Two arXiv posts caught my attention the other day, both by Siddhartha Gadgil. The first paper  shows that an embedded surface in a 3-manifold is incompressible if and only if it is isotopic to an (embedded) minimal area surface, i.e. minimal over its homotopy class, for every Riemannian metric for the 3-manifold. This seems like a very natural result, and the paper is just six pages.
The second paper  is on degree one maps between 3-manifolds. A degree one map between closed, orientable n-manifolds is a map from one into the other that induces an isomorphism of their nth homology groups (each of which are isomorphic to Z). In dimension two, its not too hard to construct a degree one map from a given surface to any lower genus surface. There are no degree one maps from a surface to a higher genus surface, though I don’t recall a proof that’s nice enough to explain in one sentence. But one can think of a degree one map as indicating that the image manifold is somehow smaller or simpler than the domain manifold.
In dimension three, things are much more complicated. One can always construct a degree one map from a closed 3-manifold into the 3-sphere. Also, a homotopy equivalence map is degree one, so all those lens spaces with the same fundamental groups have degree one maps between them. Determining in general when there is a degree one map from M to N (both 3-manifolds) was (as far as I know) a very difficult open problem. However, Gadgil gives is surprisingly simple solution: He shows that there is a degree one map from M to N if and only if M can be produced from N by Dehn surgery on a link in which each component is an unknot. The paper is, again, quite short at 11 pages, and is based on interpretting Dehn surgery in terms of 4-manifolds. (In fact, he states a version of the main theorem in terms of 4-manifolds, but I won’t try to repeat it here.)