Spectral Sequences - Cornell University

1 Spectral SequencesAllen HatcherThis is a preliminary and incomplete version of an extra fifthchapter for myAl-gebraic Topologytextbook. Its aim is to give an introduction to Spectral Sequences asthey arise in algebraic topology. The rather lengthy first section of the chapter is de-voted to the Serre Spectral sequence and some of its main applications. After this, thesecond section gives a short introduction to the more specialized Adams Spectral se-quence, which is geared toward computing stable homotopy groups, especially stablehomotopy groups of spheres. The chapter then concludes withseveral independentsections on related topics and other Spectral Sequences . Here is a Table of Contentsfor the The Serre Spectral sequence .. 520 Exact Couples Serre Spectral sequence for Homology Classes and Further Properties Serre Spectral sequence for Cohomology Homotopy Groups of Spaces of Eilenberg-MacLane Spaces Homotopy Groups of Spheres The Adams Spectral sequence .

2 580 Spectra the Adams Spectral sequence a Few Stable Homotopy Groups of Spheres Whitehead s Exact sequence The Bockstein Spectral sequence [Not yet written]. The Mayer-Vietoris Spectral sequence [Not yet written]. The EHP sequence Eilenberg-Moore Spectral Sequences .. 631 There are many situations in algebraic topology where the relationship betweencertain homotopy, homology, or cohomology groups is expressed perfectly by an exactsequence. In other cases, however, the relationship may be more complicated anda more powerful algebraic tool is needed. In a wide variety ofsituations spectralsequences provide such a tool. For example, instead of considering just a pair(X,A)and the associated long exact Sequences of homology and cohomology groups, onecould consider an arbitrary increasing sequence of subspacesX0 X1 XwithX=SiXi, and then there are associated homology and cohomology spectralsequences. Similarly, the Mayer-Vietoris sequence for a decompositionX=A Bgeneralizes to a Spectral sequence associated to a cover ofXby any number of this great increase in generality comes, not surprisingly, a correspondingincrease in complexity.

3 This can be a serious obstacle to understanding Spectral se-quences on first exposure. But once the initial hurdle of believing in Spectral se-quences is surmounted, one cannot help but be amazed at first Spectral sequence that appeared in algebraic topology, and still the mostimportant one, is the Serre Spectral sequence which relatesthe homology or cohomol-ogy groups of the fiber, base, and total space of a fibration. The homotopy groups ofthese three spaces fit into a long exact sequence , but for homology or cohomology therelationship is much more complicated, as expressed in the Spectral sequence . Thisincrease in complexity can be seen already for a product fibration, where the homo-topy groups of the product are just the products of the homotopy groups of the twofactors, whereas for homology one has the more complicated K unneth first section of this chapter is devoted to the Serre Spectral sequence andsome of its main applications both to general theory and specific calculations.

4 Afterthis we give a brief introduction to the Adams Spectral sequence and its applicationto computing stable homotopy groups of 5 Spectral SequencesOne can think of a Spectral sequence as a book consisting of a sequence of pages,each of which is a two-dimensional array of abelian groups. On each page there aremaps between the groups, and these maps form chain complexes. The homologygroups of these chain complexes are precisely the groups which appear on the nextpage. For example, in the Serre Spectral sequence for homology the first few pageshave the form shown in the figure below, where each dot represents a the first quadrant of each page is shown because outside the first quadrant allthe groups are zero. The maps forming chain complexes on eachpage are known asdifferentials. On the first page they go one unit to the left, on the second page twounits to the left and one unit up, on the third page three unitsto the left and two unitsup, and in general on therthpage they gorunits to the left andr 1 units one focuses on the group at the(p,q)lattice point in each page, for fixedpandq, then as one keeps turning to successive pages, the differentials entering andleaving this(p,q)group will eventually be zero since they will either come from or goto groups outside the first quadrant.

5 Hence, passing to the next page by computinghomology at the(p,q)spot with respect to these differentials will not change the(p,q)group. Since each(p,q)group eventually stabilizes in this way, there is awell-defined limiting page for the Spectral sequence . It is traditional to denote the(p,q)group of therthpage asErp,q, and the limiting groups are denotedE p,q. In thediagram above there are already a few stable groups on pages 2and 3, the dots in thelower left corner not joined by arrows to other dots. On each successive page therewill be more such Serre Spectral sequence is defined for fibrationsF X Band relates thehomology ofF,X, andB, under an added technical hypothesis which is satisfiedifBis simply-connected, for example. As it happens, the first page of the spectralsequence can be ignored, like the preface of many books, and the important actionbegins with the second page. The entriesE2p,qon the second page are given in termsof the homology ofFandBby the strange-looking formulaE2p,q=Hp B;Hq(F;G) whereGis a given coefficient group.

