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What are the most useful fibrations that one be familiar with in order to use spectral sequences effectively in algebraic topology? There's at least the four different Hopf fibrations and $S^1\to S^{2n+1}\to \mathbb{C}\textrm{P}^n$. Anything else that's useful?

I might add that one fibration I would like someone to explain is the homotopy fiber of a map. I have trouble wrapping my head around it.

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    You should not memorize any fibrations.2012-01-15
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    (I find this question orders of magnitude more unsettling that the «How to remember the trigonometric identities» of a little time ago!)2012-01-15
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    @MarianoSuárez-Alvarez: I don't know anything about fibrations (aside from the definition) and am therefore quite curious: why shouldn't one memorize fibrations?2012-01-15
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    Perhaps you mean "be familiar with" instead of "memorize"?2012-01-15
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    Ok, let's change the word to "familiar with", since I do know how to prove that the ones I listed are fibrations. I'm interested in which ones show up in computations a lot.2012-01-15
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    I would also like to say that when doing computations some stuff should fire automatically in your brain and I do consider that to be a level of memorization...2012-01-15
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    I think this is a very very good question, and am baffled by the fact that it hasn't attracted enough attention. Having few examples to play with is the main reason I dislike fibrations somewhat.2013-06-15

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