The $RO(G)$-Graded Serre Spectral Sequence

In [Bredon], Bredon created equivariant homology and cohomology theories of $G$-spaces, now called Bredon homology and Bredon cohomology, which yield the usual singular homology and cohomology theories when the group acting is taken to be the trivial group. In [LMM], a cohomology theory for $G$-spaces is constructed that is graded on $RO(G)$, the Grothendieck ring of virtual representation of $G$. This $RO(G)$-graded theory extends Bredon cohomology in the sense that $H^{\underline{n}}(X)=H^{n}_{Br}(X)$ when $\underline{n}$ is the trivial $n$-dimensional representation of $G$.

Many of the usual tools for computing cohomolgy have their counterparts in the $RO(G)$-graded setting. These include Mayer-Vietoris sequences, Künneth theorem, suspension isomorphisms, etc. Missing from the $RO(G)$ computational tool box was an equivariant version of the Serre spectral sequence associated to a fibration $F\rightarrow E\rightarrow B$. Also, perhaps partially because of a lack of this spectral sequence, the theory of equivariant characteristic classes has not yet been developed.

The main result of this paper is to extend the spectral sequence of a $G$-fibration given in [MS] from Bredon cohomology to the $RO(G)$-graded theory with special attention to the case $G=\mathbb{Z}/2$. The restriction to $G=\mathbb{Z}/2$ is for two reasons. The first is that there is a map from Voevodsky’s motivic cohomology and $RO(\mathbb{Z}/2)$-graded equivariant cohomology, and so one can try to answer questions in motivic cohomology by considering instead the relevant equivariant cohomology. The second reason is that the general algebra of Mackey functors is extremely complicated for arbitrary groups (even compact Lie), yet the $G=\mathbb{Z}/2$ case is manageable.

A $p$-dimensional real $\mathbb{Z}/2$-representation $V$ decomposes as $V=(\mathbb{R}^{1,0})^{p-q}\oplus(\mathbb{R}^{1,1})^{q}=\mathbb{R}^{p,q}$ where $\mathbb{R}^{1,0}$ is the trivial representation and $\mathbb{R}^{1,1}$ is the nontrivial 1-dimensional representation. Thus the $RO(\mathbb{Z}/2)$-graded theory is a bigraded theory, one grading measuring dimension and the other measuring the number of “twists”. In this case, we write $H^{V}(X;M)=H^{p,q}(X;M)$ for the $V^{\text{th}}$ graded component of the $RO(\mathbb{Z}/2)$-graded equivariant cohomology of $X$ with coefficients in a Mackey functor $M$. Here is the spectral sequence:

This is a spectral sequence that takes as inputs the Bredon cohomology of the base space with coefficients in the local coefficient system ${\mathcal{H}}^{q,r}(f;M)$ and converges to the $RO(\mathbb{Z}/2)$-graded cohomology of the total space.

This is really a family of spectral sequences, one for each integer $r$. If the Mackey functor $M$ is a ring Mackey functor, then this family of spectral sequences is equipped with a tri-graded multiplication. If $a\in H^{p,0}(X;{\mathcal{H}}^{q,r}(f;M))$ and $b\in H^{p^{\prime},0}(X;{\mathcal{H}}^{q^{\prime},r^{\prime}}(f;M))$, then $a\cdot b\in H^{p+p^{\prime},0}(X;{\mathcal{H}}^{q+q^{\prime},r+r^{\prime}}(f;M))$. There is also an action of $H^{*,*}(pt;M)$ so that if $\alpha\in H^{q^{\prime},r^{\prime}}(pt;M)$ and $a\in H^{p,0}(X;{\mathcal{H}}^{q,r}(f;M))$, then $\alpha\cdot a\in H^{p,0}(X;{\mathcal{H}}^{q+q^{\prime},r+r^{\prime}}(f;M))$.

Under certain connectivity assumptions on the base space, the local coefficients ${\mathcal{H}}^{q,r}(f;M)$ are constant, and the spectral sequence becomes the following. This result is restated and proved as Theorem 3.1.

The coefficient systems ${\mathcal{H}}^{q,r}(f;M)$ and $\underline{H}^{q,r}(F;M)$ that appear in the spectral sequence are explicitly defined in the next section. They are the equivariant versions of the usual local coefficient systems that arise when working with the usual Serre spectral sequence.

