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A spectral sequence calculating the homology groups of some spaces of mapsequivariant under compact group actions is described. For the main example, wecalculate the rational homology groups of spaces of even and odd maps $S^m \toS^M$, $m<M$, or, which is the same, the stable homology groups of spaces ofnon-resultant homogeneous polynomial maps ${\mathbb R}^{m+1} \to {\mathbbR}^{M+1}$ of growing degrees. Also, we find the homology groups of spaces of${\mathbb Z}_r$-equivariant maps of odd-dimensional spheres for any $r$. As atechnical tool, we calculate the homology groups of configuration spaces ofprojective and lens spaces with coefficients in certain local systems.

Twisted homology of configuration spaces, homology of spaces of  equivariant maps, and stable homology of spaces of non-resultant systems of  real homogeneous polynomials