Rokhlin's Theorem on Bounded Manifolds and Induced Spin Structures

This proof is about Rokhlin's Theorem in the context of manifolds. For other uses, see Rokhlin's Theorem.

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Theorem

Part 1

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

Let $\operatorname {sign} Q_M = 0$.

Then there exists a smooth oriented $5$-manifold $W$ such that:

$\partial W = M$

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Part 2

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

Let $\operatorname {sign} Q_M = 0$.

Let $M$ be endowed with a spin structure.

Then there exists a smooth oriented $5$-manifold $W$ such that:

$\partial W = M$

and the spin structure of $W$ induces the spin structure of $M$.

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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 Images of Smooth Embeddings bound Oriented Manifolds, $M'$ must bound a $5$-manifold $W'$.

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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 $\R^6$.

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Part 2

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Source of Name

This entry was named for Vladimir Abramovich Rokhlin.