It is a common social construct, especially in the 19th century, that a woman's place was within the house, making it a home and being there to care of her husband and children. Marriage was essential, because a woman was absolutely nothing in society without a husband. The exact reason why she left her children with her husband was technically not exactly said by her. During her discussion with Torvald in Act 3 she clearly expresses that he clearly challenged her ability to be a suitable mother and that she did not have the skills to be one. As for the fake, it was a labor of love. Without her forging that signature to get a loan, Torvald's life would no longer exist. If anything, that kind of risk she took for her family's well-being proves nothing other than that she is a fit mother and wife for her.