Mr. Emanuel contended that he had always maintained a home in Chicago, the city where he was born, and that his time at the White House was a matter of national service. But Mr. Emanuel’s opponents said that Mr. Emanuel did not meet the state’s residency requirements to run for a mayoralty, one of which is to have lived in the city for a year before the day of the election. His return to Chicago in the fall, they argued, was too late to qualify for a Feb. 22 ballot.