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What books/articles do you recommend for learning spectral sequences? I am interested in their applications to algebraic topology, particularly to understand the homology of fibre bundles. I have a good backgroud on differential geometry and a reasonable background on modules and algebras.

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My favourite reference for understanding spectral sequences is

Boardman, J. Michael. "Conditionally convergent spectral sequences." Contemporary Mathematics 239 (1999): 49-84 (pdf).

I don't think I really understood a spectral sequence before reading it. After that, examples examples examples. The Serre spectral sequence is a good place to start (and Serre's original paper not the worst place to learn it).

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The article by Timothy Chow, "You Could Have Invented Spectral Sequences", Notices of the AMS 53 2006 pp. 15-19 (pdf) is a start.

Given your background and interests, you could try John McCleary's A User's Guide to Spectral Sequences, (Cambridge, 2001) doi:10.1017/CBO9780511626289.

More at the following question: Spectral Sequences reference.

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    $\begingroup$ Doug Ravenel "accidentally" put his personal copy of the pdf of McCleary's book on an internet-facing location on his computer... $\endgroup$ – theHigherGeometer Mar 4 at 6:31
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I have learnt it from Mosher and Tangora. But, perhaps John McCleary’s book is a reference to be mentioned.

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