nLab Rokhlin's theorem

Idea

Rokhlin’s theorem states that the signature of a smooth orientable closed spin 4-manifold is divisible by $16$. It therefore serves as an obstruction for the existence of smooth structures. A famous example is the E₈ manifold with intersection form the E₈ lattice and therefore signature $8$. Since it is an orientable closed spin 4-manifold, Rokhlin’s theorem states that it cannot carry a smooth structure.

Formulation

For a smooth orientable closed spin 4-manifold $M$, hence with vanishing second Stiefel-Whitney class $w_2(M)=0$, one has:

$$\sigma(M) \equiv 0 mod 16.$$

Generalizations

Kervaire-Milnor theorem

Proposition

(Kervaire-Milnor theorem) For a smooth compact 4-manifold $M$ and a characteristic sphere $S^2\hookrightarrow M$, one has:

$$\sigma(M) =S^2\cdot S^2 mod 16.$$

A characteristic sphere $i\colon S^2\hookrightarrow M$ represents the second Stiefel-Whitney class $w_2(TM)=PD(i_*[S^2])\in H^2(M,\mathbb{Z}_2)$ using Poincaré duality $PD\colon H_2(M,\mathbb{Z}_2)\rightarrow H^2(M,\mathbb{Z}_2)$ and the fundamental class $[S^2]\in H_2(S^2,\mathbb{Z}_2)\cong\mathbb{Z}_2$, which is the unique generator.

Freedman-Kirby theorem

Proposition

(Smooth Freedman-Kirby theorem) For a smooth compact 4-manifold $M$ and a characteristic surface $\Sigma\hookrightarrow M$, one has:

$$\sigma(M) =\Sigma\cdot\Sigma +8 Arf(M,\Sigma) mod 16$$

with the Arf invariant.

A characteristic surface $i\colon\Sigma\hookrightarrow M$ represents the second Stiefel-Whitney class $w_2(TM)=PD(i_*[\Sigma])\in H^2(M,\mathbb{Z}_2)$ using Poincaré duality $PD\colon H_2(M,\mathbb{Z}_2)\rightarrow H^2(M,\mathbb{Z}_2)$ and the fundamental class $[\Sigma]\in H_2(\Sigma,\mathbb{Z}_2)\cong\mathbb{Z}_2$, which is the unique generator.

Proposition

(Topological Freedman-Kirby theorem) For a topological compact 4-manifold $M$ and a characteristic surface $\Sigma\hookrightarrow M$, one has:

$$\sigma(M) =\Sigma\cdot\Sigma +8 Arf(M,\Sigma) +8 KS(M) mod 16$$

with the Arf invariant and Kirby-Siebenmann invariant.

References

See also:

• Wikipedia, Rokhlin’s theorem