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 .
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