Spectral Sequences Via Examples

Graduate J. Math. 222(2017), 10 – 28 Spectral sequences via examples

Antonio Díaz Ramos

Abstract (6) Lifting problem. We go in detail about the lifting problem and discuss several particular cases. These are lecture notes for a short course about spectral sequences that was held at Málaga, October 1820 (2016), during the Fifth (7) Edge morphisms. After introducing the rel- Young Spanish Topologists Meeting. The approach is to illustrate evant notions, a couple of examples where the the basic notions via fully computed examples arising from Algebraic edge morphisms play a central role are fully de- Topology and Group Theory. We give particular attention to extension scribed. and lifting problems in spectral sequences, which is an aspect often occulted in introductory texts. (8) Dierent spectral sequences with same target. This section is included to make the reader MSC2010. Primary 55T;Secondary 18G40,55R20,20J06 aware of this fact. (9) A glimpse into the black box. We sketch the technicalities that have been avoided in the rest of the sections. In particular, we provide some 1 Foreword detail on the construction of a spectral sequence from a ltration of a chain complex. These notes give a simplied and hands-on introduc- (10) Appendix I. Here, precise statements for the tion to spectral sequences based on examples. The following spectral sequences are given: Serre spec- aim is to introduce the basic notions involved when tral sequence of a bration, Lyndon-Hochschild- working with spectral sequences with the minimum Serre spectral sequence of a short exact sequence necessary background. We go along the way bringing of groups, Atiyah-Hirzebruch spectral sequence in notions and illustrating them by concrete compu- of a bration. All examples in these notes are tations. The target reader is someone not acquainted based on these spectral sequences. Also, sketch with this topic. For full details the reader should con- of proofs of the two rst spectral sequences are sult [17], [21] or [16]. included and the ltrations employed are de- The following list describes the contents of the dif- scribed. ferent sections of these notes: (11) Appendix II. We list some important results (2) Spectral sequences and ltrations. We in- in Topology whose proofs involve spectral se- troduce the intimate relation between spectral quences and we briey comment on the role they sequences, ltrations, quotients and (co)homology. play in the proofs. (3) Dierentials and convergence. We give de- (12) Solutions. Answers to the proposed exercises. nitions of what a (homological type) spectral se- Some of them are just a reference to where to quence is and what is convergence in this setup. nd a detailed solution. We also talk about extension problems. We denote by R the base ring (commutative with (4) Cohomological type spectral sequences. We unit) over which we work. introduce cohomological type spectral sequences and say a few more words about the extension Acknowledgements: I am grateful to Oihana problem. Garaialde, David Méndez and Jose Manuel Moreno for reading a preliminary version of these notes. Also, (5) Additional structure: algebra. We equip thanks to Urtzi Buijs for its nice version of the bridge- everything with products, yielding graded alge- river-tree picture using the TikZ package. I thank bras and bigraded algebras. We start discussing the referee and the editors for their careful reading the lifting problem. and their many helpful suggestions and recommen- dations.

10 2 Spectral sequences and ltrations 11

2 Spectral sequences and ltrations A spectral sequence is a tool to compute the homol- P ogy of a chain complex. It arises from a ltration of a b the chain complex and it provides an alternative way A to determine the homology of the chain complex. A Q spectral sequence consists of a sequence of intermedi- 0 1 2 3 ate chain complexes called pages, E ,E ,E ,E ,..., The homology H = H∗(C, d) of the following chain with dierentials denoted by d0, d1, d2, d3,..., such complex (C, d) computes the integral homology of the that Er+1 is the homology of Er. The various pages sphere S2: have accessible homology groups which form a ner and ner approximation of the homology H we wish d d to nd out. This limit process may converge, in which 0 / Z{P,Q} / Z{A, B, C} / Z{a, b, c} / 0. case the limit page is denoted by E∞, see Section d(P ) = C − B + A d(A) = b − a 3 for more details. Even if there is convergence to d(Q) = C − B + A d(B) = c − a E∞, reconstruction is still needed to obtain H from E∞. This reconstruction procedure involves solving d(C) = c − b the extension and lifting problems, see Sections 3, 4, 5 and 6. Instead of directly calculating H, we instead consider the following submodules on each dimension: H 0 {P,Q} {A, B, C} {a, b, c} 0 d4 ... / Z / Z / Z / d ············· 3 ······················ E∞ d ············ 0 / 0 / Z{A, B} / Z{a, b, c} / 0 E2 2 0 / 0 / Z{A} / Z{a, b} / 0. As you can see, the dierential restricts to each row, Fig. 1: You want to cross the bridge and then to climb the making each row into a chain complex, and each row tree. Don't fall into the river of madness! is contained in the row above. A family of such subcomplexes is known as a ltration of (C, d). We may Although the dierentials dr's cannot always be all then consider the quotient of successive rows. The computed, the existence of the spectral sequence of- object obtained is known as the associated graded ob- ten reveals deep facts about the chain complex. The ject with respect to the given ltration. We call it spectral sequence and its internal mechanisms can E0 and it inherits a dierential d from d: still lead to very useful and deep applications, see 0 Appendix II 11. 0 / Z{P,Q} / Z{C} / 0 / 0 Our rst two examples illustrate how the Er's and 0 / 0 / {B} / {c} / 0 the dr's arise from a ltration. To read the precise Z Z denition of what a ltration is and how exactly it 0 / 0 / {A} / {a, b} / 0 gives rise to a spectral sequence, see Section 9. Z Z

