Purity of quaternionic conjugation spaces

Conjugation spaces relate the cohomology of a space and its fixed points via a degree-halving isomorphism and admit a characterization in terms of homological purity. We extend this framework to the Klein four group, where the corresponding structures exhibit a degree-quartering behavior governed by Dickson invariants. Under a mild assumption, we prove that quaternionic conjugation spaces are homologically pure. As an application, we show that such spaces are both ${\mathcal{K}}_4$-maximal and ${\mathcal{K}}_4$-Galois maximal, establishing a connection with Smith–Thom type inequalities in real algebraic geometry.