In relation to "The Shield of Achilles," what is a specific heroic deed for which Achilles is known?
Homer's opening to his epic poem invokes Achilles before any other soldier, testament to his heroism and sense of greatness on the battlefield. Achilles' greatness in the epic is represented when his absence translates into losses for the the Greeks. Part of the reason that Achilles is such a great subject for the poem is that he truly represents the greatest of Greek warriors.
Achilles' bravery on Arete on the battlefield is the basis of Homer's Iliad. His ability to defeat any foe head on is confirmed with his heroic deed in slying Hector, his worthy adversary from Troy. Achilles' ability on the battlefield is why he commands so much respect from his soldiers who follow him without hesitation.
It is also the reason why he angers Agamemnon, as the King knows that he will never be equal to the mighty Achilles. Auden's poem takes this essence and projects the tragic condition within it. Achillles' greatness is the only reason why Hephaestus would make a shield for him in the first place, something that his mother, Thetis, knows very well. Yet, she is horrified by what visions are present on the shield, believing that the great deeds of her son will outweigh that which is present on it.
Yet, it is through Auden's strength as a thinker and poet that he concludes, through Achilles' death, that his greatness is cut short by the fact that war kills everyone at some point. What hope can there be in such a condition if the great warrior Achilles is victim to it?