2 problems

Connected-fibres conjecture. Connectedness of the pre-images alone should imply that $F$ is injective.

Let $f:\mathbb{S}^3\to\mathbb{S}^2$ be a harmonic map, where both spheres have their round metrics. Let $\pi:\mathbb{S}^3\to\mathbb{S}^2$ be the Hopf fibration. Eells' conjecture.…