Rokhlin's Theorem on Bounded Manifolds and Induced Spin Structures

From ProofWiki
Jump to navigation Jump to search

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





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:

$\partial W = M$



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 $M$.



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 $W'$.



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 $\R^6$.



Part 2




Source of Name

This entry was named for Vladimir Abramovich Rokhlin.