In paper holomorphic disks and 3-manifold invariants, Ozsvath and Szabo connstruct two homeomoephisms $\mathcal {f} : H_{1}(\Sigma)\rightarrow H_{1}(Sym^{g}(\Sigma))$ and $\mathcal {g} : H_{1}(Sym^{g}(\Sigma))\rightarrow H_{1}(\Sigma)$. Then they says these two maps are inverses of each other.

Sorry for my weak ability, I can not see how this work. So can somebody explain it, like what f or g maps the generator to ? And why these two maps are inverses of each other? Thank you.