Fixed points on torus fiber bundles over the circle
Volume 183 / 2004
Fundamenta Mathematicae 183 (2004), 1-38
MSC: Primary 55M20; Secondary 55R10.
DOI: 10.4064/fm183-1-1
Abstract
The main purpose of this work is to study fixed points of fiber-preserving maps over the circle $S^1$ for spaces which are fibrations over $S^1$ and the fiber is the torus ,$T$. For the case where the fiber is a surface with nonpositive Euler characteristic, we establish general algebraic conditions, in terms of the fundamental group and the induced homomorphism, for the existence of a deformation of a map over $S^1$ to a fixed point free map. For the case where the fiber is a torus, we classify all maps over $S^1$ which can be deformed fiberwise to a fixed point free map.
