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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.

Eduardo Longa
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  • 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. – Ryan Budney Mar 04 '21 at 04:10
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    I think Hatcher has some good notes. – Anubhav Mukherjee Mar 04 '21 at 04:15
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    Mosher and Tangora is a good (but perhaps old-fashioned) way to get started with the Serre spectral sequence. – John Palmieri Mar 04 '21 at 06:03
  • I really like homotopical topology by fomenko and fuchs – Thomas Rot Mar 04 '21 at 07:05
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    Relevant questions: https://mathoverflow.net/questions/45036/spectral-sequences-opening-the-black-box-slowly-with-an-example/, https://mathoverflow.net/questions/23297/simple-examples-for-the-use-of-spectral-sequences, https://mathoverflow.net/questions/22188/introductory-book-on-spectral-sequences – David White Mar 04 '21 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).

Denis Nardin
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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.

David Roberts
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Phil Harmsworth
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    Doug Ravenel "accidentally" put his personal copy of the pdf of McCleary's book on an internet-facing location on his computer... – David Roberts Mar 04 '21 at 06:31
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I have learnt it from Mosher and Tangora. But, perhaps John McCleary’s book is a reference to be mentioned.

user51223
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