He does not invite Odysseus back to the island.
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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.