With smoke still rising from the World Trade Center ruins, New Yorkers chose their candidates for mayor Tuesday in primaries thrown into turmoil by the disaster and the possibility that Rudolph Giuliani may try to stay on to steer the city through its crisis.

The Republican mayor is barred by law from running for a third term this fall, and he has not announced his plans. But his popularity has soared since the Sept. 11 attack and some voters say he ought to stay on past Dec. 31, when his term is up.

"The last thing the city needs is a change in government," said Fran Kane, a Democrat who said she would have voted for Giuliani if she could.

Lorraine Fittipaldi, a Republican, cast a write-in vote for the mayor. "Even if they keep him for a year, I think he's been wonderful," she said.

On a day of on-and-off rain, four Democrats and two Republicans sought their parties' nominations to move a step closer to the most prominent municipal job in the country. The winners of each primary meet Nov. 6.

With at least 15,000 absentee ballots cast -- and the possibility of a sizable write-in vote for Giuliani by Democratic or Republican voters -- there was a chance the winners would not be known until the official count is completed next week.

The Democratic candidates -- Public Advocate Mark Green, City Comptroller Alan Hevesi, City Council Speaker Peter Vallone and Bronx Borough President Fernando Ferrer -- all were seeking the 40 percent of the vote needed to win.

If no one gets 40 percent, the top two vote-getters advance to an Oct. 11 runoff. Michael Bloomberg, the billionaire media mogul, and former Rep. Herman Badillo were on the ballot for the GOP nomination.

The primaries began Sept. 11 but were postponed within hours after two hijacked airliners brought down the trade center, burying thousands of people in the rubble. Votes cast by machine were thrown out.

The candidates immediately suspended campaigning and only resumed making low-profile appearances over the weekend. Nine polling places were damaged or destroyed by the attack, and some voters were assigned different locations to cast ballots.

According to early exit poll results, four in 10 Democratic voters and a strong majority of Republicans said they would vote for Giuliani in November if he somehow sought a third term.

Two-thirds of Democrats and nearly all Republicans approved of the job the mayor is doing, according to the exit poll samples from 561 Democratic voters and 109 Republican voters by Edison Media Research. Few doubted the city would rebound.

The polling also found that terrorism and the recovery effort did not dislodge traditional political issues as most important for Democrats -- a third cited education as the top issue in their vote, while fewer than one in five cited preventing future attacks or rebuilding the financial district. Republicans -- who are outnumbered 5-1 in the city -- seemed more likely to cite terrorism or rebuilding as most important in their vote.

Giuliani has not ruled out trying to stay on as mayor, saying Monday that he needed more time to think about his political future. Sources who spoke on the condition of anonymity said Giuliani's advisers were exploring ways the state Legislature or City Council could get around term limits.

The mayor urged people to choose from the six candidates and said Tuesday's outcome would not affect his plans.

In other elections around the country:

-- Arkansas voters chose candidates for a special election to fill the congressional seat vacated by new Drug Enforcement Administration chief Asa Hutchinson.

-- Voters decided whether Oklahoma should become the 22nd state to ban labor contracts requiring employees to pay union dues. Business and labor interests spent more than $10 million battling for votes.

-- Boston Mayor Thomas Menino was favored to win a nonpartisan runoff in his bid for a third term.