# Orientable 3-manifolds are parallelizable

##### March 5, 2006 mathematics

Here’s a very easy theorem.

Theorem. All closed orientable 3-manifolds are parallelizable. All closed orientable 3-manifolds are the boundary of a 4-manifold.

Proof. Let be an orientable -manifold. Recall that the Wu class is the unique cohomology class such that , and Wu’s theorem says that . The up-shot is that Stiefel-Whitney classes are homotopy invariants, even though they are defined using the tangent bundle.

Since is orientable, we have . Since , the Steenrod squares and kill everything, so and . By Wu’s theorem, , and . In other words, all the Stiefel-Whitney classes vanish.

Orientability matters; after all, being orientable is the same thing as vanishing. For example, is not parallelizable, since .