Vol. 6, No. 2, 2021

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The real cycle class map

Jens Hornbostel, Matthias Wendt, Heng Xie and Marcus Zibrowius

Vol. 6 (2021), No. 2, 239–317

The classical cycle class map for a smooth complex variety sends cycles in the Chow ring to cycles in the singular cohomology ring. We study two cycle class maps for smooth real varieties: the map from the I-cohomology ring to singular cohomology induced by the signature, and a new cycle class map defined on the Chow–Witt ring. For both maps, we establish compatibility with pullbacks, pushforwards and cup products. As a first application of these general results, we show that both cycle class maps are isomorphisms for cellular varieties.

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I-cohomology, singular cohomology, Chow–Witt rings, real realization, real cellular varieties
Mathematical Subject Classification 2010
Primary: 14F25
Secondary: 19G12
Received: 5 December 2019
Revised: 5 November 2020
Accepted: 22 November 2020
Published: 1 August 2021
Jens Hornbostel
Fachgruppe Mathematik und Informatik
Bergische Universität Wuppertal
Matthias Wendt
Fachgruppe Mathematik und Informatik
Bergische Universität Wuppertal
Heng Xie
Fachgruppe Mathematik und Informatik
Bergische Universität Wuppertal
Marcus Zibrowius
Mathematisches Institut
Heinrich Heine University Düsseldorf