Obstruction theory for the $\mathbb{Z}_2$-index of $4$-manifolds

We develop a complete obstruction theory for the $\mathbb{Z}_2$-index of a compact connected 4-dimensional manifold with free involution. This $\mathbb{Z}_2$-index, equal to the minimum integer $n$ for which there exists an equivariant map with target the $n$-sphere with antipodal involution, is computed in two steps using cohomology with twisted coefficients. The key ingredient is a spectral sequence computing twisted cohomology of the orbit space of a free involution on odd complex projective spaces. We illustrate the main results with various examples including computation of the secondary obstruction.