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Chow-Witt rings of Grassmannians

Matthias Wendt

Abstract

We complement our previous computation of the Chow-Witt rings of classifying spaces of special linear groups by an analogous computation for the general linear groups. This case involves discussion of non-trivial dualities. The computation proceeds along the lines of the classical computation of the integral cohomology of ${\rm BO}(n)$ with local coefficients, as done by Cadek. The computations of Chow-Witt rings of classifying spaces of ${\rm GL}_n$ are then used to compute the Chow-Witt rings of the finite Grassmannians. As before, the formulas are close parallels of the formulas describing integral cohomology rings of real Grassmannians.

Chow-Witt rings of Grassmannians

Abstract

We complement our previous computation of the Chow-Witt rings of classifying spaces of special linear groups by an analogous computation for the general linear groups. This case involves discussion of non-trivial dualities. The computation proceeds along the lines of the classical computation of the integral cohomology of with local coefficients, as done by Cadek. The computations of Chow-Witt rings of classifying spaces of are then used to compute the Chow-Witt rings of the finite Grassmannians. As before, the formulas are close parallels of the formulas describing integral cohomology rings of real Grassmannians.

Paper Structure

This paper contains 34 sections, 37 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Chow--Witt rings of infinite Grassmannians
  3. Chow--Witt rings of finite Grassmannians
  4. Decomposition of $\mathbf{I}$-cohomology
  5. Recollection on Chow--Witt rings
  6. Twisted coefficients and cohomology operations
  7. Oriented intersection product and total Chow--Witt ring
  8. The fundamental square
  9. Decomposing $\mathbf{I}$-cohomology into $\mathbf{W}$-cohomology and the image of $\beta$
  10. Characteristic classes for vector bundles
  11. Setup of localization sequence
  12. Euler class
  13. Chern classes
  14. Pontryagin classes
  15. Stiefel--Whitney classes and their (twisted) Bocksteins
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $F$ be a perfect field of characteristic $\neq 2$.

Theorems & Definitions (86)

  • Theorem 1.1
  • Remark 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 3.1
  • Remark 3.2
  • ...and 76 more