The Plot is about Private Detective solving the mystery behind Sean Regan whereabouts & to get rid of Geiger blackmail from General Sternwood.
Why the movie title The Big Sleep?
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From the Wikipedia page of the novel,
The Big Sleep (1939) is a hardboiled crime novel by Raymond Chandler, the first to feature the detective Philip Marlowe. It has been adapted for film twice, in 1946 and again in 1978. The story is set in Los Angeles, California.
It mainly deals with hardcore crimes and murders. So it should be understood that The Big sleep refers to Death.
In the novel, the author of the novel Raymond Chandler quotes:
You were dead, you were sleeping the big sleep, you were not bothered by things like that, oil and water were the same as wind and air to you. You just slept the big sleep, not caring about the nastiness of how you died or where you fell. Me, I was part of the nastiness now. Far more a part of it than Rusty Regan was.
As the movies are book adaptations and the most of part is taken from the book, the movie title suggests the same that the big sleep here is Death.