Is every 2 dimensional manifold whose boundary is a cycle, a continuous image of the unit disc? Maybe it happens if the space is good enough? I wanted to prove an equality between two definitions I've seen to simply-connectedness. (Every continuous image the circle is null homotopic, and every cycle is a boundary of another manifold). I also want to know if the vice versa holds: if any continuous image of the unit disc is a manifold. Please keep in mind I don't have a lot of knowledge on the subject. I wasn't sure about which tags I should add to this question. Thanks!
Any compact, connected, locally connected second-countable Hausdorff space (in particular any compact connected manifold with boundary) is a continuous image of the unit interval, and thus of the unit disk. See the Hahn–Mazurkiewicz theorem.