A Note on the Periodic Structure of Transversal Maps on the Torus and Products of Spheres

Abstract

Let X be a compact differentiable manifold. A $$C^1$$ map $$f:X\rightarrow X$$ is called transversal if for all positive integers m, the graph of $$f^m$$ intersects transversally the diagonal of $$X\times X$$ at (x, x) for any x fixed point of $$f^m$$. In the present article, we describe the periodic structure of transversal maps on the n-dimensional torus. In particular, we give conditions on the eigenvalues of the induced linear map on the first homology, in order that all sufficiently large odd numbers are periods of the map. We present similar results for transversal maps on products of spheres of the same dimension. Later we generalize these results for transversal self-maps on rational exterior spaces of rank n.