was this tunnel built under a river?
Yes, the Hudson River.
Lincoln Tunnel
In 1949, due to increased traffic demand, New Jersey Governor Alfred E. Driscoll suggested building a third crossing under the Hudson River. 
where does the lincoln tunnel exit in manhattan New York City how long is the tunnel? UNANSWERABLE when was this constructed? Construction of the central tube started in 1934 and it opened in 1937. was there any controversy related to the planning? Yes. The city wanted the Port Authority to build part of a proposed Mid-Manhattan Expressway above 30th Street from Eighth Avenue to the West Side Elevated Highway near 12th Avenue.
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Lincoln Tunnel [SEP] Planning and controversy
In 1949, due to increased traffic demand, New Jersey Governor Alfred E. Driscoll suggested building a third crossing under the Hudson River. He met with the Port Authority's board of commissioners and told them that the Holland and Lincoln Tunnels had now reached their full capacity of 15 million annual vehicles apiece (though the Port Authority noted that the Lincoln Tunnel had only seen 11.1 million vehicles in the past year). The same year, the Port Authority conducted a study of 135,000 motorists who used Hudson River crossings. It found that much of the traffic on the Holland and Lincoln Tunnels could be lessened if another tunnel between New Jersey and Midtown Manhattan were built either north or south of the Lincoln Tunnel.