Height h detection and connective real k-theory of elementary abelian 2-groups

In this paper, we determine the connective K-cohomology with reality of elementary abelian $2$-groups as a module over $\mathbb{Z}[v_1,a]$, where $v_1$ is the equivariant Bott class and $a$ the Euler class of the sign representation. This gives in particular a new approach to the computation of the connective real K-theory of such groups. The originality here is to make all computations in the $\mathbb{Z}/2$-equivariant stable category, considering only $\mathbb{Z}/2$-equivariant cohomology theories, and to use relative homological algebra over certain subalgebras of the equivariant Steenrod algebra to perform explicit computations.