The real cycle class map

TL;DR

This paper investigates two cycle class maps for smooth real varieties, establishing their compatibility properties and proving they are isomorphisms for cellular varieties, thus extending classical cycle class theory to real algebraic geometry.

Contribution

It introduces a new cycle class map on the Chow-Witt ring and proves both maps are isomorphisms for cellular varieties, enhancing understanding of cycle classes over real numbers.

Findings

Both cycle class maps are compatible with pullbacks, pushforwards, and cup products.

The cycle class maps are isomorphisms for cellular varieties.

A new cycle class map on the Chow-Witt ring is defined and studied.

Abstract

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 basic compatibility results like 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.