Bukhari

Hadith Commentary:
(1)
A woman who has received a revocable divorce (talaq raj‘i) remains her husband’s wife during her waiting period (‘iddah), therefore, her residence and other expenses remain the responsibility of her husband, as is mentioned in the hadith. The Messenger of Allah sallallahu alayhi wa sallam said:
“Maintenance and residence are for the woman as long as the husband has the right to take her back.”
(Musnad Ahmad: 6/373)
However, after the third divorce, the husband is not responsible for the residence and maintenance of his wife, as is explicitly stated in the hadith.
The Messenger of Allah sallallahu alayhi wa sallam said to Fatimah bint Qays radi Allahu anha:
“There is no residence or maintenance for you.”
(Sahih Muslim, al-Talaq, Hadith: 3698 (1480))
Umar radi Allahu anhu used to reject these narrations.
Umar radi Allahu anhu used to say that we cannot abandon the Book of Allah and the Sunnah of our Prophet alayhis salam on the statement of a single woman. We do not know whether she remembers correctly or has forgotten.
When Marwan said the same thing to Fatimah radi Allahu anha, she replied:
“The Book of Allah will decide between me and you.”
Allah the Exalted has said:
“Divorce the women for their waiting period...” until He said:
“You do not know; perhaps Allah will bring about a new matter after that.” So, after the third divorce, what new matter has Allah brought about? (Sahih Muslim, al-Talaq, Hadith: 3704, 3710 (1480))

(2)
Although Umar radi Allahu anhu rejected it according to his understanding, Fatimah radi Allahu anha is a noble Companion and she narrates the clear command of the Messenger of Allah sallallahu alayhi wa sallam, therefore, her statement will be accepted as proof that for a woman who has received a third divorce, there is neither residence nor is the husband obligated to provide maintenance. However, if at the time of the third divorce she is pregnant, then her expenses are the responsibility of the husband, as the Messenger of Allah sallallahu alayhi wa sallam said to Fatimah bint Qays radi Allahu anha:
“There is no maintenance for you except if you are pregnant.”
(Musnad Ahmad: 6/414)

(3)
In summary, separation between husband and wife occurs either through divorce, death, or annulment of marriage (faskh nikah). If separation occurs through divorce, then it will either be a revocable divorce (talaq raj‘i) or an irrevocable divorce (talaq bain). If it is a revocable divorce, then both maintenance and residence are obligatory upon the husband, whether she is pregnant or not. If it is an irrevocable divorce, then neither maintenance nor residence is obligatory upon the husband, except if the woman is pregnant, in which case maintenance is obligatory but residence is not.
If separation occurs through death, then neither maintenance nor residence is obligatory upon the man, but if she is pregnant, then maintenance is obligatory upon the man.
If separation occurs through annulment, it will be either through li‘an (mutual cursing) or khul‘ (divorce at the wife’s request). If separation is through li‘an, then neither maintenance nor residence is obligatory upon the man, even if the woman is pregnant, because the man has denied even the pregnancy. If annulment is through khul‘, then neither maintenance nor residence is obligatory, except if the woman is pregnant, then, due to the generality of the verse, only maintenance will be obligatory as in the case of an irrevocably divorced woman.
And Allah knows best.