## Serre Spectral Sequence of Representations

Suppose that $G$ is a group acting on a fibre bundle $(F,E,B)$ by bundle automorphisms. In this case, the action automorphisms $E\to E$ give the integral homology $H_\ast(E;\mathbb{Z})$ the structure of a $G$-module. Also, the action automorphisms $F\to F$ and $B\to B$ give each module $H_p(B;H_q(F;\mathbb{Z}))$ the structure of a $G$-module. Can the Serre spectral sequence of the fibre bundle be made $G$-equivariant in the sense of being a spectral sequence of $G$-modules converging to $H_\ast(E;\mathbb{Z})$, and with second page $E_{p,q}=H_p(B;H_q(F))$ (considered as $G$-modules in the above sense)?

Thanks!

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Be careful: G does not act on the fibers. For b in B, only the isotropy group G_b acts on the fiber over b. – Peter May Feb 14 at 0:21
You are quite correct. My original description was definitely misleading in that regard. Thanks! – PDC Feb 14 at 0:30
The Serre spectral sequence is functorial. Doesn't this give it to you? – Lennart Meier Feb 14 at 5:25

Dylan, I am afraid that you are missing something. It is true that each element of the group gives a map of spectral sequences. But look at the fibers and my comment about isotropy groups above. If you construct the Serre spectral sequence in the usual way, there is a non-equivariantly inconsequential choice of base point in the base space. But that is no longer inconsequential equivariantly. You do not get to collate your actions one element of the group at a time into an algebraically meaningful action of the group on "the" spectral sequence. If you try to express the action of $G$ on the $E_2$ term algebraically, you will see the point. Unfortunately for calculations, the "right" way to think about the question is to switch to Bredon cohomology, say with the constant integer coefficient system. Then there is an equivariant Serre spectral sequence, developed by Moerdijk and Svensson: The equivariant Serre spectral sequence. Proc. Amer. Math. Soc. 118 (1993), no. 1, 263–278. While there has been more recent theoretical work, there have been few serious calculations with it.

Edit: Dylan, in answer to your question below, the incorporation of varying fixed point data (as in the base space here) is just the kind of thing Bredon cohomology is designed for. For a simple but serious use of Bredon cohomology to obtain information about ordinary cohomology see "A generalization of Smith theory" (#57 on my web page). For a nice exposition and a worked example of the equivariant Serre spectral sequence, see Megan Shulman's "Equivariant spectral sequences for local coefficients" arXiv:1005.0379

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 Uh oh! Thank you for pointing that out (hopefully PDC sees this and accepts your answer). Is there no way to avoid this at the possible cost of keeping track of local coefficient systems? – Dylan Wilson Feb 15 at 16:46

: This is wrong!! See Peter's answer below.

Unless I'm missing something, this follows from the naturality of the Serre spectral sequence. That is, each element of $g$ gives a map of fiber sequences, whence a map of Serre spectral sequences converging to the map on $H_*(E;\mathbb{Z})$.

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 I suspected that some form of naturality would prove relevant. My interpretation of naturality in this context is as functor from some category of fibrations to spectral sequences of Z-modules. Could you direct me to a reference? Thanks! – PDC Feb 13 at 23:54 See, for example, page 18 here: math.cornell.edu/~hatcher/SSAT/SSch1.pdf – Dylan Wilson Feb 14 at 1:19 (Though Hatcher makes unnecessary assumptions on the base... a general statement can be found in May and Ponto's new book Theorem 24.5.1, or in "A user's guide to spectral sequences", or in myriad other places... It's also a good exercise!) – Dylan Wilson Feb 14 at 1:22 Excellent! Thank you! – PDC Feb 14 at 15:18

Personal comment: I feel that the two previous answers may be together creating some confusion on the subject of the question. I wish to address that with my answer, which would be more suited as a comment, were it not so long. I certainly hope my contribution will not result in even more confusion. $\newcommand{\Ab}{\mathrm{Ab}} \newcommand{\To}{\longrightarrow} \newcommand{\rightset}[2]{#2\rlap{\scriptstyle #1}} \newcommand{\leftset}[2]{\llap{\scriptstyle #1}#2} \newcommand{\label}[1]{\qquad\qquad \text{#1}}$

Peter May's answer is quite interesting and, incisively, cautions us to pay attention to the requisite details. On the other hand, I believe his answer brings some unnecessary sophistication to a rather simple situation. More precisely, I think that Dylan Wilson's original answer, while it ignored all details, was essentially correct (if one replaces "fibre sequences" by "fibrations" in what he writes). The only subtle point is the functorial identification of the $E^2$-term of a Serre fibration $f:E\to B$, which fundamentally requires doing without a basepoint for $B$, and without the corresponding distinguished fibre of $f$.

As Peter May indicates, in the description of the Serre spectral sequence for a fibration $f:E\to B$, one usually assumes:

1. $B$ is path connected, and pointed;

2. the monodromy action of $\pi_1 B$ on $H_\ast F$ is trivial, where $F$ is the fibre of $f$ over the basepoint of $B$.

The second condition is a useful simplification which applies in most cases. Moreover, together with hypothesis 1, it allows us to write the $E^2$-term in the following neat, well-known form: $$E^2_{p,q}=H_p(B,H_q(F)) \label{(I)}$$ Obviously, without choosing a basepoint for $B$ that expression does not even make sense! Regardless, we can construct the Serre spectral sequence more generally: see, for example, the book More concise algebraic topology by Peter May and Kate Ponto for a generalization in which condition 2 does not necessarily hold.

