Fixed Points, Coincidences and Kervaire Invariants

Ulrich Koschorke (Siegen)

In the 1920s Solomon 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). 
Similarly other central notions of  topology turn out to play 
a crucial role here, e.g. various versions of 
Hopf  invariants (a la James, Hilton, Ganea...).