The title of this short story refers to how many of the people in the story refer to Bovanne. There is an irony in this because they just say "my man Bovanne" to sound cool and hip, but they do not actually care one way or the other about the elderly blind man. Many people at the fundraiser address Bovanne with a friendly greeting, but none of them go any further than that. Nobody makes conversation with him or offers to get him any refreshments.
Conversely, the story's narrator, Miss Hazel, is the only one in the story to take a genuine and empathetic interest in Bovanne and his wellbeing. However, she makes it a point to tell the reader that he "ain’t my man."
Although people are nice to Bovanne on a surface level, nobody really cares about him. Many even seem to harbor ill feelings toward him just beneath the surface. Hazel mentions the many frowns and looks of disapproval she receives while dancing with Bovanne. Later on, Elo, Hazel's daughter, states that she hates that Bovanne does not hide his blind eyes behind sunglasses. Her other children agree that their mother was an embarrassment for dancing the way she did with Bovanne. To them, saying "my man Bovanne" is empty of any actual meaning.