The goal is to lift information (aka cohomology) from the associated graded complex to that of the original complex. The first approximation is . The cycles are what lands in a lower layer (or upper index plus one) of the filtration. Thus it is natural to let be the elements that drop levels down the filtration, and be the (decreasing) intersection of all of them. The boundary are the image of elements that comes from levels above the filtration, and be the increasing union of all of them. Let . Note that has a differential, but what is the grading? Note that the filtration number increase by but the grading on the complex only increase by . So to make things more symmetric we define so then . This grading is natural if is the total complex of some double complex.
A more general way in which spectral sequence arise is via exact couples. Some spectral sequences cannot be constructed from filtered complexes, see this discussion. The disadvantage is that it can be the case that our objects of study are not explicitly filtered or do not come from a filtered differential object. In this section we present another general algebraic setting, exact couples, in which spectral sequences arise. The ease of definition of the spectral sequence and its applicability make this approach very attractive. Unlike the case of a filtered differential graded module, however, the target of the spectral sequence coming from an exact couple may be difficult to identify.
It is easy to see for double complexes concentrated in upper left quadrant, stablize as long as . For Serre’s spectral sequence we use pullback of a cellular filtration of the base and get a filtration of . The theorem is that there is a filtration on with associated graded . But we still need to solve some extension problem over if we want to determine .
Example:
For the path space fibration ( simply connected) Since the path space is contractible, we know what the spectral sequence is converging to, and we immediately get . Similarly, implies is injective. Using this we can show that and is for odd and for even.
The same argument shows and is 0 for not congruent to 0 mod $(n-1) and otherwise. More generally, if the base is , then we get the Wang sequence, which says we have a filtration
As an application, we prove Hurewicz theorem: If is a space and for every , then is abelianization.
If , this is Poincare’s theorem. If , we look at the loop-path fibration . Note that for , so by induction hypothesis, we have . It remains to show .
Aside: Serre exact sequence. From the vanishing of the terms in and of the -page of the Serre spectral sequence, we see that for the differential starting from to be nonzero, we must have and . Thus if , there is only one possibly non-zero differential on . Similarly, for the differential to be nonzero, we must have and . Thus if , then there is only one possibly non-zero differential to . In these two segments, we have and . If , we have a SES . Note that . Splicing these together we get LES . Transgression: This is the last possible differential . There is a geometric constructino of this differential: If we look at the LES for the pair , we get Since the pair maps to , it induces a map of LES. It includes . We have a partially defined map . Serre shows the transgression is this map. This can be used to show that the map from the Serre exact sequence is the same as the one that we need to prove Hurewicz theorem. An immediate corollary is that if is simply connected and for , then (simply connectedness is essential since could be a perfect group). Take the inverse image of in in , it is called the binary icosahedral group and it is perfect, so is a counterexample. Another corollary is if is simply connected and CW and for all , then is contractible. Similarly we have the homology Whitehead theorem, which is very useful in practice. The spaces with all trivila homology groups are called acyclic spaces. Another corollary is we can compute the -th fundamental group of . We can also construct the Moore space with -th reduced homology group and zero otherwise. The idea is to present as the cokernel of some and let be the mapping cone of .
Eilenberg-Maclane spaces : They are spaces with only one non-trivial homotopy group. There are multiple techniques for constructing higher Eilenberg–MacLane spaces. One of which is to construct a Moore space for an abelian group . Note that the lower homotopy groups are already trivial by construction. Now iteratively kill all higher homotopy groups by successively attaching cells of dimension greater than , and define as direct limit under inclusion of this iteration. If we attach an -cell to via an attaching map , the -th homotopy groups for are unchanged. Moreover, we have and .
We also have the cohomological version of Serre spectral sequnce and there is a cup product on the -th page for every . As an example, consider and note that . Let . Writing for some integer , we can figure out by hitting both sides with the differential . More generally we can use it to show that and where and are divided power algebra and exterior algebra respectively. Combining this with the homology whitehead theorem we deduce . More generally for a connected CW complex , the James reduced product has the same homotopy type as , the loop space of the suspension of .
Another application is to calculate the cohomology of . The result is for symmetric algebra for even and exterior algebra for odd. First we need to get a concrete model of . One explicit model is the mapping telescope defined by where the maps are multiplication by . Using homotopy groups of filtered homotopy colimits, we can show this is a . Now look at the . Look at the Serre spectral sequence, we know everything on the first column (the cohomology of the fiber). From this we can inductivelly figure out the page and also the ring multiplication, the conclusion being where . By induction on we can show that for even we have two rows and for odd we have two columns.
Let be an element representing . This induces a map on rational cohomology and it turns out to be an isomorphism by the above calculation of . Serre shows something more, , i.e. the rational homotopy group of odd dimensional sphere is at degree and 0 otherwise. It turns out that all the nontorsion elements in the homotopy groups of spheres are detected either by degree or by the Hopf invariant. See Hatcher’s Spectral sequence Theorem 5.22 for a proof. The idea is to find a space admitting a fibration with all reduced rational cohomology zero (The idea is that should be close to but with -th homotopy group killed using ), which then implies for all by Hurewicz theorem. From the LES for homotopy groups for fibration we then deduce is finite for .
To quote from wikipedia:
The method of killing homotopy groups”, due to Cartan and Serre involves repeatedly using the Hurewicz theorem to compute the first non-trivial homotopy group and then killing (eliminating) it with a fibration involving an Eilenberg–MacLane space. In principle this gives an effective algorithm for computing all homotopy groups of any finite simply connected simplicial complex, but in practice it is too cumbersome to use for computing anything other than the first few nontrivial homotopy groups as the simplicial complex becomes much more complicated every time one kills a homotopy group. The Serre spectral sequence was used by Serre to prove some of the results mentioned previously. He used the fact that taking the loop space of a well behaved space shifts all the homotopy groups down by , so the nth homotopy group of a space is the first homotopy group of its -fold repeated loop space, which is equal to the first homology group of the -fold loop space by the Hurewicz theorem. This reduces the calculation of homotopy groups of to the calculation of homology groups of its repeated loop spaces. The Serre spectral sequence relates the homology of a space to that of its loop space, so can sometimes be used to calculate the homology of loop spaces. The Serre spectral sequence tends to have many non-zero differentials, which are hard to control, and too many ambiguities appear for higher homotopy groups. Consequently, it has been superseded by more powerful spectral sequences with fewer non-zero differentials, which give more information
Reference:
Sepctral sequencces that everyone should know
https://mathoverflow.net/questions/45036/spectral-sequences-opening-the-black-box-slowly-with-an-example
https://www.math.ucla.edu/~mikehill/Teaching/Math885/Lecture12.pdf
https://www.math.ucla.edu/~mikehill/Teaching/Math885/Lecture08.pdf
https://math.stackexchange.com/questions/50377/homotopy-groups-of-s2
https://www.college-de-france.fr/media/jean-pierre-serre/UPL7235285843586540944_Serre_The_se.pdf