Are there any simply connected parallelizable 4-manifolds?

Question

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

score 21 · Accepted Answer · answered Jun 14 '13 at 05:49

A simply connected smooth compact 4-manifold has its homology concentrated in even degrees, by Poincare duality. Thus its Euler characteristic is positive. By the Poincare-Hopf index theorem, such a manifold can have no nowhere vanishing vector field, and so is certainly not parallelizable.

score 5 · Answer 2 · answered Jun 13 '13 at 19:10

5

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

answered Jun 13 '13 at 19:10

Xipan Xiao

2,781

2

More generally, the orthogonal groups have simply connected compact universal covering spaces. – Mariano Suárez-Álvarez Jun 14 '13 at 05:51
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The question is specifically about 4-manifolds. We could say now that, as a corollary, that there are no 4-dimensional simply connected Lie groups. – Turion May 05 '16 at 18:06

Are there any simply connected parallelizable 4-manifolds?

2 Answers2