Why do the townspeople in "A Rose for Emily" believe the boyfriend/husband-to-be has left?
Homer Barron's disappearance was a big deal in a small town like William Faulkner's Jefferson in the short story, "A Rose for Emily," but there was never any sinister connection until after Emily Grierson had died. Homer had earlier left Jefferson after Emily's relatives had appeared to try and reason with her concerning her relationship with the "Yankee" Barron.
So she had blood-kin under her roof again and we sat back to watch developments. At first nothing happened. Then we were sure that they were to be married... We were really glad...
... Sure enough, after another week they departed. And, as we had expected all along, within three days Homer Barron was back in town. A neighbor saw the Negro man admit him at the kitchen door at dusk one evening.
And that was the last we saw of Homer Barron. And of Miss Emily for some time.
In the minds of the people of Jefferson, Miss Emily had simply been jilted by the lifelong bachelor. They had probably had a falling out, and Homer had left in the middle of the night. Miss Emily's absence was her way of mourning her best and last chance at marriage. As for "the smell," it was a foregone conclusion that it was, as Miss Emily had implied by purchasing the poison, simply a rat.