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

DOI: http://dx.doi.org/10.12775/TMNA.2009.019

#### 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.

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.

#### Keywords

Fixed point; fiber bundle; fiberwise homotopy; abelianized obstruction

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