There is a famous theorem about when $S^n$ has the structure of a Lie group. What about the complex projective space $\mathbb CP^n$? For example, why $\mathbb CP^2$ is not a Lie group (without using classification for low dimension compact Lie groups)?
Lie group structure on the complex projective space
algebraic-topologylie-groupsprojective-space
Best Answer
$\mathbb{CP}^n$ has Euler characteristic $n+1$, but a compact (positive-dimensional) Lie group has Euler characteristic $0$, for example by the Lefschetz fixed point theorem.
Alternatively, you can show in various ways that the rational cohomology ring of a compact Lie group must be an exterior algebra on a finite number of odd generators. But $H^{\bullet}(\mathbb{CP}^n, \mathbb{Q})$ is concentrated in even degrees, so doesn't admit odd generators.