Symmetric products of Galois-Maximal varieties

The main result of this paper is the proof that all the symmetric products of a finite Galois-Maximal space are Galois-Maximal spaces. This applies to the special case of real algebraic varieties, solving the problem first stated by Biswas and D'Mello in biswas&d'mello: symmetricₚroductsM-curves about symmetric products of Maximal curves. We also give characterisations of these spaces and state a new definition that generalises to a larger class of spaces. Then, we extend the characterisation in terms of the Borel cohomology given in us to the new family. Finally, we introduce the notion of cohomological stability and cohomological splitting, provide a systematic treatment and relate them with the properties of being a Maximal or Galois-Maximal space. These cohomological properties play an important role in the proof of our main theorem.