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The $RO(C_4)$ cohomology of the infinite real projective space

Nick Georgakopoulos

Abstract

Following the Hu-Kriz method of computing the $C_2$ genuine dual Steenrod algebra $(H\mathbf F_2)_{\bigstar}(H\mathbf F_2)$, we calculate the $C_4$ equivariant Bredon cohomology of the classifying space $\mathbf R P^{\infty ρ}=B_{C_4}Σ_{2}$ as an $RO(C_4)$ graded Green-functor. We prove that as a module over the homology of a point (which we also compute), this cohomology is not flat. As a result, it can't be used as a test module for obtaining generators in $(H\mathbf F_2)_{\bigstar}(H\mathbf F_2)$ as Hu-Kriz do in the $C_2$ case.

The $RO(C_4)$ cohomology of the infinite real projective space

Abstract

Following the Hu-Kriz method of computing the genuine dual Steenrod algebra , we calculate the equivariant Bredon cohomology of the classifying space as an graded Green-functor. We prove that as a module over the homology of a point (which we also compute), this cohomology is not flat. As a result, it can't be used as a test module for obtaining generators in as Hu-Kriz do in the case.

Paper Structure

This paper contains 29 sections, 18 theorems, 166 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Conventions and notations
  3. The Tate diagram for C2 and C4
  4. The Tate diagram for C2
  5. The RO(C4) homology of a point
  6. The Tate diagram for C4
  7. Equivariant classifying spaces
  8. The case of C2
  9. The cohomology of the C4 classifying space of O(1)
  10. A cellular decomposition of the C4 classifying space of O(1)
  11. A decomposition using trivial spheres
  12. The spectral sequence for the C4 classifying space of O(1)
  13. The E1 page
  14. The d1 differentials
  15. Bottom level computation
  16. ...and 14 more sections

Key Result

Proposition 5.1

There exist elements $e^a,e^u,e^{\lambda},e^{\rho}$ in degrees $\sigma+\lambda,\sigma+\lambda-2, \lambda, \rho$ of $k^{\bigstar}_{C_4}(B_{C_4}\Sigma_{2+})$ respectively, such that The relation set $S$ consists of two types of relations (we use indices $i,j\ge 0$): The middle level of $k^{\bigstar}(B_{C_4}\Sigma_{2+})$ is generated by the restrictions of $e^a,e^u,e^\lambda,e^\rho$, which we denot

Figures (5)

  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (36)

  • Proposition 5.1
  • Proposition 5.2
  • Proposition 5.3
  • proof
  • Remark 7.1
  • Remark 7.2
  • Proposition 7.3
  • proof
  • Remark 7.4
  • Proposition 7.5
  • ...and 26 more