An example of a smooth manifold with a group structure which is not a Lie group

Question

I'm looking for an example of a smooth manifold which is also a group but the group operations are not smooth. Most introductory books on differential geometry don't discuss such examples which makes me conjecture that the examples are not easy to describe.

Answer 1

Let $M$ denote any manifold of positive dimension which can not be given the structure of a Lie group, e.g. $S^2$, or any nonorientable manifold.

Pick your favorite set theoretic bijection with $\mathbb{R}$. Such a bijection exists: any chart has cardinality $|\mathbb{R}^n|=|\mathbb{R}|$ and since manifolds are second countable, $M$ can be covered by countably many charts, $|M|\leq |\mathbb{R}|^{|\mathbb{N}|}=|\mathbb{R}|$.

Call such a bijection $f:M \rightarrow \mathbb{R}$. Now, for $a,b\in M$, define $a+b= f^{-1}(f(a)+f(b))$. This gives $M$ the structure of a group isomorphic to $\mathbb{R}$.