Purity of quaternionic conjugation spaces

TL;DR

This paper extends the concept of conjugation spaces to quaternionic settings, demonstrating homological purity and maximality properties related to Klein four group actions and real algebraic geometry.

Contribution

It introduces quaternionic conjugation spaces, proving their homological purity and maximality, and connects these structures to Smith--Thom inequalities.

Findings

Quaternionic conjugation spaces are homologically pure.

Such spaces are both $\\mathcal{K}_4$-maximal and $\\mathcal{K}_4$-Galois maximal.

Establishes a link with Smith--Thom inequalities in real algebraic geometry.

Abstract

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.