What is the significance of the grandfather's deathbed speech who has he betrayed?
The Invisible Man's grandfather says, on his deathbed:
“overcome ‘em with yeses, undermine ‘em with grins, agree ‘em to death and destruction, let ‘em swoller you till they vomit or bust wide open”
The grandfather, a member of the "slave generation," says that the Civil Rights' generation must look like an Uncle Tom (house Negro), but secretly be rebellious to everyone: all blacks, all whites, all black stereotypes (field and house Negroes). The theme of appearance vs. reality suggests that the Invisible Man must be a traitor to one's own race, family, and heritage.
As you know, in the "Battle Royal" the Invisible Man wants to be a Booker T. Washington (field Negro), but the grandfather laughs at his grandson. His words haunt him throughout the novel, hanging over his every decision like Hamlet's father's ghost hangs over his.
In the end, the Invisible Man will heed his grandfather's advice: he will drop out of society and hide in his hole in the basement. He will betray everyone as an existential means to find himself. The narrator must reshape his racial identity from one obedient to authoritarian structures to one free to reject all of them. He is in the process of reinventing himself, casting off both the house and field Negro stereotypes. His authenticity then is not defined by being inauthentic to any ascribed stereotype, but by relishing what he wants to relish, regardless of stereotype.