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.
