I am currently taking the equivalent of a first semester graduate course in algebraic topology (roughly covering the first two chapters of Hatcher and possibly more depending on time). Part of the assessment is learning a result above and beyond the scope of the course and delivering a 20 minute presentation of its proof, or something of similar mathematical content. I was wondering if people had any suggestions for a significant result that would be (mostly) accessible to other graduate students toward the end of such a course. I am particularly interested in learning cohomology, so ideally something in that direction. Given that I have a lot of time, I am not afraid to be fairly ambitious and take on something difficult. If it helps, I am fairly interested in algebraic geometry, although I am not limiting myself to that by any means.
Any suggestions would be appreciated.