What does the outcome of the narrator's relationships with the white men suggest about the possibility of friendship/ cooperation btwn races?
I'm not sure to which "white men" exactly you are referring (there are several), but here are the major ones:
1) The white leaders in the Battle Royal: they blindfold him with white cloth. They want him to see the world with white eyes. They dangle a naked white woman in front of him, to taunt him with what he cannot have. They are characterized as wolves--not exactly images of friendship and cooperation.
2) Mr. Norton becomes fascinated with Trueblood's story of incest. He gets drunk at the Golden Day and is instrumental in getting the Invisible Man expelled from the college.
3) The supervisors at Liberty Paints: they are proud of Optic White, a paint that can cover up any blemish. This is a racial metaphor: they want to negate black and white differences so that one only sees white.
In the end, the Invisible Man secludes himself underground. He is alienated and exploited by everyone: white, black, young, and old. This is not a comedy: there is no happy ending or cooperation at the end. Granted, the narrator may come out of his hole to become an integrationist, but for now he must remain, in the words of his grandfather, a "traitor" to both races in order to find himself.