Gemara

(Beraisa): If Reuven dug and opened a pit and gave it to the public, he is exempt;

Nechunya the pit-digger used to dig and open pits and give them to the public. Chachamim praised him for his actions.

51b - Question: From when is a partner in a pit exempt?

Version #1 (R. Mani): He is exempt if he left the other partner using it.

Version #2 (R. Mani): He is exempt from when he gave the cover to his partner.

Bava Basra 60a (Mishnah): One may not make a pit under a Reshus ha'Rabim;

R. Eliezer permits this, if the cover can support a wagon laden with rocks.

One may not extend ledges or beams to a Reshus ha'Rabim. He may recess his house from the Reshus ha'Rabim, and extend them up to the Reshus ha'Rabim.

If one buys a Chatzer that has ledges extending to the Reshus ha'Rabim, he has a Chazakah (he may keep them).

23a: We claim for him that perhaps the area under them was really part of the Chatzer, or the public allowed the previous owner to build them.

60a: The first Tana argues with R. Eliezer, for sometimes the cover will get weaker, and he will not know.

Ploni had a ledge extending into the Reshus ha'Rabim. People complained that it was impeding them. R. Ami ordered him to remove it.

Ploni: You also have a ledge that juts out!

R. Ami: My ledge extends to an alley. The people pardon it. Yours extends to a Reshus ha'Rabim. How will you get everyone's pardon?!

Rishonim

Rif and Rosh (Bava Kama 21b and 5:7): If one dug a pit in Reshus ha'Rabim for public needs and handed it (Rosh - the cover) over to them, he is exempt.

The Rif and Rosh (Bava Basra 32b and 3:77) bring the Gemara in Bava Basra.

Rambam (Hilchos Nizkei Mamon 13:23): One may not make a hollow under Reshus ha'Rabim, and not round, long or square pits. This is even if a wagon laden with rocks can pass over them, lest it be diminished without his knowledge. One may dig a pit for public needs.

Poskim

Shulchan Aruch (CM 410:7): If one dug a pit in Reshus ha'Rabim for public needs and handed over the cover to them, or told Beis Din that he wants to withdraw from it and they will cover it, he is exempt. This is if he was allowed to dig there.

Beis Yosef (DH u'Mah she'Chosav u'Mosar): The Rosh learned that he must give the cover to them, just like a partner in a pit is exempt from when he gave over the cover (51b).

Beis Yosef (DH u'Mah she'Chosav b'Shem): The Ramah says that if he has no cover, it suffices to tell Beis Din.

SMA (10): The Ramah says that if he dug it for himself, and then give it to the Rabim, he is liable until he fills it in. The Tur disagrees. Since the Rabim need it, and he gave it to them, it does not matter that he initially dug it for himself.

Shulchan Aruch (417:1): One may not make a hollow under Reshus ha'Rabim, and not round, long or square pits. This is even if a wagon laden with rocks can pass over them, lest it be diminished without his knowledge.

Beis Yosef (DH Ein): The Rashbam forbids even if he accepts to pay for all damage that will result, for people do not want to be damaged and then seek compensation in Beis Din. Chachamim forbid a hollow under Reshus ha'Rabim with a strong cover, lest the cover diminish and one who passes over will not know (and it will collapse under him).

SMA (1): The Rambam and Shulchan Aruch forbid a hollow and a round, long or square pit. It seems that they forbid open pits. The Mishnah did not say 'and'. The Rashbam explains that the hollow forbidden is a round, long or square pit.

Prishah (2): The Tur forbids even if the pit is covered by a stone dome. Even though there is no concern lest it become wormy, the rocks can be worn away through many wagons passing over.

Rema: Some say that even though this is the law, the custom is to make sewage pipes and cellars under Reshus ha'Rabim, and everyone pardons this, since it is the custom. Also, Reshus ha'Rabim belongs to the mayor. We follow whatever he authorizes, and we follow the custom.

SMA (3): People pardon this, for they know that they will need to do similarly at some time.

Rashba (2:292, cited in Beis Yosef DH v'Chosav ha'Rashba): If one made a hollow under Reshus ha'Rabim, even if many saw this and were silent, this does not help, like R. Ami said. One could say about an old hollow that the people of the city gathered and pardoned it. Outsiders who pass through cannot protest. People of the city can open and close roads. Bava Basra 23a says that explicit pardon of the Rabim helps. However, one could say that one cannot appease the Rabim; we assume that the seller recessed and built a ledge in his property. Nowadays, all cities have hollows under Reshus ha'Rabim for rainwater and people's wastewater to flow. No one refrains from making pits for silos in Reshus ha'Rabim. Everyone needs to do so, pardons it and wants it. They permitted what is forbidden. Also, the king authorized it. Even Stam, it is permitted, and all the more so if he authorized someone. The rulers own the roads. They may close them and open new ones in their place.

Gra (1): Silence of the Rabim does not help, like R. Ami said (about Reshus ha'Rabim) 'how will you get the pardon of everyone?!' However, explicit pardon of the seven leaders of the city, in the assembly of the residents helps, like Tosfos (23a DH Achulei) explains the Gemara there. R. Ami said that silence does not help.

Shulchan Aruch (ibid.): One may dig a pit for public needs.

Source (Gra 3): The Tosefta (Bava Kama 6:2) permits this. We find that Nechunya dug pits and gave them to the Rabim.