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Realizing doubles: a conjugation zoo

Published online by Cambridge University Press:  08 April 2020

Wolfgang Pitsch
Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain (
Jérôme Scherer
EPFL, Mathematics, Station 8, CH-1015 Lausanne, Switzerland (


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.

Research Article
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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