Example 2.1. Consider a (semi-simplicial) model of d0(P ) = C , d0(Q) = C , d0(B) = c , d0(A) = b−a. 2 the 2-sphere S with vertices {a, b, c}, edges {A, B, C} If we compute the homology with respect to d0 we and solid triangles {P,Q} and with inclusions as fol- obtain E1: lows: 0 / Z{P − Q} / 0 / 0 / 0 0 / 0 / 0 / 0 / 0 P 0 / 0 / 0 / {a¯} / 0. P G4 A H a Z 4GG HH ww 4 GG HH ww 44 GG HwHw c Clearly, the dierential d , which is also induced by 44 G G ww HH B 1 4 GG ww HH C , is zero. This implies that 2 1. For the 4 G B Gww H b a d E = E 44 xx GG ww same reason, for all and ∞ 4xx GG ww dn = 0 n ≥ 2 E = ... = xx4 GwGw A b 3 2 1. The two classes and cor- xx 44 ww GG E = E = E P − Q a¯ xx 4 ww GG respond respectively to generators of the homology Q x x 4 C ww G c groups 2 and 2 . Q H2(S ; Z) = Z H0(S ; Z) = Z

Of course in Example 2.1, H = H∗(C, d) is easy to compute directly and the use of a spectral sequence is not needed. However and in practice, H∗(C, d) is very hard to obtain directly and passing to a spectral sequence is necessary. 3 Dierentials and convergence 12

