On the homology of spaces of equivariant maps

A spectral sequence calculating the homology groups of some spaces of maps equivariant under compact group actions is described. For the main motivating example, we calculate the rational homology groups of spaces of even and odd maps S-m ? S-M, m < M, or, which is the same, the stable homology groups of spaces of non-resultant homogeneous polynomial maps Rm+1 ? RM+1 of increasing degree. Also, we calculate the homology groups of spaces of Z(r)-equivariant maps of odd-dimensional spheres for any r. In auxiliary calculations, we find the homology groups of configuration spaces of projective and lens spaces with coefficients in several local systems.