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Apartments lose heat in the winter. Buildings are bedeviled with leaks, mold and lead paint. Entire housing developments are overrun with rats.

Conditions for the more than 400,000 low and moderate-income New Yorkers living in buildings managed by the New York City Housing Authority have worsened to the point that the federal government is considering taking control.

Now, two men from opposite ends of the political spectrum, Mayor Bill de Blasio and Ben Carson, are trying to hammer out a deal to save the housing authority, also known as Nycha.

Mayor de Blasio and Mr. Carson, the secretary of housing and urban development, met last week to continue negotiating what sort of oversight would be imposed on the city’s housing authority, which is under scrutiny for years of mismanagement and a dilapidated housing stock in need of $32 billion in repairs.