On minimal fixed points set of fiber preserving maps of S1-bundles over S1

In this work, we describe the minimal fixed points set of fiberwise maps of S 1 -bundles over S 1 , up to fiberwise homotopies. There are two fibrations under these conditions, one orientable, the torus, and the other non-orientable, the Klein bottle. For fiberwise maps the minimal fixed points set can be empty, otherwise it is described as the finite union of disjoint circles. We present models for which the fixed points set are minimal, where minimal means that no proper subset can be realized as the fixed points set in the same fiberwise homotopy class.