This has been an age long question in every home. The answer remains spontaneous in such a way that as the world is now dynamic and events unfold on dialy basis, responsibilities of men and women are sometimes interchangeable. Some men now run their offices at home while some women have chosen to take up jobs that engage them during the day.
The jobs and careers of the man and his wife in one family may not be the same with another family. This is why a question like this can be rhetoric at times.
It is naturally the duty of both parents to care for the home, depending on their schedules. If the man on the other hand takes up a job that's highly engaging while the woman has a lesser work, it should be clear that the woman handles the home.