Let $G$ be a left topological group, i.e. a topological space with group operation such that left multiplication $L_g : x \mapsto gx$ is continuous (but right multiplication and inversion are not required to be). Assume also $G$ to be a topological manifold. Does this imply that $G$ is a topological group?

EDIT 1: as pointed out by Yves Cornulier, the aswer is no if $G$ is not required to be connected. Hence, I would like to know if the statement still holds when $G$ is assumed to be connected.

EDIT 2: as discussed in the comments, we are also assuming $G$ to be paracompact, with countable atlas say.

4more comments