  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.
