# Are there any simply connected parallelizable 4-manifolds?

On pp. 166 of Scorpan's "The Wild World of 4--manifolds", he gives an example of a parallelizable 4-manifold ($S^1\times S^3$) and then asserts: "there are no simply connected examples". It's confusing to me whether he means that there are no such examples in general, or examples that are 4-manifolds. In any case, here is my question:

Do there exist (compact, smooth, oriented) simply connected manifolds that are parallelizable? 4-manifolds?

(Recall that a manifold is called parallelizable if it has a trivial tangent bundle.)

• At the very least "compact" should be added, otherwise just take $\mathbb{R}^n$. In fact, any contractible manifold will be both simply connected and parallelizable. – Matt Jun 13 '13 at 18:34
• @Matt you're absolutely right. I've edited the question. – Aru Ray Jun 13 '13 at 18:59

At least $S^3$ is a lie group so parallelizable.