Abelianized Obstruction for fixed points of fiber-preserving maps of Surface bundles

Abstract

Let $f\colon M \to M$ be a fiber-preserving map where $S\to M \to B$ is
a bundle and $S$ is a closed surface. We study the abelianized
obstruction, which is a cohomology class in dimension 2, to deform $f$
to a fixed point free map by a fiber-preserving homotopy.
The vanishing of this obstruction is only a necessary
condition in order to have such deformation, but in some cases it
is sufficient. We describe this obstruction and we prove that the
vanishing of this class is equivalent to the existence of solution
of a system of equations over a certain group ring with coefficients
given by Fox derivatives.