For example, given a Lie group, its fundamental group must be Abelian. So $\Sigma_g$ ($g>1$) can't have Lie group structure. We also know for $S^n$ only $n=0,1,3$ can have Lie group structures.

In general, what's the sufficient **or** necessary conditions for a manifold to have Lie group structure?

necessaryconditions: the second homotopy group must be trivial, and the third torsion-free, according to mathoverflow.net/questions/8957/… . Also, the (co)homology with field coefficients must carry a Hopf algebra structure. $\endgroup$ – Mark Grant Sep 11 '17 at 6:09