Does divorce occur if the wife travels to her home country for exams, while the husband has forgotten the details of an oath he previously swore?

The general principle is that the bond of marriage remains intact and an oath is not broken unless there is certainty. Therefore, if you are not certain about what your wife mentioned regarding the conditional divorce, then nothing is binding upon you. Assuming you made your wife's divorce conditional upon her descending alone in a specific year, then your oath is not broken if she descends alone in another year, because an oath is based on the intention of the one who makes it.