Cohomology of $\mathbb{N}$-graded Lie algebras of maximal class over ${\mathbb{Z}}_2$

We compute the cohomology with trivial coefficients of Lie algebras ${\mathfrak{m}}_0$ and ${\mathfrak{m}}_2$ of maximal class over the field ${\mathbb{Z}}_2$. In the infinite-dimensional case, we show that the cohomology rings $H^{\ast }({\mathfrak{m}}_0)$ and $H^{\ast }({\mathfrak{m}}_2)$ are isomorphic, in contrast with the case of the ground field of characteristic zero, and we obtain a complete description of them. In the finite-dimensional case, we find the first three Betti numbers of ${\mathfrak{m}}_0(n)$ and ${\mathfrak{m}}_2(n)$ over ${\mathbb{Z}}_2$.