What is unknown about the sphere $S^2$?

Question

The standard sphere $S^2$ is (arguably) the simplest symmetrical geometric object. We can view $S^2$ as a smooth manifold in the category of smooth manifolds, or a Riemann sphere $\mathbb{P}^1$ in the category of complex manifolds, or a genus zero curve (which is also often called a line) in the category of complex algebraic varieties.

We can ask different questions about $S^2$ depending on the category we put it into. For example, we can ask whether there is a nowhere vanishing vector field on $S^2$, how many complex structures there are on $S^2$, or what the Gromov-Witten invariants of $\mathbb{P}^1$ are. All these questions have been answered.

What is unknown about $S^2$?

If we replace $S^2$ by $S^6$, then we don't know if there is a complex structure on $S^6$. Although $S^2$ is a simpler object, I guess there should still be open problems involving $S^2$. In other words, I'm asking what we know we don't know about $S^2$.

Answer 1

1. Nirenberg's problem is still wide-open:

For which (smooth) functions $K: S^2\to {\mathbb R}$ there exists a Riemannian metric on $S^2$ whose Gaussian curvature is $K$?

See the paper from the last year

Michael T. Anderson, The Nirenberg problem of prescribed Gauss curvature on $S^2$,

for a survey of the status of the problem.

2. We still do not know all the homotopy groups $\pi_n(S^2)$. All what we know is that they are all nontrivial (for $n\ge 2$), and this is a relatively recent result:

S. Ivanov, R. Mikhailov, J. Wu, "On nontriviality of homotopy groups of spheres", 2015.