My apartment complex refused to fix a staircase and I fell and injured myself, can I sue the apartment complex?

Yes. The injured party need only show that the apartment complex knew or should have known that the staircase was dangerous and posed a risk of injury to tenants or other parties visiting the complex, that the apartment complex breach the duty by not repairing or maintaining the staircase, that the failure of the apartment complex to fix the staircase caused of the injuries, and that the injured party actually sustains medical and/or physical damage.