# Lie group structure on exotic $\mathbb{R}^4$

Are there Lie group structures on exotic $$\mathbb{R}^4$$s?

By the theorem that every continuous group homomorphism of two Lie groups is smooth, we can conclude that if $$G$$ and $$H$$ are two Lie groups that are isomorphic as groups, then if they are homeomorphic, they are also diffeomorphic.

From this we can conclude that the topological group $$(\mathbb{R}^4,+)$$ only admits one smooth structure turning it into a Lie group (namely the usual one). Now there are uncountably many other smooth structures on $$\mathbb{R}^4$$ (but still homeomorphic), namely the so called exotic $$\mathbb{R}^4$$s. By our above arguments we know that exotic $$\mathbb{R}^4$$s do not admit a Lie group structure when $$+$$ is the operation.

But does there exist a different group operation on some exotic $$\mathbb{R}^4$$ turning it into a Lie group?

• Which 4-dimensional Lie algebras do you know? Commented May 23 at 9:57

No, there are no such Lie groups. One way to answer is to observe that if $$G$$ is a contractible (as a topological space) Lie group then its exponential map is a diffeomorphism, see for instance Yves' answer here and references therein. In dimension 4 one can prove the result you are asking about without quoting any deep theorems and using the basic structure theory of Lie groups and Lie algebras. Namely, if $$G$$ is a simply-connected Lie group, consider its Levi-Malcev decomposition $$S\rtimes H$$, where $$S$$ is solvable and $$H$$ is semisimple. Then check that both $$S$$, $$H$$ have to be simply-connected and if $$G$$ is contractible, so are $$S$$ and $$H$$. There are few cases to consider then, the most interesting one is when $$H$$ is nontrivial. There are just two nontrivial semisimple Lie algebras of dimension $$\le 4$$, namely, $$o(3)$$ and $$sl(2,\mathbb R)$$ (it is a nice exercise which you can solve without appealing to root systems). The first one will not give you a contractible Lie group. The second one will give you Lie groups locally isomorphic to $$SL(2,\mathbb R)$$, leaving you with the universal cover of this group. The latter is easily seen to be diffeomorphic to $$\mathbb R^3$$ (start by proving that $$SL(2,\mathbb R)$$ is diffeomorphic to $$S^1\times \mathbb R^2$$). Lastly, taking a semidirect product with $$\mathbb R$$ (which is actually a direct product in this case) gives you a group diffeomorphic to $$\mathbb R^4$$.