Let f: M → M be a C1 map on a C1 differentiable manifold. The map f is called transversal if for all m ∈ N the graph of fm intersects transversally the diagonal of M × M at each point (x, x) such that x is a fixed point of fm. We study the set of periods of f by using the Lefschetz numbers for periodic points. We focus our study on transversal maps defined on compact manifolds such that their rational homology is $H_0 \approx \mathbb{Q}, H_1 \approx \mathbb{Q} \oplus \mathbb{Q}$ and Hk ≈ {0} for k ≠ 0, 1.