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Fix $n\!\in\!\mathbb{N}$. Let $X$ be a topological space and $X_0\subseteq X_1\subseteq X_2\subseteq \ldots$ subspaces of $X$. Let $\iota_k:X_k\rightarrow X$ be the inclusion. Let $F_k:=\mathrm{Im}\,(\iota_{k\ast}: H_nX_k\rightarrow H_nX)$ and $F^k:=\mathrm{Ker}\,(\iota_k^\ast: H^nX\rightarrow H^nX_k)$.

Question: Is $(F_k)_{k\in\mathbb{N}}$ a filtration on $H_nX$ and $(F^k)_{k\in\mathbb{N}}$ a filtration on $H^nX$, i.e. does there hold $$F_0\subseteq F_1\subseteq F_2\subseteq\ldots \text{ and } F^0\supseteq F^1\supseteq F^2\supseteq\ldots\text{, and why?}$$

If not, then for a fibration $F\rightarrow E\rightarrow B$, how is the convergence of $H_p(B; H_q(F))$ to $H_{p+q}E$ in the Leray-Serre spectral sequence defined when homology is over a ring $R$? I am using McCleary's A User's Guide to Spectral Sequences, 2nd edition. Could you find a specific reference in that book? I've searhed through chapter 5, but to no avail.

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I wouldn't look at the Serre spectral sequence part. I'd just look up the section on the spectral sequence associated to a filtered complex. – Drew May 16 '13 at 0:42
up vote 1 down vote accepted

Of course!

For homology: We have that $X^k \subset X^{k+1} \subset X$, so by functoriality we get $H_n(X^k)\rightarrow H_n(X^{k+1}) \rightarrow H_n(X)$. So if I'm in the image of the composite, certainly I'm in the image of the second map. Indeed, denote the first map by $i$, then we have $i_{k} = i_{k+1} \circ i$, so if $i_k(x) = y$ then $i_{k+1}(i(x)) = y$, whence $\text{im}(i_{k+1})$ contains $y$. So $F_k \subset F_{k+1}$.

For cohomology: We have $H^n(X) \rightarrow H^n(X^{k+1}) \rightarrow H^n(X^k)$ and, by a similar argument, if I'm in the kernel of the first map then certainly I'm in the kernel of the composite - if I'm zero after the first application then I better stay zero under the second! So $F^k \supset F^{k+1}$.

This is how you get the filtration on $H^*(X)$, in fact on $C^*(X)$, by the same argument... and now you can apply the general formalism of a spectral sequence associated to a filtered complex to get a spectral sequence converging to $H^*(X)$ with the given filtration and with $E_1$ term $H^*(X^k, X^{k-1}).$

In the special case when the filtration comes from lifting the skeletal filtration of the base to the total space of a fibration, one has to check that the $E_2$-page is what you want for the Serre spectral sequence, and this takes a little work but isn't too bad (it's in McCleary, for example.)

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You're right, functoriality of (co)homology is all what's needed for filtrations. I was incredibly clumsy by trying to prove this via the definition of singular (co)homology (dealing with elements in $\mathrm{Ker\partial}/\mathrm{Im}\partial$) and got lost in the details. Thanks! – Leon May 17 '13 at 2:40

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