Exercise 2.2. Cook up a dierent ltration to that 3 Dierentials and convergence in Example 2.1 and compute E0 and E1. Obtain again the integral homology of S2. A (homological type) spectral sequence consists of a sequence of pages and dierentials r , such We can learn much from this example. The rst {E , dr}r≥2 r+1 r fact is that, in general, that E is the homology of (E , dr). Each page is a dierential bigraded module over the base ring R. E0 is the associated graded object for This means that at each position (p, q), there is an the ltration on C, -module r . Moreover, the dierential squares R Ep,q dr 1 to zero and has bidegree . This last bit and d0 is induced by d. Then E is the homology of (−r, r − 1) 0 means that it maps r r , as in these (E , d0) and d1 is induced by d again. In fact, this is dr : Ep,q → Ep−r,q+r−1 the general rule to pass from En to En+1: pictures: En+1 is the homology of (En, d ) and n ••••• ••••• •••••dH d is induced from d. HH n+1 •••••gP •••••eK •••••HH PPP KK HH Although there exists a closed expression for •••PP •• •••••KK •••••HH dn+1 KKK HH in terms of d and C, it is so complicated that, in ••••• ••••K • •••••H 0 ••••• ••••• ••••• practice, you are only able to fully describe (E , d0), 1 2 2 2 3 4 (E , d1) and E . Then E is what you write in the (E , d2) (E , d3) (E , d4) statement of your theorem about your spectral se- 0 One does not need to know neither (E , d0) nor quence. You are seldom capable of guring out what 1 to start computing with a spectral sequence. are. (E , d1) d2, d3,... You can proceed even if you do not know C or the This gives you some hints about the bridge in Fig- ltration that gave rise to the spectral sequence. If ure 2. The next example provides information on how you know H and you are told that the spectral se- the tree in that gure looks like. quence converges to , written as 2 , this H Ep,q ⇒ Hp+q Example 2.3. Recall the ltration of chain com- roughly means that: plexes we gave in Example 2.1: (a) the value at each position (p, q) stabilizes after a nite number of steps, and the stable value is 0 {P,Q} {A, B, C} {a, b, c} 0 / Z / Z / Z / denoted by ∞ , and that Ep,q 0 / 0 / Z{A, B} / Z{a, b, c} / 0 (b) you can recover H from E∞. 0 / 0 / {A} / {a, b} / 0. Z Z Before unfolding details about point (b), we work Now, instead of taking quotient followed by homol- out an example. ogy, we do it the other way around, i.e., we rst take n homology, Example 3.1. The sphere of dimension S is a Moore space , i.e., its homology is given by n M(n, Z) Hi(S ; Z)= if and otherwise. Moreover, the spheres 0 / Z{P − Q} / 0 / Z{a¯} / 0 Z i = n, 0 0 of dimensions 1, 3 and 2 nicely t into the Hopf - 0 / 0 / 0 / Z{a¯} / 0 bration: 1 3 2 0 / 0 / 0 / Z{a¯} / 0, S / S / S The Hopf map S3 → S2, we recall, takes (z , z ) ∈ obtaining a ltration of , and then we take quotient 1 2 H S3 ⊆ × to z1 ∈ ∪ {∞} = S2. Associated by successive rows, C C z2 C to a bration over a CW -complex, there is the Serre 0 {P − Q} 0 0 0 spectral sequence which converges to the homology of / Z / / / the total space. The ltration on the chain complex 0 / 0 / 0 / 0 / 0 for E is obtained by taking preimages of the skeleta 0 / 0 / 0 / Z{a¯} / 0. of the base space. See Theorems 10.1 and 10.2 in Appendix I 10 for more details. As explained there, So we obtain the associated graded object with re- the second page of the spectral sequence consist of the spect to the induced ltration of H. We call it E∞. homology of the base space with (local) coecients As you can observe, in this particular example we 2 ∞ 1 ∞ in the homology of the ber. In this case, π1(S ) = have E = E of Example 2.1. In general, E is the and the second page consists of the homology of limit of the pages 0 1 2 3 . 1 E ,E ,E ,E ,... S2 with coecients in the homology of S1, and it What happened in the above example is not a co- converges to the homology of S3: incidence and, in general, 2 2 1 3 Ep,q = Hp(S ; Hq(S ; Z)) ⇒ Hp+q(S ; Z). E∞ is the associated graded object for the induced ltration on H. So we have: All notions discussed above will now be made pre- , if p = 0, 2 and q = 0, 1, E2 = H (S2; H (S1; )) = Z cise. p,q p q Z 0, otherwise, 3 Dierentials and convergence 13 and E2 looks like this: Remark 3.2. Note that the double indexing in examples 2.1 and 3.1 are dierent. The one used in the 3 0 0 0 0 latter case is customary. 2 0 0 0 0 Exercise 3.3. Deduce using the same arguments that 1 0 0 Z Z if Sl → Sm → Sn is a bration then n = l + 1 and 0 Z 0 Z 0 m = 2n − 1. 0 1 2 3 Next we unravel what point (b) above means with 2 more detail: You can recover the -module from The only possible non-trivial dierential d2 in E is R Hn 2 2 . All higher dierentials the diagonal R-modules {E∞ } via a nite num- d2 : E2,0 = Z → E0,1 = Z p,q p+q=n ber of extensions. d3, d4,... must be zero because of the location of the Z's in the diagram. This ensures that point (a) above holds. As is a -homomorphism, it is • ` ` •••• d2 : Z → Z Z ` determined by . Depending on this value, we ••••• ` d2(1) ••••• ` ` get the following page 3 ∞: ` E = E ••••• ` ` ••••• ` `

