Odysseus is crafty, brave, intelligent, and powerful these character traits serve as an example of the model of a man. He is the perfect warrior learning most of his knowledge through his adventures in his travels. However, these traits and Odysseus' ability are constantly challenged by the temptation of women. In the Odyssey, countless examples of these kind of temptations reveal the importance of gender and the role of women. Odysseus' shows the importance women play with his numerous interactions with women.
Odysseus' marriage to Penelope is another clear manifestation of gender roles within the epic. Penelope, wife to Odysseus and mother to Telemachus, is an important female character in the story, because Odysseus' homecoming is all for her love. Penelope is faithful to Odyssus the entire time he is away on his journey, she even is faithful when many suitors try and marry her. Although Odysseus slept with many of the women he encounters Penelope is faithful while waiting for him. While it may seem unfair, Penelope's only reason for remaining loyal to Odysseus is to state her independence from men and to show them that she is capable of doing it on her own. Telemachus is now technically the man of the house while Odysseus is away, but he lacks the experience he needs to run the household. Penelope needs a man around, but instead she tricks them by saying she will find a suitor once her shroud is finished and every night she undoes the weaving that she had previously worked on. The trick works until some of her house servants catch her in the act. She then proceeds to trick them yet again, by her promise to marry the suitor that can string and shoot Odysseus's bow knowing that only Odysseus can string it. Homer, was trying to show us the scheming nature of women, because Penelope shows many of the traits that Odysseus, a man, possesses.
This story is reliant on the role of women. We cannot entirely judge the ideals about gender roles in Greek culture only...