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

TL;DR

This paper develops a spectral sequence approach to compute the homology of equivariant map spaces, including stable homology of polynomial maps and homology of configuration spaces with local systems.

Contribution

It introduces a spectral sequence method for calculating homology of equivariant maps and applies it to polynomial and sphere map spaces, extending known results.

Findings

Computed rational homology of spaces of even and odd maps between spheres.

Determined homology groups of spaces of ${f Z}_r$-equivariant maps of spheres.

Calculated homology of configuration spaces of projective and lens spaces with local system coefficients.

Abstract

A spectral sequence calculating the homology groups of some spaces of maps equivariant under compact group actions is described. For the main example, we calculate the rational homology groups of spaces of even and odd maps $S^m \to S^M$, $m<M$, or, which is the same, the stable homology groups of spaces of non-resultant homogeneous polynomial maps ${\mathbb R}^{m+1} \to {\mathbb R}^{M+1}$ 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 a technical tool, we calculate the homology groups of configuration spaces of projective and lens spaces with coefficients in certain local systems.