What is the significance of the grandfather's deathbed speech who has he betrayed?
After slavery was made illegal, generations of African Americans had to make decisions about how to respond to continued inequalities. Even though slavery was officially abolished with the Emancipation Proclamation, given in 1863, racial inequalities still abounded.
The protagonist of The Invisible Man is a character who learns to find power in his invisibility. He explains his invisibility at the start of the novel:
I am an invisible man. No, I am not a spook like those who haunted Edgar Allan Poe; . . . I am a man of substance, of flesh and bone, fiber and liquids—and I might even be said to possess a mind. I am invisible, understand, simply because people refuse to see me. . . . Nor is my invisibility exactly a matter of a bio-chemical accident to my epidermis. That invisibility to which I refer occurs because of a peculiar disposition of the eyes of those with whom I come in contact. . . . I am not complaining, nor am I protesting either. It is sometimes advantageous to be unseen, although it is most often rather wearing on the nerves. . . .
This passage shows how the character is not literally invisible, but his skin color leads people to ignore him and to think him less significant than a white person. His invisibility is a product of the racism still prominent in his society in the 1930s. Notably, the invisible man does not even have a name in the text, again showing his invisibility.
Before the invisible man's grandfather dies, he tells his grandson:
"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."
His grandfather suggests that the best way to live peacefully and happily in America, which is still severely divided by racism, is to act the way white people ask black people to act. He suggests that his grandson maintain the appearance of a passive life; he should avoid any appearance of rebellion and accept the instructions of the white people he interacts with.
The grandson reflects on his grandfather's words at several points in the novel. He meditates on these words, trying to discern whether or not a life of external submission is the method of social equality that he believes to be most beneficial in achieving equal rights.
Early in the story, the invisible man attempts to obey the words of a white man, a school trustee named Mr. Norton. He is asked, as one of the college's students, to chauffeur Mr. Norton around the community. He listens obediently when Mr. Norton sees the home of Mr. Trueblood and asks to stop. He learns that Mr. Trueblood was accused of impregnating both his wife and his daughter. The idea of this incestuous relationship upsets Mr. Norton greatly and leads to continued conflict. When the invisible man returns to school, he is reprimanded by Mr. Bledsoe, the president of the college. He is expelled for listening to a white man when this situation might hurt the university financially. After being expelled, he ends up moving to New York City. Mr. Bledsoe sends him with letters, under the guise of helping the invisible man find a job, that harm the invisible man's work prospects in New York City.
This unfortunate situation makes the invisible man question his grandfather's theory of outer obedience to white authorities. He listened to Mr. Norton, and it led to great amounts of trouble for him. By the end, he witnesses more rebellious characters, such as Ras the Exhorter, who oppose the expectations of white society. He has to make a decision about whether or not he will obey his grandfather's instructions—will he act out submission to white authorities? Is that really what is best to aid the Civil Right's Movement? Much of the internal conflict that the invisible man faces in the novel stems from his grandfather's dying words.
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.