On the Motivic K-Theories of Classifying Spaces of Cyclic 2-Groups

We compute the $2$-completed integral motivic homology, effective algebraic $K$-theory, and very effective Hermitian $K$-theory of the geometric classifying spaces $Bμ_{2^n}$, $n\geq 2$, over algebraically closed and real closed fields. We first carry out the corresponding classical calculations and, then obtain the motivic results using motivic spectral sequences and comparison techniques. The results reveal the classical $2$-torsion together with motivic $τ$-periodicity and $ρ$-torsion.