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The $C_2$-equivariant ordinary cohomology of $BU(2)$

Steven R. Costenoble, Thomas Hudson

Abstract

We calculate the ordinary $C_2$-cohomology, with Burnside ring coefficients, of $BU(2)$, the classifying space for $C_2$-equivariant complex 2-plane bundles, using an extended grading that allows us to capture a more natural set of generators. This allows us to define characteristic classes for such bundles. Combined with earlier calculations, it also allows us to define characteristic numbers for equivariant complex lines and surfaces and we give some sample computations.

The $C_2$-equivariant ordinary cohomology of $BU(2)$

Abstract

We calculate the ordinary -cohomology, with Burnside ring coefficients, of , the classifying space for -equivariant complex 2-plane bundles, using an extended grading that allows us to capture a more natural set of generators. This allows us to define characteristic classes for such bundles. Combined with earlier calculations, it also allows us to define characteristic numbers for equivariant complex lines and surfaces and we give some sample computations.

Paper Structure

This paper contains 20 sections, 20 theorems, 155 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A useful monomorphism
  3. The cohomology of the fixed set $B U(2)^{C_2}$
  4. Freeness of the proposed structure
  5. Change of grading
  6. The cohomology of $B U(2)$
  7. Restriction to nonequivariant cohomology
  8. Restriction to fixed points
  9. The cohomology of $B U(2)$
  10. Examples of bases
  11. Units and dual elements
  12. Characteristic numbers of lines and surfaces
  13. Characteristic numbers for lines
  14. Characteristic numbers of $\mathbb{P}({\mathbb C}^{2})$
  15. Characteristic numbers of $\mathbb{P}({\mathbb C}^{1+\sigma})$
  16. ...and 5 more sections

Key Result

Theorem 2.1

Suppose that $B$ is a ${C_2}$-space and suppose that $B^{C_2}$ has a component $B^0$ such that the inclusion $B^0\to B$ is a nonequivariant equivalence. Then we have the following commutative diagram with exact rows and columns and with the indicated isomorphisms, monomorphisms, and epimorphisms. (I

Figures (1)

  • Figure 1: ${\mathbb H} = H_{C_2}^{RO({C_2})}(S^0)$

Theorems & Definitions (42)

  • Theorem 2.1
  • proof
  • Definition 2.2
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 32 more