Degree-one cohomology of metacyclic $2$-groups: a low-bit dichotomy

Abstract

We study the subalgebra generated by degree-one mod-$2$ cohomology for metacyclic $2$-groups $G=\langle a,b\mid a^{2^α}=1,\ b^{2^m}=a^c,\ b^{-1}ab=a^r\rangle$ with unique normal forms, where $α,m\ge2$ and $r\equiv-1\pmod4$. Presentation compatibility forces $c=0$ or $2^{α-1}$ and $G^{\mathrm{ab}}\cong C_2\times C_{2^m}$. If $c\equiv0\pmod4$, the degree-one-generated algebra is $\mathbb{F}_2 [x,y]/(y^2,x^2+xy)$, a four-dimensional algebra shared by the four nonmodular types of Aaboun--Zekhnini. The unique exceptional case $α=2$, $c=2$ is modular and has infinite-dimensional degree-one algebra $\mathbb{F}_2[x,y]/(y^2,x^2y)$. The regular algebra is PBW, Koszul, universally Koszul, and strongly Koszul, whereas the modular algebra is not Koszul. We realize the regular algebra by inflation from the common quotient $C_4\rtimes_{-1}C_4$ and identify its quadratic dual with the characteristic-two Jordan plane. For the inversion tower $C_4\rtimes_{-1}C_{2^m}$, we give uniform presentations of the full mod-$2$ cohomology rings and determine the kernels and images of inflation in every degree. Their direct limit yields the continuous cohomology ring of $C_4\rtimes_{-1}\mathbb Z_2$. We also record an arithmetic consequence for Galois groups arising over cyclotomic $\mathbb Z_2$-extensions of real quadratic fields.