# Serre classes and the Serre spectral sequence

Let $$C$$ be a Serre class which satisfies the additional axioms about $$\otimes, \mathrm{Tor}, K(A,1)$$'s.

It is then easy to check that if $$F\to X\to B$$ is a Serre fibration with $$\pi_1(B)$$ acting trivially on the homology of $$F$$, and the (positive) homologies of $$B,F$$ are in $$C$$, then the same is true of $$X$$.

(For that we don't even need $$K(A,1)$$'s)

Indeed it follows easily from the Serre spectral sequence and the universal coefficient theorem applied on $$E^2$$.

I'm wondering if this holds when $$\pi_1(B)$$ acts nontrivially.

Is it true that under the hypotheses (minus the hypothesis on the action) $$H_p(B, H_q(F)) \in C$$ ? (Here it's therefore homology with local coefficients) If not, does $$H_p(X) \in C$$ still hold for some other reason ?

(I'm only interested in homology in strictly positive degrees)

EDIT : more specifically, here are all the axioms for $$C$$:

1) if $$0\to M\to N\to L\to 0$$ is a short exact sequence, $$N\in C \iff M,L \in C$$

2) If $$A,B\in C, A\otimes B, \mathrm{Tor}(A,B) \in C$$

3) If $$A\in C$$, for all $$k>0, H_k(K(A,1))\in C$$

Note that if there is a counterexample where $$C$$ doesn't satisfy 3), I'm also interested

2nd Edit : William answered the question I asked, but I'm wondering if anyone has an example with a connected fiber, so I'll leave this as unaccepted for a couple of days to see if one can find an example with connected fiber.

• can you be more explicit about what "the additional axioms" are? Or provide a source where we can go read them? – William Jun 7 at 14:33
• @William : sure, I'll edit (some authors include them in the definition of Serre class) – Max Jun 7 at 14:34

What about the bundle $$\mathbb{Z}/2 \to S^n \to \mathbb{R}P^n$$ where $$n>0$$ is even? Then for all $$k>0$$ the groups $$H_k(\mathbb{Z}/2)$$ and $$H_k(\mathbb{R}P^n)$$ are all in the Serre class of finite groups, but $$H_n(S^n)\cong \mathbb{Z}$$.
• Sure, it's up to you. I'm wondering if it's possible to come up with an example where you just replace the $\mathbb{Z}/2$ fibres in this principal bundle with a connected space $F$ whose positive homology groups are finite and which has an appropriate $\mathbb{Z}/2$ action. Specifically I'm wondering about things like $F = \mathbb{R}P^2 \times \mathbb{R}P^2$ where the action switches the fibres, or something like $F = K(A,1)$ where $A$ is a finite abelian group and the $\mathbb{Z}/2$ action is induced by the inversion homomorphism. I haven't managed to work anything out yet. – William Jun 9 at 1:48