[REOPENED]
This question was heavily downvoted and closed: When a compact topological manifold with boundary is a ball? Color me ignorant, perhaps, but I wondered if it were as trivial as the downvotes and closure might suggest.
Although it is a question for the topological category, a very similar question but for the smooth category was upvoted and answered last year, and the answers suggested there was interesting mathematics: The boundary of a domain whose interior is diffeomorphic to the ball.
Update: OP added some context.