Let φ be a C1 flow without fixed points on a closed manifold. We show that generically a small C0ö perturbation of φ does not modify the homology directions. We determine these directions for a C0 open dense set of flows and examine the explosion of them. To do this we introduce the notion of homotopy direction for φ.
We give an elementary proof of Kida's formula on Hurwitz-type relations of cyclotomic λp-invariants of abelian number fields, using the theory of Γ-transforms of rational functions. Our proof will be carried out only in terms of rational functions.