Realizing doubles: a conjugation zoo

TL;DR

This paper explores conjugation spaces, constructs exotic examples, and investigates which spaces can be realized as fixed points of conjugation involutions, identifying obstructions and providing new insights into their structure.

Contribution

It introduces exotic conjugation spaces and analyzes the realization problem, showing which spaces can or cannot be realized as real loci of conjugation spaces.

Findings

Constructed new examples of exotic conjugation spaces.

Identified obstructions to realizing certain spaces as real loci.

Provided examples of spaces and manifolds not realizable as fixed points.

Abstract

Conjugation spaces are topological spaces equipped with an involution such that their 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 two, generalizing the classical examples of complex projective spaces under complex conjugation. Spaces which are constructed from unit balls in complex Euclidean spaces are called spherical and are very well understood. Our aim is twofold. We construct "exotic" conjugation spaces and study the realization question: which spaces can be realized as real loci, i.e., fixed points of conjugation spaces. We identify obstructions and provide examples of spaces and manifolds which cannot be realized as such.