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Perhaps this should be obvious but why is it that one may associate to a fibration exact sequences of topological spaces a long exact sequence of fundamental groups, but in (co)homology, one only has a spectral sequence instead?

I'd love to read any form of explanation.

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    $\begingroup$ What kind of answer are you looking for? Both 'sequences' are constructed in standard references on algebraic topology. You can easily follow their proofs if you have enough background. $\endgroup$
    – Fernando Muro
    Commented May 10 at 13:53
  • $\begingroup$ Thanks for your reply. I mean to ask if there is a certain particular reason that instead of having a fibration exact sequence in cohomology, there is a spectral sequence. What is the source for this asymmetry? $\endgroup$
    – kindasorta
    Commented May 10 at 14:31
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    $\begingroup$ As I said, any relatively advanced monograph on algebraic topology is a possible source. I wonder, though, whether you want a mathematical source or a speculative one. I don't have a taste for the latter but many people seem to like them. The homotopy LES is very easy to construct and you've probably met it before. The total space of a fibration is a twisted cartesian product of the base and the fiber, its singular complex is a twisted tensor product, and such things have a SS, like bicomplexes. $\endgroup$
    – Fernando Muro
    Commented May 10 at 14:45
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    $\begingroup$ Homotopy groups are obtained by mapping out of a fixed space, cohomology groups are obtained by mapping into a fixed space. Fibration sequences behave well when mapped into (and cofibration sequences behave well when mapped out of). $\endgroup$
    – John Palmieri
    Commented May 10 at 15:26

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