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Answer:
The Rapture is a belief that prior to the final judgment God will take away from the world all those who are saved, and those left behind will have one last chance to turn to God during the reign of the Antichrist.
This is not a Catholic belief. The Church teaches that there is but one moment for the final judgment when the living and dead will be judged. There will be no period of time where the righteous are taken from the Earth while others remain behind.
There is no definitive answer from the Church as to what happens to those in purgatory and those in need of purification at the time of the final judgment. It is indeed possible that at the time of the final judgment all the faithful on Earth may have undergone such trials as to not need any more purification for their sins. It is also possible that God can choose an alternate means of purifying the souls of the faithful prior to the final judgment.