Rumfoord himself came from New England wealth and as a space explorer entered a phenomenon known as a chrono-synclastic indifibula. This manifestation between Earth and Mars gave Rumfoord the ability to see the past, present, and future but bounced his physical pattern between Earth, Mars, and Titan. It is in this state that Rumfoord established the Church of God the Utterly Indifferent on Earth to unite the planet against the Martian invasion that he also instigated. On Titan, the only place he can exist as a solid human being and not as a broadcast image, Rumfoord befriends an explorer from Tralfamadore (a world that also figures in Vonnegut's Slaughterhouse-Five) named Salo who needs a "good-luck piece" to repair his damaged spaceship. The "piece" is brought to Salo by Constant and Rumfoord's ex-wife's son Chrono, who repairs his ship. Chrono chooses to live among the Titanian birds, Rumsfoord is spun out of the solar system, his ex-wife dies, and Salo returns Constant to Earth where he dies. He's welcomed into what we assume is heaven, and Constant responds, "I guess someone up there likes me."
The title is derived from the three-woman design on a pack of cigarettes that Constant uses, a design that Rumfoord's incorporated into his house on Titan.
The novel itself treats religion by its Marxian definition (as an "opium of the people") that drives its followers to often insane acts. Consider Malachi's banishment from Earth by the Church as a result of Rumsfoord's teachings as well as Rumsfoord's instigation of the Earth-Mars war leading to the church's formation.
The author also decries how humans not only use religious and ethical systems and to manipulate others but also how humans allow themselves to be manipulated by those systems. Consider Rumsfoord himself; he can see the past, present, and future but he can't see Salo's message, and he is genuinely hurt by that fact. Salo's ship crashed on Titan in Earth's prehistory, and his people, the Tralfamadorians, have been sending him messages through Earth's great architectural achievements such as the Egyptian Pyramids, Stonehenge, and the Great Wall of China. So the manipulators (of Earth) themselves have been manipulated throughout their history. With Salo now gone to deliver his message from his one end of the universe to the other, it is hoped that humanity will finally make its own destiny, without outside manipulation.