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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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  • $\begingroup$ The Serre spectral sequence is the starter for most people, as far as I know. I found Spanier to be a good reference, but likely there are better and more modern references now. $\endgroup$
    – Ryan Budney
    Commented Mar 4, 2021 at 4:10
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    $\begingroup$ I think Hatcher has some good notes. $\endgroup$
    – Anubhav Mukherjee
    Commented Mar 4, 2021 at 4:15
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    $\begingroup$ Mosher and Tangora is a good (but perhaps old-fashioned) way to get started with the Serre spectral sequence. $\endgroup$
    – John Palmieri
    Commented Mar 4, 2021 at 6:03
  • $\begingroup$ I really like homotopical topology by fomenko and fuchs $\endgroup$
    – Thomas Rot
    Commented Mar 4, 2021 at 7:05
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    $\begingroup$ Relevant questions: mathoverflow.net/questions/45036/…, mathoverflow.net/questions/23297/…, mathoverflow.net/questions/22188/… $\endgroup$
    – David White
    Commented Mar 4, 2021 at 12:28

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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... $\endgroup$
    – David Roberts ♦
    Commented Mar 4, 2021 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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