Kairvaire Problems in Stable Homotopy Theory

The Kervaire Problem is an unsolved problem in Stable Homotopy Theory. The first interesting example is in dimension $30$. There exists a closed stably-parallelizable manifold ${\tilde{M}}^{30}$ with Arf-Kervaire invariant 1. It is unknown, if such a manifold exists in dimension $126$? The goal of the paper is to recall the construction of the manifold $M^{30}$ and investigate its properties.