In the Serre spectral sequence, is it true that we can replace homology by reduced homology? That is: If $f:X\rightarrow B$ is a Serre fibration,with $F$ the fiber, then if $\tilde E^2_{pq}=\tilde H_p(B,\tilde H_q F) \longrightarrow \tilde H_{p+q}(X)$?

I think it is fine: we use complexes involving "augmentation", then filtration, do the usual things in the spectral sequence, and finally we get a sequence that converges to the homology of the original complex. Only now it becomes reduced homology. But I am not precisely sure.

I ask this question because I want to know the answer of another problem (asked by me) "homology dimension of mapping class group of surface with boundary". (I am sorry I don't know how to insert a link). I need some help for that problem. Thanks!

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    $\begingroup$ Hint for the Serre spectral sequence: is the Kuenneth formula valid in reduced homology? (Work over a field and count dimensions.) $\endgroup$ – Tim Perutz Jul 31 '10 at 14:18
  • $\begingroup$ Hi,Tim Perutz, it is not valid. $\endgroup$ – HYYY Jul 31 '10 at 14:43
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    $\begingroup$ As Tim Perutz indicates, this version is not valid. However, consider: for spaces with a chosen basepoint, reduced homology is isomorphic to homology relative to the basepoint. For p: E -> B with subfibration D -> E, together with a subspace A of B, there is a Serre spectral sequence involving relative homology of the base and fiber, computing the homology of E relative to $D \cup p^{-1}A$. $\endgroup$ – Tyler Lawson Jul 31 '10 at 15:18

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