3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 Hn 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 More precisely, there exist numbers s ≤ r and a nite 1 Z 0 Z 0 1 0 0 Z 0 1 Zn 0 Z 0 increasing ltration of Hn by R-modules, 0 Z 0 Z 0 0 Z 0 0 0 0 Z 0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 0 = As−1 ⊆ As ⊆ As+1 ⊆ ... ⊆ Ar−1 ⊆ Ar = Hn, d2(1) = 0 d2(1) = ±1 d2(1) = n together with short exacts sequences of R-modules: n 6= 0, ±1 A → H → E∞ (3.1) Now, point (b) above means that each Z-module r−1 n r,n−r 3 can be recovered from the diagonal - ∞ Hn(S ; Z) Z Ar−2 → Ar−1 → Er−1,n−r+1 modules ∞ : {Ep,q}p+q=n ... ∞ As → As+1 → Es+1,n−s−1 3 0*j *j 0 0 0 *j *j *j ∞ 2 0 0*j *j 0 0 0 → As → Es,n−s. *j *j *j *j *j *j *j *j *j *j *j *j 1 ?)i )i 0 *j *j Z*j *j 0 So, starting from the bottom, ∞ , one ex- )i *j *j *j *j *j As = Es,n−s )i )i )i *j *j *j *j *j 0 Z'g 'g 0 )i )i ?*j *j *j *j 0*j *j pects to nd out from these extensions what the R- 'g 'g )i )i *j *j *j *j 0'g 1)i )i 2*j *j *j 3 *j *j *j modules are. This is in gen- 'g ' )i ) * *j * As+1,...,Ar−1,Ar = Hn H (S3; ) H (S3; ) H (S3; ) H (S3; ) eral not possible without further information. For in- 0 Z 1 Z 2 Z 3 Z stance, in Example 3.1 we could deduce the homology groups because there was only a non-zero entry In the three dierent cases discussed above, d (1) = H∗ 2 on each diagonal of E∞. In the particular case of R 0, d2(1) = ±1, d2(1) = n, n 6= 0, ±1, we would get the following values: being a eld, all extensions are trivial being extensions of free modules. In particular, Hn is the direct r ∞ 3 sum H ∼ ⊕ E . The next example exhibits the H (S3; ) = H (S3; ) = H (S ; ) = n = l=s l,n−l 0 Z Z 0 Z Z 0 Z Z kind of extension problems one can nd when R is 3 3 3 H1(S ; Z) = Z H1(S ; Z) = 0 H1(S ; Z) = Zn not a eld. 3 3 3 H (S ; ) = H (S ; ) = 0 H2(S ; ) = 0 2 Z Z 2 Z Z Remark 3.4. If H is graded and a ∈ Hp is homo- 3 3 3 geneous then is called its degree or its total H3(S ; ) = H3(S ; ) = H3(S ; Z) = Z |a| = p Z Z Z Z degree. If is bigraded and then (d (1) = 0) (d (1) = ±1) (d (1) = n E a ∈ Ep,q |a| = (p, q) 2 2 2 is called its bidegree and |a| = p + q is called its total n 6= 0, ±1). degree. So Equation (3.1) says that elements from As we know the homology of 3, we deduce that ∞ S E of a given total degree contribute to H∗ on the ∞ ∗,∗ we must have d2(1) = ±1 and that E contains ex- same total degree. actly two entries dierent from zero. This brings up another aspect: in order to compute some key dier- Remark 3.5. Already in the previous examples 2.1 entials in the Serre spectral sequence, it is necessary and 3.1, plenty of the slang used by spectral se- to have a deep insight into the geometry of the - quencers becomes useful. For instance, regarding Example 2.1, one says that dies killing and bration. In the case above, d2 is determined by the B c holonomy of the bration and it is possible to argue that c dies killed by B, being the reason that some geometrically why d (1) = ±1. This then recovers dierential (d0) applied to B is exactly c. Also, as 2 for all , one says that the spectral se- our knowledge of H (S3; ). dn = 0 n ≥ 1 ∗ Z quence collapses at E1. The terminology refers to 4 Cohomological type spectral sequences 14 the fact that n 1 for all . In Example Again there are 's in diagonals contributing to E = E n ≥ 2 Z2 3.1, the in 2 dies killing the in 2 and the with even. A careful analysis shows Z E2,0 Z E0,1 Hn(C4; Z) n > 0 that all must die killed by : spectral sequence collapses at E3. d3 Example 3.6. The groups and t into the C2 C4 Z2 Z2 Z2 Z2 Z2 Z2 Z2 short exact sequence fLLL fLLL 0 0 0LLL 0 0LLL 0 0 LLL LLL C → C → C . Z2 Z2 Z2 Z2 Z2 Z2 Z2 2 4 2 L L 0 0f L 0L 0f L 0L 0 0 Associated to a short exact sequence of groups there LLL LLL LLL LLL is the Lyndon-Hochschild-Serre spectral sequence 10.3, Z2 0 Z2 0 Z2 0 Z2 and we are going to study some extension problems ZZ2 0 0 0 0 0 arising in this situation. The second page of the spectral sequence consists of the homology of C2 with co- 4 ecients in the homology of C2, and it converges to Hence, E looks as follows. the homology of C4: 2 Z2 0 Z2 0 0 0 0 Ep,q = Hp(C2; Hq(C2; Z)) ⇒ Hp+q(C4; Z). 