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.
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
Wine and food. The cyclops gets drunk, and then Odysseus stabs the cyclops in the eye with a sharpened log, then he gets away! Odysseus gives the cyclops 4 bowls of wine. As a parting gift, Odysseus and his men give the Cyclops a burning hole in its eye.