In a mayor’s race crammed with celebrity razzle-dazzle, historic candidacies and tabloid turns, a gangly liberal from Brooklyn is quietly surging into the top tier of the field by talking about decidedly unglamorous topics: neglected hospitals, a swelling poverty rate and a broken prekindergarten system.

Now that Anthony D. Weiner’s campaign has imploded, Bill de Blasio, the public advocate, is drawing new energy and voter interest to a candidacy that presents the most sweeping rejection of what New York City has become in the past 12 years — a city, he says, that is defined by its yawning inequities.

“We are not, by our nature, an elitist city,” he told a group of young Democrats a few nights ago at a cramped bar in Brooklyn. “We are not a city for the chosen few.”

It is the campaign season’s riskiest calculation: that New Yorkers, who have become comfortably accustomed to the smooth-running, highly efficient apparatus of government under Michael R. Bloomberg, are prepared to embrace a much different agenda for City Hall — taxing the rich, elevating the poor and rethinking a Manhattan-centric approach to city services.