Topology of $TS^n$

Question

I know that the tangent bundle $TS^n$ is not very often trivial, i.e. isomorphic to $S^n\times\mathbb{R}^n$. If it is not, is it still homeomorphic to $S^n\times\mathbb{R}^n$? It is clear that they are homotopy equivalent, but I am not too sure whether they are also homemorphic.

Answer 1

The Euler class is a topological invariant and the Euler class of $S^2$ is $2$, so it is homeomorphic to the trivial bundle.

https://en.wikipedia.org/wiki/Euler_class#Properties