If the criterion is “results using algebraic topology which shocked the mathematical community in the last 10 years”, then how about Abouzaid and Blumberg’s proof of the Arnol’d conjecture using Morava $K$-theory? Morava $K$-theory has been around since the ‘70’s or so and has become a central part of our understanding of stable homotopy theory via the chromatic picture, but it was always notoriously hard to find direct geometric applications of this stuff. Not only did Abouzaid and Blumberg find a geometric application, but the application they found resolved a longstanding open conjecture in symplectic geometry.