In "A Rose for Emily" why does Emily have basically a nonexistent relationship with her town?
Actually, in Faulkner's "A Rose for Emily," Emily has quite a constant relationship with the town. The relationship just stops at Emily's front door. But the house is really a town focal point. The townspeople, apparently, often watch the house and talk about it.
In fact, one of the characteristics of the story that gives it unity is the focus on the house. Almost everything in the story revolves around it.
Of course, Emily doesn't come out of the house much and doesn't socialize at all, but that just makes the relationship of a different kind, it doesn't eliminate it.
Those who enter the home to talk Emily into paying taxes, tell the rest of the town about it. Those who try to get Emily to give up her father's body, talk about it. Those who try to put a mailbox on her home, talk about it. The pharmacist talks about Emily's purchase of poison. The town is almost obsessed with her. This is a relationship, although it's an unusual one.
In short, though, if you need a specific answer, Emily is so isolated from the town because she wants to be and has to be. You don't invite people into your home when you have a corpse of a murdered human being upstairs in your bed--and you're sleeping with it.
In the short story "A Rose for Emily" Miss Emily was raised by a protective father. He had put her on a pedestal and governed everything that she had done. He is a regal man who sees himself as being above the rest of the townspeople and because of his wealth the people probably looked at things the same way.
Miss Emily was heavily influenced by her father's will. He made sure that she did not go out unless she was accompanied by him where she rode on the side of the carriage, seated next to him.
It stands to reason that after her father died she had continued to stay withdrawn and doing what she could o eek out a living without socialize with those beneath her.