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The $C_2$-equivariant ordinary cohomology of $BT^2$

Steven R. Costenoble, Thomas Hudson

Abstract

We calculate the ordinary $C_2$-cohomology of $BT^2$ with Burnside ring coefficients, using an extended grading that allows us to capture a more natural set of generators. We discuss how this cohomology is related to those of $BT^1$ and $BU(2)$, calculated previously, both relationships being more complicated than in the nonequivariant case.

The $C_2$-equivariant ordinary cohomology of $BT^2$

Abstract

We calculate the ordinary -cohomology of with Burnside ring coefficients, using an extended grading that allows us to capture a more natural set of generators. We discuss how this cohomology is related to those of and , calculated previously, both relationships being more complicated than in the nonequivariant case.

Paper Structure

This paper contains 16 sections, 29 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. A preliminary calculation
  3. The proposed structure
  4. The cohomology of the fixed set $(B T^2)^{C_2}$
  5. The cohomology of $B T^2$
  6. Restriction to nonequivariant cohomology
  7. Restriction to fixed points
  8. The cohomology of $B T^2$
  9. Examples of bases
  10. Units and dual elements
  11. Notes on the relations
  12. Euler classes of tensor products
  13. $B T^2$ as a projective bundle over $B U(2)$
  14. It is not free
  15. The pushforward map to the cohomology of $B U(2)$
  16. ...and 1 more sections

Key Result

Theorem 1.4

As a module, $H_{C_2}^{RO(\Pi BT^1)}(BT^1_+)$ is free over ${\mathbb H}$, and as a commutative algebra we have where $I$ is the ideal generated by the relations ∎

Theorems & Definitions (68)

  • Theorem 1.4: Co:InfinitePublished
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • proof
  • Definition 3.1
  • Proposition 3.3
  • ...and 58 more