

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