In the next decades, the gravitational-wave (GW) standard siren observations and the neutral hydrogen 21 cm intensity mapping (IM) surveys, as two promising non-optical cosmological probes, will play an important role in precisely measuring cosmological parameters. In this work, we make a forecast for cosmological parameter estimation with the synergy between the GW standard siren observations and the 21 cm IM surveys. We choose the Einstein Telescope (ET) and the Taiji observatory as the representatives of the GW detection projects and choose the Square Kilometre Array (SKA) phase I mid-frequency array as the representative of the 21 cm IM experiments. We find that the synergy of the GW standard siren observations and the 21 cm IM surveys could break the cosmological parameter degeneracies. The joint ET+Taiji+SKA data give $sigma(H_0)=0.28 {rm km s^{-1} Mpc^{-1}}$ in the $Lambda$CDM model, $sigma(w)=0.028$ in the $w$CDM model, which are better than the results of $Planck$+BAO+SNe, and $sigma(w_0)=0.077$ and $sigma(w_a)=0.295$ in the CPL model, which are comparable with the results of $Planck$+BAO+SNe. In the $Lambda$CDM model, the constraint accuracies of $H_0$ and $Omega_{rm m}$ are less than or rather close to 1%, indicating that the magnificent prospects for non-optical precision cosmology are worth expecting.
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