[Paper Review] Equivariant Spectral Sequences for Local Coefficients

Megan Guichard Shulman|arXiv (Cornell University)|May 3, 2010

Homotopy and Cohomology in Algebraic Topology7 references3 citations

TL;DR

This paper develops equivariant spectral sequences for computing Bredon cohomology with local coefficients, generalizing Eilenberg's nonequivariant spectral sequence using the universal cover and group ring actions. It applies the method to compute the equivariant cohomology of $B_{C_p}O(2)$, showing it is isomorphic to the $\mathbb{Z}/2$-invariant subalgebra of $H^{*}_{C_p}(B_{C_p}SO(2);A)$, generated by specific classes including $D_{2k}$, $D_{2k-1}C$, and $C^2$. The spectral sequence collapses at $E_2$ with no extension problems due to projectivity of homology groups.

ABSTRACT

We recall how a description of local coefficients that Eilenberg introduced in the 1940s leads to spectral sequences for the computation of homology and cohomology with local coefficients. We then show how to construct new equivariant analogues of these spectral sequences and give a worked example of how to apply them in a computation involving the equivariant Serre spectral sequence. This paper contains some of the material in the author's Ph.D. thesis, which also discusses some results of L. Gaunce Lewis on the cohomology of complex projective spaces and corrects some flaws in his paper.

Motivation & Objective

• To develop equivariant analogues of Eilenberg's spectral sequence for homology and cohomology with local coefficients.
• To address the lack of tools for computing equivariant cohomology with nontrivial local systems, especially in Serre spectral sequence applications.
• To correct flaws in Lewis's work on the cohomology of complex projective spaces and provide a rigorous framework for such computations.
• To demonstrate the method via a detailed computation of $H^{*}_{C_p}(B_{C_p}O(2);A \otimes \mathbb{F}_q)$ using the equivariant Eilenberg spectral sequence.
• To show that the Serre spectral sequence collapses at $E_2$ with no extension problems due to projectivity of the homology of $S^\infty$.

Proposed method

• Generalizes Eilenberg's definition of local coefficients via the universal cover $\tilde{X}$ and $R[\pi_1(X)]$-module structures to the equivariant setting.
• Constructs an equivariant Eilenberg spectral sequence using the category $\Pi = \Pi_{C_p} B_{C_p} \mathbb{Z}/2$ and the homology of $S^\infty$ as a $\Pi$-module.
• Applies the spectral sequence to the fibration $f: B_{C_p}O(2) \to B_{C_p}\mathbb{Z}/2$, using the local coefficient system $h_{C_p}^{V+t}(f;A)$.
• Uses the fact that $\mathcal{H}_*(S^\infty)$ is either 0 or $\underline{\mathbb{F}_q}$, both projective, to show $\operatorname{Ext}^u$ vanishes for $u > 0$, implying collapse at $E_2$.
• Identifies $H^*_{C_p}(B_{C_p}O(2);A)$ as the fixed points of $H^*_{C_p}(B_{C_p}SO(2);A)$ under the $\mathbb{Z}/2$-action, using the Hom functor $\operatorname{Hom}_\Pi(\underline{\mathbb{F}_q}, \cdot)$.
• Establishes that the Green functor $\underline{H}^*_{C_p}(B_{C_p}O(2))$ is a sub-Green functor of $\underline{H}^*_{C_p}(\mathbb{C}P(\mathcal{U}))$ on generators $D_{2k}, D_{2k-1}C, C^2$.

Experimental results

Research questions

• RQ1 How can Eilenberg's nonequivariant spectral sequence for local coefficients be generalized to the equivariant setting with $C_p$-actions?
• RQ2 What conditions ensure the collapse of the equivariant Serre spectral sequence with local coefficients, and how can extension problems be avoided?
• RQ3 How does the cohomology of $B_{C_p}O(2)$ relate to that of $B_{C_p}SO(2)$ under the $\mathbb{Z}/2$-action?
• RQ4 What is the structure of the Green functor $\underline{H}^*_{C_p}(B_{C_p}O(2))$ in terms of generators and relations?
• RQ5 Can the equivariant Eilenberg spectral sequence be used to compute $H^*_{C_p}(B_{C_p}O(2);A \otimes \mathbb{F}_q)$ explicitly?

Key findings

• The equivariant Eilenberg spectral sequence collapses at $E_2$ with no extension problems because $\mathcal{H}_*(S^\infty)$ is projective as a $\Pi$-module.
• The $E_2$ page of the Serre spectral sequence is isomorphic to $\operatorname{Hom}_\Pi(\mathcal{H}_*(S^\infty), h_{C_p}^{V+t}(f;A))$, which simplifies due to the vanishing of $\operatorname{Ext}^u$ for $u > 0$.
• The cohomology $H^*_{C_p}(B_{C_p}O(2);A)$ is isomorphic to the $\mathbb{Z}/2$-fixed subalgebra of $H^*_{C_p}(B_{C_p}SO(2);A)$, i.e., $H^*_{C_p}(B_{C_p}O(2);A) \cong H^*_{C_p}(B_{C_p}SO(2);A)^{\mathbb{Z}/2}$.
• The Green functor $\underline{H}^*_{C_p}(B_{C_p}O(2))$ is a sub-Green functor of $\underline{H}^*_{C_p}(B_{C_p}SO(2))$, generated by $D_{2k}$, $D_{2k-1}C$, and $C^2$.
• The computation confirms that $H^*_{C_p}(B_{C_p}O(2);A \otimes \mathbb{F}_q)$ is generated by the even powers of the nonequivariant generator $x$ of $H^*(\mathbb{C}P^\infty)$, restricted to the $\mathbb{Z}/2$-fixed part.
• The map $\operatorname{Hom}_\Pi(\underline{\mathbb{F}_q}, \scr{N})$ is in bijection with $\scr{N}(C_p, x_0)^{\mathbb{Z}/2}$, allowing explicit identification of the cohomology as fixed points.