Rokhlin's Theorem on Bounded Manifolds and Induced Spin Structures

This proof is about Rokhlin's Theorem for zero-signature manifolds as boundaries. For other uses, see Rokhlin's Theorem.

Theorem

Part 1

Let $M$ be a smooth oriented 4-manifold.

If $\operatorname {sign} Q_M = 0$, then $\exists$ a smooth, oriented 5-manifold $W$ such that $\partial W = M$.

Part 2

If $M$ is endowed with a spin structure and satisfies the other criteria of Part 1, then $W$ is such that $\partial W = M$ and the spin structure of $W$ induces the spin structure of $M$.

Proof

Part 1

By the Whitney Immersion Theorem, there exists an immersion of $M$ into $\R^7$.

Suppose $\exists M'$ such that $M'$ embeds in $\R^6$ and that $M'$ and $M$ are cobordant.

By a proof due to R. Thom, $M'$ must bound a 5-manifold $W'$.

The union of the cobordism and $W'$ are necessarily a 5-manifold $W$ which satisfy the theorem.

Hence it suffices to show that for any smooth, orientable 4-manifold, there exists a similar manifold which is cobordant to the original and embeds in $\R^6$.

Source of Name

This entry was named for Vladimir Abramovich Rokhlin.