For reference, I will briefly describe a common construction of the Serre spectral sequence for any Serre fibration $f:E\to B$; no assumptions are made on $B$ or $f$. We will construct it, without loss of generality, when $B$ is a CW-complex. Then, for a general topological space $B$, we first replace it with a functorial CW-approximation, and pull back the fibration $f:E\to B$ along the natural map from the approximation to $B$. Importantly, the CW-approximation gives a functor from the category of topological spaces to the category of CW-complexes and cellular maps.

For a CW-complex $B$ the spectral sequence is easy to construct. Since $B$ is a CW-complex, we can filter it by its skeleta. The inverse image by $f:E\to B$ of these skeleta gives a filtration on $E$. The Serre spectral sequence is then the spectral sequence associated with the corresponding filtration on the singular chain complex of $E$.

Observe that the functoriality of the above spectral sequence (even starting from the $E^1$-term) with respect to maps of Serre fibrations is assured by:

• the naturality of the CW-approximation of a space;

• the functoriality of the spectral sequence associated with a filtered chain complex.

A map of Serre fibrations $f\to \overline{f}$ is simply a commutative square $$\begin{matrix} E & \To & \overline{E} \\ \leftset{f}{\Big\downarrow} & & \rightset{\overline{f}}{\Big\downarrow} \\ B & \To & \overline{B} \\ \end{matrix}$$ In particular, if a group $G$ acts on a Serre fibration by such maps, then we automatically obtain an induced action of $G$ on the corresponding Serre spectral sequence. Therefore, it becomes a spectral sequence of $G$-modules converging to the homology of the total space seen as a $G$-module. This is an immediate consequence of the exactness of the forgetful functor from $G$-modules to abelian groups. That conclusion should answer the question, apart from the identification of the $E^2$-term, which is discussed below.

Indispensably, the fibre of the Serre fibration did not actually figure into the above construction of the Serre spectral sequence. In fact, the fibres only make an appearance once one tries to compute the $E^2$-term. Nevertheless, one can also identify the $E^2$-term in this general case, and moreover, in a functorial manner. Given a Serre fibration $f:E\to B$, the $E^2$-term of the general Serre spectral sequence described above is naturally isomorphic to (which I write here as equal) $$E^2_{p,q}=H_p(B,H_q(F_\bullet)) \label{(II)}$$ with the naturality holding with respect to maps of Serre fibrations. Here is a description of the right hand side of (II). Let $F_x=f^{-1}(x)$ be the fibre of $f$ over $x\in B$. In the above expression, $H_q(F_\bullet)$ represents the system of local coefficients on $B$ corresponding to the functor $$H_q(F_\bullet):\Pi_1(B)\To\Ab$$ from the fundamental groupoid of $B$ to the category of abelian groups. This functor takes a point $x\in B$ to $H_q(F_x)$, and a path homotopy class $[\gamma]$ of paths in $B$ to the map induced by the monodromy along $\gamma$ on the homology of the fibres over the endpoints of $\gamma$. Then the right hand side of (II) is the homology of $B$ for this system of local coefficients. It is straightforward to check this homology is indeed functorial with respect to maps of Serre fibrations.

Disclaimer: I believe that the functorial identification (II) of the $E^2$-term follows the same recipe as the usual derivation of the isomorphism (I) obtained under conditions 1 and 2. To be honest, I have not carefully checked the details myself! Please let me know if I am in error. It appears to be considered folklore knowledge. Nevertheless, I cannot actually provide a complete reference to the identification (II) of the $E^2$-term. The only textbooks I have found which refer to local coefficients in the Serre spectral sequence are the aforementioned book by May and Ponto, and McCleary's A user guide to spectral sequences. The former skips the proof in the general case, but at first sight the proof in McCleary's book does seem to give (II) as a natural isomorphism, even if the result is not stated explicitly. Can anyone provide a better reference?

In particular, this functorial identification of the $E^2$-term finishes the answer to the question. The $E^2$-term, identified as the right hand side of (II), becomes a $G$-module in the canonical manner when $G$ acts on a Serre fibration. The fundamental ingredient was that (II) removed any reference to a basepoint of the base space of the fibration.

Finally, when we do have a basepoint for $B$, the naturality of the identification of the $E^2$-term as in (I) follows from the general case (II). After all, under conditions 1 and 2 above, the local coefficients over $B$ appearing in (II) are trivial, and canonically identifiable with the homology of the fibre $F$ of $f$ over the basepoint of $B$. The functoriality of the right hand side of (I) is the obvious one, at least with respect to maps of Serre fibrations which preserve the basepoint of the base spaces. For maps which do not preserve the basepoints, the functoriality makes use of the canonical identification of the homology of any two fibres of $f:E\to B$, which is a consequence of condition 2. In case condition 2 does not hold, this identification is impossible, and one is stuck working with local coefficients: it is then best to forego the basepoint of $B$ entirely.

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You ask for a reference on local coefficients: I like Whitehead: Elements of Homotopy Theory. – Ralph Mar 10 at 13:51
Ricardo, thanks for the nice addition. Here is a bet: I bet that if you look at what you wrote and, say, Moerdijk and Svensson, you will see that your answer is giving a special case of the Serre spectral sequence in Bredon cohomology, which I do contend is more natural to the equivariant situation. For a nice complete exposition I again refer you to Megan Shulman's "Equivariant spectral sequences for local coefficients" arXiv:1005.0379. She uses a beautiful old treatment of local coefficients that goes back to Eilenberg and fits perfectly. – Peter May Mar 10 at 14:00
@Ralph: Thank you. I will take a look at it. @Peter: That seems very plausible, indeed. Thank you for the references. As I mentioned, my answer was more suited as a comment, but I always write way too much. I just wanted to make clear that the ordinary Serre spectral sequence does indeed work, as long as one has a precise enough statement of it which also provides functoriality. – Ricardo Andrade Mar 10 at 21:27