The most successful model of the cohomology of a homogeneous space \(G/K\), due to Cartan, can be reformulated as a pure Sullivan algebra, essentially due to Borel’s thesis, which extracts this algebra from the Serre spectral sequence of \(G \to G/K \to BK\). This model generalizes to a model for biquotients \(H \backslash G/K\), first discussed by V. Kapovitch, that can also be used to compute Borel cohomology. Equivariant K-theory and Borel cohomology are connected by the Atiyah–Segal completion theorem and the Chern character, allowing us to transport some questions about one to the other.

On the other hand, the the cohomology of a biquotient can be seen as \(\mathrm{Tor}^*_{H*BG}(H*BH,H*BK)\), which is the one and only page of a Künneth spectral sequence in Borel cohomology (really an Eilenberg–Moore spectral sequence). We explain a related spectral sequence in equivariant K-theory due to Hodgkin and a map between the two which seems not to have been discussed previously.