Is the husband obligated to divorce his wife who asks to live in her parents' home and refuses to return to the marital home, while the wife's father refuses her return to her husband unless the wife and her husband live in her father's home, and the husband insists on not leaving his mother alone?

If you have provided your wife with an independent dwelling, she has no right to demand another home, unless your mother is harming her in it and you cannot prevent that harm from her. In that case, you must provide her with another dwelling, based on the saying of the Prophet, peace and blessings be upon him: "There should be no harm nor reciprocating harm."

If there is no unavoidable harm to her, then neither she nor her father has the right to stipulate another home. If they insist, refer the matter to the judiciary.

If her staying in your home involves unavoidable harm, then you must provide her with another dwelling, even if it is at her father's house, while still honoring your mother. If it is impossible to combine honoring your mother and your wife's right after exhausting all avenues, then you have no choice but to divorce your wife; because your mother's right takes precedence, based on the Prophet's saying, peace and blessings be upon him: "Your mother," three times when asked about who is most deserving of good companionship, and his saying: "The most deserving person of a man's good companionship is his mother."