What is the Euler characteristic of the compact real form of the Lie group $E_8$?
2026-05-14 13:47:29.1778766449
What is the Euler characteristic of $E_8$?
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Let $X$ be a nontrivial left-invariant vector field on $E_8$. Since $X$ is never zero, it has vanishing index sum. By Poincare-Hopf, $\chi(E_8)$ is zero. This works for any compact Lie group.