This spectral sequence is rich with information about the fibration involved, even in the case of the trivial fibration $id\colon X\rightarrow X$. In this case, the $E_{2}$ page takes the form $E^{p,q}_{2}=H^{p,0}(X;\underline{H}^{q,r}(pt;M))\Rightarrow H^{p+q,r}(X;M)$. Set $M=\underline{\mathbb{Z}/2}$ and consider the case $r=1$. Then $H^{p,0}(X;\underline{H}^{q,r}(pt;\underline{\mathbb{Z}/2}))=0$ if $q\neq 0,1$. The case $q=0$ gives $H^{p,0}(X;\underline{H}^{0,1}(pt;\mathbb{Z}/2))=H^{p,0}(X;\underline{\mathbb{Z}/2})$, and if $q=1$, $H^{p,0}(X;\underline{H}^{1,1}(pt;\underline{\mathbb{Z}/2}))=H^{p}_{sing}(X^{G};\mathbb{Z}/2)$. The spectral sequence then has just two non-zero rows as shown in Figure 1 below.

As usual, the two row spectral sequence yields the following curious long exact sequence:

$0\rightarrow H^{0,0}(X)\rightarrow H^{0,1}(X)\rightarrow 0\rightarrow H^{1,0}(X)\rightarrow H^{1,1}(X)\rightarrow H^{0}_{sing}(X^{G})\rightarrow H^{2,0}(X)\rightarrow H^{2,1}(X)\rightarrow H^{1}_{sing}(X^{G})\rightarrow\cdots$

Now, to any equivariant vector bundle $f\colon E\rightarrow X$, there is an associated equivariant projective bundle $\mathbb{P}(f)\colon\mathbb{P}(E)\rightarrow X$ whose fibers are lines in the fibers of the original bundle. Applying the above spectral sequence to this new bundle yields the following result, which appears later as Theorem LABEL:thm:localss.

Here, when we say the spectral sequence collapses, we do not mean it collapses in the usual sense. Each fibration $f\colon E\rightarrow X$ maps to the trivial fibration $id\colon X\rightarrow X$ in an obvious way. Naturality then provides a map from the spectral sequence for $id\colon X\rightarrow X$ to the spectral sequence for $f\colon E\rightarrow X$. In the above theorem, the spectral sequence “collapses” in the sense that the only nonzero differentials are those arising from the trivial fibration $id\colon X\rightarrow X$.

In non-equivariant topology, the Leray-Serre spectral sequence gives rise to a description of characteristic classes of vector bundles. Consider the universal bundle $E_{n}\rightarrow G_{n}$ over the Grassmannian of $n$-planes in $\mathbb{R}^{\infty}$. Forming the associated projective bundle $\mathbb{P}(E_{n})\rightarrow G_{n}$ yields a fiber bundle with fiber $\mathbb{R}\mathbb{P}^{\infty}$. Applying the Leray-Serre spectral sequence to this projective bundle yields characteristic classes of $E_{n}$ as the image of the cohomology classes $1,z,z^{2},\dots\in H^{*}_{sing}(\mathbb{R}\mathbb{P}^{\infty};\mathbb{Z}/2)$ under the transgressive differentials. Since this universal bundle classifies vector bundles, characteristic classes of arbitrary bundles can be defined as pullbacks of the characteristic classes, $c_{i}\in H^{i}(G_{n};\mathbb{Z}/2)$, of the universal bundle. It would be nice to adapt this construction to the $\mathbb{Z}/2$ equivariant setting. However, the equivariant space $G_{n}((\mathbb{R}^{2,1})^{\infty})=G_{n}({\mathcal{U}})=G_{n}$ is not 1-connected, and so the spectral sequence is not as easy to work with. It seems that there is no way to avoid using local coefficient systems in this setting.

Section 2 provides some of the definitions and basics that are needed for this paper. The main theorem is stated and proved in section 3, making use of some technical homotopical details that are provided in section LABEL:sec:homotopical. In section LABEL:sec:ProjBundles, the spectral sequence is then applied to compute the cohomology of a projective bundle $\mathbb{P}(E)$ associated to a vector bundle $E\rightarrow X$. Section LABEL:sec:AH provides an application of the $RO(\mathbb{Z}/2)$-graded Serre spectral sequence to loop spaces on certain spheres.

The section contains some of the basic machinery and notation that will be used throughout the paper. In this section, let $G$ be any finite group.

