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Realizing doubles: a conjugation zoo

Part of: Homotopy theory Operations and obstructions

Published online by Cambridge University Press:  08 April 2020

Wolfgang Pitsch  and
Jérôme Scherer
Wolfgang Pitsch
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193Bellaterra, Spain ([email protected])
Jérôme Scherer
Affiliation:
EPFL, Mathematics, Station 8, CH-1015Lausanne, Switzerland ([email protected])
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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.

Keywords

Conjugation spacesrealizationHopf invariant

MSC classification

Primary: 55P91: Equivariant homotopy theory
Secondary: 57S17: Finite transformation groups 55S10: Steenrod algebra 55S35: Obstruction theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 2 , April 2021 , pp. 509 - 524
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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