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 M be an orientable 3 -manifold. Recall that the Wu class v is the unique cohomology class such that \langle v \cup x, [M] \rangle = \langle Sq(x), [M] \rangle , and Wu’s theorem says that w(M) = Sq(v) . The up-shot is that Stiefel-Whitney classes are homotopy invariants, even though they are defined using the tangent bundle.

Since M is orientable, we have w_1(M) = 0 . Since \dim M = 3 , the Steenrod squares Sq^2 and Sq^3 kill everything, so v_2 = 0 and v_3 = 0 . By Wu’s theorem, w_2(M) = Sq^1(v_1) + v_2 = 0 , and w_3(M) = Sq^1(v_2) + v_3 = 0 . In other words, all the Stiefel-Whitney classes vanish.

Orientability matters; after all, being orientable is the same thing as w_1 vanishing. For example, RP^2 \times S^1 is not parallelizable, since w_1(RP^2 \times S^1) = w_1(RP^2) + w_1(S^1) \neq 0 .