Let $\Sigma$ be a genus $g$ closed Riemann surface. Recall that the symmetric product $\mathrm{Sym}^g\Sigma$ is the quotient of the $g$-fold product $\Sigma \times \dots \times \Sigma$ under the action of the symmetric group $\mathfrak{S}_g$ on $g$ letters.
In Lemma 2.6 of Osvath and Szabo's first paper
http://arxiv.org/PS_cache/math/pdf/0101/0101206v4.pdf
on Heegaard Floer homology, they prove that
$$H_1(\Sigma)\cong H_1(\mathrm{Sym}^g\Sigma).$$
Unfortunately I do not understand their proof. In particular, I do not understand their construction of the map $H_1(\mathrm{Sym}^g\Sigma) \rightarrow H_1(\Sigma)$, nor why this map inverts the (obvious map) $H_1(\Sigma) \rightarrow H_1(\mathrm{Sym}^g\Sigma)$. Could anyone explain this please?