Realizing doubles: a conjugation zoo

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.

Floyd's manifold is a conjugation space

“…We give here an affirmative answer to this conjecture, after several attempts. This problem appears as an open question in our previous article on the subject, [7], where this was only proved for the suspension spectrum in the stable homotopy category. The main problem is to endow F 10 with an action by the group of two elements C 2 in such a way that it becomes a conjugation space with fixed points F 5 .…”

Conjugation spaces are cohomologically pure

“…In this article, we address the question whether a conjugation frame is purely algebraic or if the maps and have some geometric meaning. Even if one can construct ‘exotic’ conjugation spaces, which we do in a separate paper [30], the best‐known and most common examples of conjugation spaces are cellular, in the sense that they arise from conjugation spheres, [12, Example 3.6], by attaching conjugation cells. The two‐dimensional sphere is the one‐point compactification of the field of complex numbers endowed with complex conjugation and higher, even dimensional, conjugation spheres are obtained analogously from .…”