Rokhlin's Theorem on Bounded Manifolds and Induced Spin Structures


 * ''This article is on Rokhlin's Theorem for zero-signature manifolds as boundaries. If you are looking for Rokhlin's Theorem on intersection forms, see this page.

Part 1
If a smooth oriented 4-manifold $$M \ $$ has $$\text{sign } Q_M=0 \ $$, 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 $$W \ $$ such that $$\partial W = M$$ and the spin structure of $$W \ $$ induces the spin structure of $$M \ $$.

Part 1
By the Whitney Immersion Theorem, there exists an immersion of $$M \ $$ into $$\mathbb{R}^7$$.

Suppose $$\exists M'$$ such that $$M' \ $$ embeds in $$\mathbb{R}^6$$ 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 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 $$\mathbb{R}^6$$.