Abstract: In the 1920s Salomon Lefschetz and Jakob Nielsen presented groundbreaking work on fixed points of continuous maps. This inspired much topological research in the subsequent decades. We will review some of the classical results and then turn to very recent developments concerning fixed points and, more generally, coincidences.

Given two maps between manifolds,we study the geometry of their coincidence locus (using nonstabilized normal bordism theory and pathspaces). We extract an invariant which must necessarily be trivial if the two maps can be deformed away from one another. Often this is also sufficient. Surprisingly however, in certain cases the full answer involves also the Kervaire invariant (which was originally introduced and used in an entirely different area of topology, namely: manifolds without smooth structures and exotic spheres).