The new academic building was glamorous, its perforated metal skin shooting up dramatically from the streets of the East Village, then swerving around a daring gash of glass. It made a statement about just how far the Cooper Union for the Advancement of Science and Art had come, from its 19th-century origins as a charity for the poor to one of the most selective colleges in the nation.

But that was before market convulsions shook the school’s finances, and before the truth about its dire budgetary situation came to light. Now the audacious building, at 41 Cooper Square, completed in 2009, has become the most visible symbol of a debate about the future of Cooper Union on the eve of what could be the most important decision in its history.

The university, which offers world-class instruction in art, architecture and engineering, but no expensive athletic programs, no tricked-out student centers, no plush lawns to sprawl on between classes, is currently losing $12 million a year, about a fifth of its overall budget. So 153 years after the inventor and industrialist Peter Cooper founded a school long rhapsodized as “free as air and water,” it is considering whether to end its most famous tradition, and start making undergraduates pay to attend.

Many students and alumni, shocked at the possible loss of their school’s defining trait, have criticized the decision to erect the $177 million building, which was designed by Thom Mayne of the firm Morphosis. But Cooper Union’s president, Jamshed Bharucha, who arrived after construction was completed, said that the discussion about the building was a red herring, and that the real problems date to at least the 1970s.