6 (One can begin to feel comfortable with spectralThe Serre Spectral SequenceSection when this formula no longer looks bizarre.) AftertheE2page the spectralsequence runs its mysterious course and eventually stabilizes to theE page, and thisis closely related to the homology of the total spaceXof the fibration. For example,if the coefficient groupGis a field thenHn(X;G)is the direct sumLpE p,n pof theterms along thenthdiagonal of theE page. For a nonfieldGsuch asZone canonly say this is true modulo extensions the fact that in a short exact sequenceof abelian groups 0 A B C 0 the groupBneed not be the direct sum of thesubgroupAand the quotient groupC, as it would be for vector an example, supposeHi(F;Z)andHi(B;Z)are zero for oddiand free abelianfor eveni. The entriesE2p,qof theE2page are then zero unlesspandqare the differentials in this page go up one row, they must all be zero, so theE3page is the same as theE2page. The differentials in theE3page go three units tothe left so they must all be zero, and theE4page equals theE3page.

7 The samereasoning applies to all subsequent pages, as all differentials go an odd number ofunits upward or leftward, so in fact we haveE2=E . Since all the groupsE p,n pare free abelian there can be no extension problems, and we deduce thatHn(X;Z)is the direct sumLpHp B;Hn p(F;Z) . By the universal coefficient theorem this isisomorphic toLpHp(B;Z) Hn p(F;Z), the same answer we would get ifXweresimply the productF B, by the K unneth main difficulty with computingH (X;G)fromH (F;G)andH (B;G)ingeneral is that the various differentials can be nonzero, andin fact often are. Thereis no general technique for computing these differentials, unfortunately. One eitherhas to make a deep study of the fibration in question and reallyunderstand the in-ner workings of the Spectral sequence , or one has to hope for lucky accidents thatyield purely formal calculation of differentials. The situation is somewhat better forthe cohomology version of the Serre Spectral sequence .

8 Thisis quite similar to thehomology Spectral sequence except that differentials go in the opposite direction, asone might guess, but there is in addition a cup product structure which in favorablecases allows many more differentials to be computed purely is also possible sometimes to run the Serre Spectral sequence backwards, ifone already knowsH (X;G)and wants to deduce the structure ofH (B;G)fromH (F;Z)or vice versa. In this reverse mode one does detective work todeduce thestructure of each page of the Spectral sequence from the structure of the followingpage. It is rather amazing that this method works as often as it does, and we will seeseveral instances of CouplesLet us begin by considering a fairly general situation, which we will later specializeto obtain the Serre Spectral sequence . Suppose one has a spaceXexpressed as theunion of a sequence of subspaces Xp Xp+1 . Such a sequence is called522 Chapter 5 Spectral SequencesafiltrationofX.

9 In practice it is usually the case thatXp= forp <0, butwe do not need this hypothesis yet. For example,Xcould be a CW complex withXpitspskeleton, or more generally theXp s could be any increasing sequence ofsubcomplexes whose union isX. Given a filtration of a spaceX, the various long exactsequences of homology groups for the pairs(Xp,Xp 1), with some fixed coefficientgroupGunderstood, can be arranged neatly into the following largediagram:The long exact Sequences form staircases, with each step consisting of two arrows tothe right and one arrow down. Note that each groupHn(Xp)orHn(Xp,Xp 1)appearsexactly once in the diagram, with absolute and relative groups in alternating will call such a diagram of interlocking exact Sequences astaircase may write the preceding staircase diagram more conciselyasthe triangle at the right, whereAis the direct sum of all the absolutegroupsHn(Xp)andEis the direct sum of all the relative groupsHn(Xp,Xp 1). The mapsi,j, andkare the maps forming the long exact sequencesin the staircase diagram, so the triangle is exact at each of its three corners.

10 Such atriangle is called anexact couple, where the word couple is chosen because there areonly two groups involved, the exact couple arising from the filtration withXpthepskeleton of a CWcomplexX, the mapd=jkis just the cellular boundary map. This suggests thatdmay be a good thing to study for a general exact couple. For a start, we haved2=jkjk=0 sincekj=0, so we can form the homology group Kerd/Imd. Infact, something very nice now happens: There is aderived coupleshown in the diagram at the right, with E =Kerd/Imd, the homology ofEwith respect tod. A =i(A) A. i =i|A . j (ia)=[ja] E . This is well-defined:ja Kerdsincedja=jkja=0; andifia1=ia2thena1 a2 Keri=Imksoja1 ja2 Imjk=Imd. k [e]=ke, which lies inA =Imi=Kerjsincee Kerdimpliesjke=de= ,k is well-defined since[e]=0 E impliese Imd Imj= derived couple of an exact couple is Serre Spectral SequenceSection : This is an exercise in diagram chasing, which we present in condensed form. j i =0:a A a =ia j i a =j ia =[ja ]=[jia]=0.