Is there a compact manifold having Euler characteristic 0 which cannot be given a Lie group structure?

Question

I realized that a (compact) Lie group must have Euler characteristic 0 due to Poincare-Hopf index theorem. Now I'm thinking of its converse. Is there a compact manifold having Euler characteristic 0 which cannot be given a Lie group structure?

Answer 1

Yes, lots. There are plenty of other obstructions to having a Lie group structure: for example, the fundamental group must be abelian (by the Eckmann-Hilton argument), and the rational cohomology must be an exterior algebra on odd generators (this is due to Hopf).

In particular, every closed $3$-manifold has Euler characteristic $0$, but most of them, such as $S^1 \times \Sigma_g, g \ge 2$, have both nonabelian fundamental group and cohomology that is not an exterior algebra.