The Arf-Kervaire invariant of framed manifolds as an obstruction to embeddability

We prove that no $14$-connected (resp. $30$-connected) stably parallelizable manifold $N^{30}$ (resp. $N^{62}$) of dimension $30$ (resp. $62$) with the Arf-Kervaire invariant 1 can be smoothly embedded into ${\mathbb{R}}^{36}$ (resp. ${\mathbb{R}}^{83}$).