A conjugation space, as introduced by J.-C. Hausmann et al. [Algebr. Geom. Topol. 5, 923–964 (2005; Zbl 1081.55006)] is a space equipped with an involution such that the fixed points have the same mod \(2\) cohomology (as a graded vector space, a ring, and even an unstable algebra) but with all degrees divided by \(2\). The best known and most common examples of conjugation spaces are cellular, in the sense that they arise from conjugation spheres, by attaching conjugation cells.
To understand how close arbitrary conjugation spaces are from being spherical, the authors follow the guiding principle brought up by the second author and recast the definition of conjugation spaces in the equivariant stable world. They make use of [M. A. Hill et al., Ann. Math. (2) 184, No. 1, 1–262 (2016; Zbl 1366.55007)] to introduce the notion of homological purity of an equivariant space. The main result of the present work is a stable equivariant characterization of conjugation spaces in terms of this notion of purity.
This conceptual viewpoint is used to exhibit some properties of conjugation spaces. In particular, the compatibility of the conjugation frame with Steenrod operations is shown and the reduced mod \(2\) cohomology group \(H^\ast(X)\), and the Borel cohomology \(H^\ast(X_{h\mathbb{C}_2})\) of \(X\) for the cyclic group \(\mathbb{C}_2\) of order \(2\), are related via the “derived functor of the destabilization functor” of [J. Lannes and S. Zarati, Math. Z. 194, 25–59 (1987; Zbl 0627.55014)].