FUNDAMENTA MATHEMATICAE 183 (2004) Fixed points on torus fiber bundles over the circle by D. L. Gon¸ calves (S˜ ao Paulo), D. Penteado (S˜ ao Carlos) and J. P. Vieira (Rio Claro) 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 homo- morphism, 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. INTRODUCTION Given a fibration E → B and a fiber-preserving map f : E → E over B, the question if f can be deformed over B to a fixed point free map has been considered by many authors (see for example [Do-74], [F-H-81] and [Go-87]). In [F-H-81], E. Fadell and S. Husseini showed that the above problem can be stated in terms of obstructions (including higher ones). This is obtained under the hypothesis that the base, the total space and the fiber F are manifolds, and the dimension of F is greater than or equal to 3. The case where the fiber has dimension 2 is not considered. This case, even when the base is a point, is still a main open problem; when the total space is a surface with negative Euler characteristic it is known that the vanishing of the Nielsen number is not equivalent to the existence of a deformation to a fixed point free map (see [Ni-27], [Ji-85] and [Ke-87]). Consider a fiber-preserving map f : M → M , where M is a fiber bundle over the circle S 1 and the fiber is a closed surface S. Such fiber bundles are obtained from S × [0, 1] by identifying (x, 0) with (φ(x), 1), where φ is a homeomorphism of S. The main purpose of this work is to study in detail the case where the fiber is a closed surface. We develop a few generalities when S = S 2 and S = RP 2 . For the cases when the fiber is either the 2000 Mathematics Subject Classification : Primary 55M20; Secondary 55R10. Key words and phrases : fixed point, fiber bundle, fiberwise homotopy, abelianized obstruction. [1]