Diffeomorphism of $S_n = \{(x,y,z,w) \in \mathbb{R}^4 \mid x^n+y^n+z^n+w^n=1\}$
When $n$ is odd, $S_n$ contains the line
$$L = \{ (-s, s, 0, 1) : s\in \mathbb R\}$$
is thus is not compact. Hence $S_n$ is not homeomorphic to $S^2$, which is compact.
When $n$ is odd, $S_n$ contains the line
$$L = \{ (-s, s, 0, 1) : s\in \mathbb R\}$$
is thus is not compact. Hence $S_n$ is not homeomorphic to $S^2$, which is compact.