Abstract: A conjecture of Fred Wilhelm is the following: Given a Riemannian manifold M with positive sectional curvature, if M admits a Riemannian submersion to another manifold B, then the dimension of M is less than twice the dimension of B. We discuss some examples supporting this conjecture and some evidence for it. I will then discuss new evidence for the conjecture obtained using topological methods. This is joint work with Manuel Amann (KIT).