Halves of a real Enriques surface

The real partE ℝ of a real Enriques surfaceE admits a natural decomposition in two halves,E ℝ=E ℝ (1) ∪E ℝ (2) , each half being a union of components ofE ℝ. We classify the triads (E ℝ;E ℝ (1) ,E ℝ (2) ) up to homeomorphism. Most results extend to surfaces of more general nature than Enriques surfaces. We use and study in details the properties of Kalinin's filtration in the homology of the fixed point set of an involution, which is a convenient tool not widely known in topology of real algebraic varieties.