A $G$-CW complex is a $G$-space $X$ with a filtration $X^{(n)}$ where $X^{(0)}$ is a disjoint union of $G$-orbits and $X^{(n)}$ is obtained from $X^{(n-1)}$ by attaching cells of the form $G/H_{\alpha}\times\Delta^{n}$ along maps $f_{\alpha}\colon G/H_{\alpha}\times\partial\Delta^{n}\rightarrow X^{(n-1)}$. The space $X^{(n)}$ is referred to as the $n$-skeleton of $X$. Such a filtration on a space $X$ is called a cell structure for $X$.

Given a $G$-representation $V$, let $D(V)$ and $S(V)$ denote the unit disk and unit sphere, respectively, in $V$ with action induced by that on $V$. A $\text{Rep}(G)$-complex is a $G$-space $X$ with a filtration $X^{(n)}$ where $X^{(0)}$ is a disjoint union of $G$-orbits and $X^{(n)}$ is obtained from $X^{(n-1)}$ by attaching cells of the form $D(V_{\alpha})$ along maps $f_{\alpha}\colon S(V_{\alpha})\rightarrow X^{(n-1)}$ where $V_{\alpha}$ is an $n$-dimensional real representation of $G$. The space $X^{(n)}$ is again referred to as the $n$-skeleton of $X$, and the filtration is referred to as a cell structure.

Let $\Delta_{G}(X)$ be the category of equivariant simplices of the $G$-space $X$. Explicitly, the objects of $\Delta_{G}(X)$ are maps $\sigma\colon G/H\times\Delta^{n}\rightarrow X$. A morphism from $\sigma$ to $\tau\colon G/K\times\Delta^{m}\rightarrow X$ is a pair $(\varphi,\alpha)$ where $\varphi\colon G/H\rightarrow G/K$ is a $G$-map and $\alpha\colon\Delta^{n}\rightarrow\Delta^{m}$ is a simplicial operator such that $\sigma=\tau\circ(\varphi\times\alpha)$.

Let $\Pi_{G}(X)$ be the fundamental groupoid of $X$. Explicitly, the objects of $\Pi_{G}(X)$ are maps $\sigma\colon G/H\rightarrow X$ and a morphism from $\sigma$ to $\tau\colon G/K\rightarrow X$ is a pair $(\varphi,\alpha)$ where $\varphi\colon G/H\rightarrow G/K$ is a $G$-map and $\alpha$ is a $G-$homotopy class of paths from $\sigma$ to $\tau\circ\varphi$.

There is a forgetful functor $\pi\colon\Delta_{G}(X)\rightarrow\Pi_{G}(X)$ that sends $\sigma\colon G/H\times\Delta^{n}\rightarrow X$ to $\sigma\colon G/H\rightarrow X$ by restricting to the last vertex $e^{n}$ of $\Delta^{n}$. A morphism $(\varphi,\alpha)$ in $\Delta_{G}(X)$ is restricted to $(\varphi,\alpha)$ in $\Pi_{G}(X)$ by restricting $\alpha$ to the linear path from $\alpha(e^{n})$ to $e^{m}$ in $\Delta^{m}$. There is a further forgetful functor to the orbit category ${\mathcal{O}}(G)$, which will also be denoted by $\pi$, as shown below.

A coefficient system on $X$ is a functor $M\colon\Delta_{G}(X)^{op}\rightarrow{\mathcal{A}b}$. We say that the coefficient system $M$ is a local coefficient system if it factors through the forgetful functor to $\Pi_{G}(X)^{op}$ (up to isomorphism). If $M$ further factors through ${\mathcal{O}}(G)^{op}$, then we call $M$ a constant coefficient system.

For the precise definition of a Mackey functor for $G=\mathbb{Z}/2$, the reader is referred to [Alaska] or [DuggerKR]. A summary of the important aspects of a $\mathbb{Z}/2$ Mackey functor is given here. The data of a Mackey functor are encoded in a diagram like the one below.

The maps must satisfy the following four conditions.

1. (1)

   $(t^{*})^{2}=id$

2. (2)

   $t^{*}i^{*}=i^{*}$

3. (3)

   $i_{*}(t^{*})^{-1}=i_{*}$

4. (4)

   $i^{*}i_{*}=id+t^{*}$

According to [LMM], each Mackey functor $M$ uniquely determines an $RO(G)$-graded cohomology theory characterized by

1. (1)

   $H^{\underline{n}}(G/H;M)=\begin{cases}M(G/H)&\text{ if }n=0\\ 0&\text{otherwise}\end{cases}$

2. (2)

   The map $H^{0}(G/K;M)\rightarrow H^{0}(G/H;M)$ induced by $i\colon G/H\rightarrow G/K$ is the transfer map $i^{*}$ in the Mackey functor.

