Two theorems of Rokhlin (Q1404787)

This paper gives new proofs of two results due to Rokhlin. They are: Theorem 1: \(\pi_{n+3}(S^n)\approx \mathbb{Z}_{24}\) for \(n\geq 5\). Theorem 2: The signature of any spin 4-manifold is divisible by 16. The proof of Theorem 1 relies on the relation between cobordism groups of immersed surfaces and stable homotopy groups, where the author profits from the knowledge of the former ones. In the proof, the Smale invariant of the embeddings plays an important rôle. Then as a byproduct of the proof of Theorem 1 one obtains Theorem 2.