She is asking for a divorce for no reason. Can he take back the mahr?

Praise be to Allah.

Firstly: 

Once the marriage contract has been done, then each spouse has rights and duties with regard to the other. One of these duties is that the wife must obey her husband and go to join him, and she should allow him to be intimate with her. He is obliged to provide her with accommodation and spend on her, and there are other duties which have been discussed in the answer to question no. 10680. 

Secondly: 

It is not permissible for a woman to ask for a divorce except when there is a reason which makes it permissible for her to do so, because of the report narrated by Abu Dawood (2226), al-Tirmidhi (1187) and Ibn Majaah (2055) from Thawbaan (may Allaah be pleased with him) who said: The Messenger of Allaah SAWS (peace and blessings of Allaah be upon him) said: “Any woman who asks her husband for a divorce when it is not absolutely necessary, the fragrance of Paradise will be forbidden to her.”

Classed as saheeh by al-Albaani in Saheeh Abi Dawood. 

“When it is absolutely necessary” means when there is some hardship and reason for resorting to asking for a divorce. 

Thirdly: 

If you do not want to divorce your wife and there is no shortcoming on your part that calls for divorce, then you may refuse to give her a divorce (talaaq) and tell your wife to seek khula’ and give up the deferred portion of the mahr, or the gold, or all the gold and other things that you gave her. 

But you should pay attention to the ties of kinship that exist between you, and not burden her family with more than they can afford. If you limit it to taking back the gold and withholding the deferred portion of the mahr, that is good. See the answer to question no. 26247. 

We ask Allaah to recompense you with good and to help you to find a righteous wife who will be a delight to you. 

And Allaah knows best.