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Equivariant real cycle class map and Witt-sheaf cohomology of classifying spaces

Lorenzo Mantovani, Ákos K. Matszangosz, Matthias Wendt

Abstract

In this paper, we study equivariant real cycle class maps for group actions on real schemes, with a view toward Witt-sheaf characteristic classes. The cycle class maps take values in singular cohomology of the real points of the quotient stack, which are identified with the homotopy fixed-points of complex conjugation on the complex points. This provides a strong relation between Witt-sheaf cohomology of the geometric classifying space of a real algebraic group and the singular cohomology of the classifying spaces of its strong real forms, which we discuss in a number of examples. As a sample application, we compute the number of Witt-sheaf cohomological invariants of spin groups over the reals.

Equivariant real cycle class map and Witt-sheaf cohomology of classifying spaces

Abstract

In this paper, we study equivariant real cycle class maps for group actions on real schemes, with a view toward Witt-sheaf characteristic classes. The cycle class maps take values in singular cohomology of the real points of the quotient stack, which are identified with the homotopy fixed-points of complex conjugation on the complex points. This provides a strong relation between Witt-sheaf cohomology of the geometric classifying space of a real algebraic group and the singular cohomology of the classifying spaces of its strong real forms, which we discuss in a number of examples. As a sample application, we compute the number of Witt-sheaf cohomological invariants of spin groups over the reals.
Paper Structure (48 sections, 48 theorems, 130 equations)

This paper contains 48 sections, 48 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Equivariant real cycle class map
  3. Realizations of quotient stacks and homotopy fixed points
  4. Examples: orthogonal and spin groups
  5. Conjectural picture of Witt-sheaf cohomology of classifying spaces
  6. Further directions
  7. Preliminaries on (quotient) stacks
  8. Groups and actions
  9. Quotient stacks, morphisms and relevant examples
  10. Stacks in motivic homotopy
  11. Stacks in motivic homotopy
  12. Homotopy orbit spaces and simplicial approach
  13. Stacks in motivic homotopy: approximation by smooth schemes
  14. Vector bundles and Picard group for stacks
  15. Real points of stacks as fixed-point groupoids
  16. ...and 33 more sections

Key Result

Theorem 1.1

Let $G$ be a linear algebraic group over $\mathbb{R}$ and let $G\looparrowright X$ be a smooth scheme with $G$-action, and let $\mathscr{L}\in {\rm CH}^1_G(X)$ be a $G$-equivariant line bundle on $X$. Then the signature map ${\rm sgn}\colon {\bf I}^q\to a_{\mathrm{r\acute{e}t}}\mathbb{Z}$ induces a which satisfies the following properties:

Theorems & Definitions (156)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 146 more