This is a request for a **reference** to a proof of a result. The result is not very hard, but I'd rather cite than reprove.

I'm looking for a generalization of the following result (Farb and Margalit, *A Primer on Mapping Class Groups*) to surfaces with boundary.

Let $Y$ be a closed surface of genus $g$, and let $\Psi$ be the representation $$\Psi: \operatorname{Mod}(Y) \rightarrow \operatorname{Sp}(H_1(Y)) \cong \operatorname{Sp}(2g, \mathbb{Z})$$ given by the action of the mapping class group on homology. Then $\Psi$ is surjective.

My question is the following.

Suppose $Y$ is a surface of genus $g$, with $r > 0$ boundary components. What is the image of $$\Psi: \operatorname{Mod}(Y) \rightarrow \operatorname{SL}(H_1(Y))?$$

Again, I'm pretty sure I know what the answer is, and I could prove it myself, but this sounds like something that is well-known and I'd rather be able to cite it.

Here's what I think the answer is: First, 'cap off' the boundary components of $Y$ to embed $Y$ in a closed genus-$g$ surface $Y'$. We get an exact sequence $$0 \rightarrow V \rightarrow H_1(Y) \rightarrow H_1(Y') \rightarrow 0,$$ where $V$ is generated by the classes of the boundary components.

The MCG action must respect this exact sequence, restrict to the identity on $V$, and induce an automorphism of $H_1(Y')$ that respects the symplectic form.

I expect that every such automorphism of $H_1(Y)$ will lie in the image of the MCG action -- and that it's not very hard to produce Dehn twists on $Y$ to prove it.

Does anyone know of a reference where this is proven?