A bit of context for this question: as a project for my master's degree my supervisor asked me to understand the construction of Milnor's exotic spheres. After learning the heavy material (I knew very little algebraic topology so learning about characteristic classes counted as "heavy" for me) the proof of existence of these spheres is surprisingly short and easy. 

I have finished early and I would like to go on and explore other ideas, ideally results which are not unreasonably difficult and preferably are "surprising" i.e. the conclusion is something other than the classification of vector bundles over a thing. Any pointers? :)