Real four-dimensional biquadrics

We consider intersections of two real five-dimensional quadrics, which are referred to for brevity as real four-dimensional biquadrics. Their rigid isotopy classes were described long ago: there are 16 such classes. We prove that the rigid isotopy class of a non-singular real four-dimensional biquadric is uniquely determined by the topological type of its real part. To do this, we calculate the dimensions of the cohomology spaces of the real part of a four-dimensional biquadric.