He does not invite Odysseus back to the island.
The cyclops was already named Polyphemus. Odysseus did not name the cyclops.
In the "Odyssey", Odysseus blinded the Cyclops Polyphemus with a red hot log.The cyclops that was blinded by Odysseus is Polyphemos.
Odysseus blinded Poseidon's son's eye who is a Cyclops. So he was cursed that can not go back home. He tried to go back home in the Odyssey.
In the cyclops cave
Odysseus was curious about the type of men the Cyclops were.
He thought it was a waist of time and that it was stupid,
The cyclops was already named Polyphemus. Odysseus did not name the cyclops.
Odysseus expects the cyclops to bear gifts to the travelers.
Odysseus did not kill the cyclops Polyphemus.
In the "Odyssey", Odysseus blinded the Cyclops Polyphemus with a red hot log.The cyclops that was blinded by Odysseus is Polyphemos.
Odysseus blinded Poseidon's son's eye who is a Cyclops. So he was cursed that can not go back home. He tried to go back home in the Odyssey.
Odysseus yells back to the Cyclops, Polyphemus, that his name is "Nobody." This clever deception is part of his plan to avoid retribution after blinding the Cyclops. When Polyphemus calls for help, he tells others that "Nobody" is hurting him, which leads them to believe he is not in danger. This moment highlights Odysseus's cunning and resourcefulness in overcoming challenges.
In the cyclops cave
Odysseus was curious about the type of men the Cyclops were.
Cyclops asked Poseiden to add to Odysseus' fate.
The Cyclops ate Odysseus' men.
Odysseus did not tell the cyclops his name cyclops promised him a gift