# Conjugation Spaces are Cohomologically Pure

## TL;DR

This paper characterizes conjugation spaces, which have involutions with fixed points sharing mod 2 cohomology, using stable equivariant homotopy theory to provide a conceptual understanding of their structural properties.

## Contribution

It offers a new characterization of conjugation spaces through the lens of purity in stable equivariant homotopy theory, simplifying their structural analysis.

## Key findings

- All known structural properties of conjugation spaces are recovered.
- A new, conceptual characterization of conjugation spaces is established.
- The approach generalizes classical examples like complex projective spaces.

## Abstract

Conjugation spaces are 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, generalizing the classical examples of complex projective spaces under complex conjugation. Using tools from stable equivariant homotopy theory we provide a characterization of conjugation spaces in terms of purity. This conceptual viewpoint, compared to the more computational original definition, allows us to recover all known structural properties of conjugation spaces.