Among the many unsuccessful attempts at solving the Poincaré conjecture, I'm wondering if there was an approach that went along the lines of showing that a closed simply connected $3$-manifold could be endowed with a Lie group structure. This might not be enough to solve it, since there are other compact $3$-dimensional Lie groups (if I'm not wrong), but it's tempting to think about since the only spheres which are Lie groups are of dimension $0, 1$ or $3$. This makes it feel like there is this special additional structure that can be used for the case $n=3$ as opposed to the generalized Poincaré conjecture, where no higher dimensional spheres are Lie groups.
So I'm wondering if someone tried this. If they did, what hurdle/subtlety did they find in the course of the attempt that prevented a solution? Did we learn something interesting from it?