Given a Mackey functor $M$, a $G$-representation $V$, and a $G$-space $X$, we can form a coefficient system $\underline{H}^{V}(X;M)$. This coefficient system is determined on objects by $\underline{H}^{V}(X;M)(G/H)=H^{V}(X\times G/H;M)$ with maps induced by those in ${\mathcal{O}}(G)$.

In this paper, $G$ will usually be $\mathbb{Z}/2$ and the Mackey functor will almost always be constant $M=\underline{\mathbb{Z}/2}$ which has the following diagram.

With these constant coefficients, the $RO(\mathbb{Z}/2)$-graded cohomology of a point is given by the picture in Figure 2.

Every lattice point in the picture that is inside the indicated cones represents a copy of the group $\mathbb{Z}/2$. The top cone is a polynomial algebra on the elements $\rho\in H^{1,1}(pt;\underline{\mathbb{Z}/2})$ and $\tau\in H^{0,1}(pt;\underline{\mathbb{Z}/2})$. The element $\theta$ in the bottom cone is infinitely divisible by both $\rho$ and $\tau$. The cohomology of $\mathbb{Z}/2$ is easier to describe: $H^{*,*}(\mathbb{Z}/2;\underline{\mathbb{Z}/2})=\mathbb{Z}/2[t,t^{-1}]$ where $t\in H^{0,1}(\mathbb{Z}/2;\underline{\mathbb{Z}/2})$. Details can be found in [DuggerKR] and [Caruso].

Given a $G$-map $f\colon E\rightarrow X$ and a Mackey functor $M$, we can define a coefficient system ${\mathcal{H}}^{q,r}(-,M)\colon\Delta_{G}(X)^{op}\rightarrow{\mathcal{A}b}$ by taking cohomology of pullbacks: ${\mathcal{H}}^{q,r}(f,M)(\sigma)=H^{q,r}(\sigma^{*}(E),M)$. In [MS], it is shown that this is a local coefficient system when $f$ is a $G$-fibration.

Given a $G$-fibration $f\colon E\rightarrow X$, we can define a functor $\Gamma_{f}\colon\Delta_{G}(X)\rightarrow{\mathcal{T}op}$. On objects, $\Gamma(\sigma)=\sigma^{*}(E)$. On morphisms, $\Gamma(\varphi,\alpha)=\overline{\varphi\times\alpha}$, where $\overline{\varphi\times\alpha}$ is the map of total spaces in the diagram

Let $f\colon E\rightarrow X$ be a $G$-fibration over an equivariantly 1-connected $G$-space $X$ with base point $x\in X$ and let $F=f^{-1}(x)$. Define a constant coefficient system $\underline{H}^{q,r}(F;M)$ as follows: $\underline{H}^{q,r}(F;M)(G/H)=H^{q,r}((G/H)\times F;M)$ and if $\varphi\colon G/H\rightarrow G/K$ is a $G$-map, then $\underline{H}^{q,r}(F;M)(\varphi)=(\varphi\times id)^{*}$. It is this coefficient system that appears in the spectral sequence of Theorem LABEL:thm:localss.

Unlike in ordinary topology, the equivariant Serre spectral sequence for a fibration $f\colon E\rightarrow X$ will not be deduced from lifting a cellular filtration of $X$ to one on $E$. Instead, the spectral sequence is a special case of the one for a homotopy colimit. Recall (from [hocolim] for example) that given a cohomology theory ${\mathcal{E}}^{*}$ and a diagram of spaces $D\colon I\rightarrow{\mathcal{T}op}_{G}$, there is a natural spectral sequence as follows:

For the case $I=\Delta_{G}(X)$, we know from [MS] that the cohomology of $\Delta_{G}(X)^{op}$ is the same as Bredon cohomology. For a $G$-fibration $f\colon E\rightarrow B$, we can consider the diagram $\Gamma_{f}\colon\Delta_{G}(X)\rightarrow{\mathcal{T}op}_{G}$ that sends $\sigma\colon G/H\times\Delta^{n}\rightarrow X$ to the pullback $\sigma^{*}(E)$. We then have the following technical lemma, whose proof is given in the next section where it appears as Lemma LABEL:lemma:hocowe.

Here is the desired spectral sequence.

The standard multiplicative structure on the spectral sequence is given by the following theorem. Recall that the analogue of tensor product for Mackey functors is the box product, denoted by $\Box$. See, for example, [FL] for a full description of the box product.