Systems of quadratic inequalities

We present a spectral sequence which efficiently computes Betti numbers of a closed semi-algebraic subset of $\mathbb{R}\mathrm{P}^n$ defined by a system of quadratic inequalities and the image of the homology homomorphism induced by the inclusion of this subset in $\mathbb{R}\mathrm{P}^n$. We do not restrict ourselves to the term $E_2$ of the spectral sequence and give a simple explicit formula for the differential $d_2$.