Rognes’ connectivity conjecture concerns the connectivity of a simplicial complex called the common basis complex. The equivariant homology of this complex is the E^1 page of a spectral sequence converging to the homology of K-theory spectra. I will describe joint work with Patzt and Wilson where we prove the connectivity conjecture for fields. I will also explain a connection between the homology the common basis complex and the André–Quillen homology of a certain equivariant ring built out of Steinberg modules.