0 0 0 0 0 0 0 Because ∞ , where ∞ is the 2 0 2 0 0 0 0 H∗(C2; Z) = H∗(RP ; Z) RP Z Z innite-dimensional projective space, it is easy to see 0 0 0 0 0 0 0 that 2 looks like this: E Z2 0 Z2 0 0 0 0 0 0 0 0 0 Z2 Z2 Z2 Z2 Z2 Z2 ZZ2 0 0 0 0 0 0 Z2 Z2 Z2 Z2 Z2 Z2 It is clear from the positions of the non-zero entries in 4 that the rest of the dierentials ZZ2 0 Z2 0 Z2 E d4, d5, d6,... must be zero. So the spectral sequence collapses at 4 ∞ E = E . For n odd, we have an extension of Z- To deduce the dierentials we would need some extra modules: information. For instance, a multiplicative structure Z2 → Hn(C4; Z) → Z2. would suce as in Example 5.2. Although it seems There are two solutions, either or a little bit articial, we are going to use the fact that Hn(C4; Z) = Z4 , and one would need extra infor even to deduce the dier- Hn(C4; Z) = Z2 × Z2 Hn(C4; Z) = 0 n > 0 entials. Then we will study the extension problems formation to decide which one is the right one. For infor with odd. The full homology of stance, bringing in some topology, we quickly deduce Hn(C4; Z) n C4 that can be computed with (periodic) resolutions, see for H1(C4; Z) = H1(K(C4, 1); Z) = π1(K(C4, 1))ab = , where is an Eilenberg-MacLane space instance [5, II.3]. C4 K(C4, 1) (see also Example 5.7). In fact, for So, assume that we know that for Hn(C4; Z) = C4 Hn(C4; Z) = 0 all odd, and to extend this fact from to , even. Then the 's in 2 , 2 , 2 , etc, n n = 1 n 6= 1 n > 0 Z2 E1,1 E3,1 E5,1 one could use results about periodic (co)homology as must die and their only chance is being killed by d2 [5, VI.9]. In any case and as aforementioned, it is from the 's in E2 , E2 , E2 , etc: easier to compute via resolutions. Z2 3,0 5,0 7,0 H∗(C4; Z) Exercise 3.7. From the pathspace bration ΩSn → Z2 Z2 Z2 Z2 Z2 Z2 PSn → Sn, where PSn ' ∗, and the Serre spectral 0 0 0 0 0 0 sequence 10.1, deduce the integral homology of the Z2 Z2 Z2 Z2 Z2 Z2 loops on the sphere ΩSn. Solution in 12.1. hQQQQ hQQQQ ZZ2 0 Q Z2 0 Q Z2 4 Cohomological type spectral sequences There are no other possible non-trivial dierentials and hence 3 is as follows: There are homological type and cohomological type d2 E spectral sequences. In Section 3, we described homological type spectral sequences. In this section, we Z2 Z2 Z2 Z2 Z2 Z2 Z2 work with cohomological type spectral sequences. 0 0 0 0 0 0 0 A cohomological type spectral sequence consists of Z2 Z2 Z2 Z2 Z2 Z2 Z2 a sequence of bigraded dierential modules {Er, dr}r≥2 0 0 0 0 0 0 0 such that Er+1 is the cohomology of (Er, dr) and such that has bidegree , r r , Z2 0 Z2 0 Z2 0 Z2 dr (r, 1 − r) dr : Ep,q → Ep+r,q+1−r as in these pictures: ZZ2 0 0 0 0 0 5 Additional structure: algebra 15

Because for every r, either r or r −1 is odd, it follows ••••• ••••• •••••H that d = 0 for all r ≥ 2. Hence the spectral sequence HH r •••••P •••••K •••••HH collapses at E = E . Because is a free abelian PPP KK HH 2 ∞ Z •••PP' •• •••••KK •••••HH group, all extension problems have a unique solution KKK HH ••••• ••••K% • •••••H$ up to isomorphism, and we deduce that: ••••• ••••• ••••• n+1, if k is even, (E2, d2) (E3, d3) (E4, d4) Kk( P n) ∼= Z In this situation, H = H∗ is the cohomology of C 0, otherwise. some cochain complex, C∗, and convergence is written Ep,q ⇒ Hp+q. As for homological type, con- vergence2 means that the pages eventually stabilize 5 Additional structure: algebra at each particular position ( ) and that you can p, q In some cases, the cohomology groups we want to recover the R-module Hn from the diagonal R- modules p,q via a nite number of exten- compute, H, have some additional structure. For {E∞ }p+q=n instance, there may be a product sions. More precisely, there exist numbers s ≤ r and a nite decreasing ltration of n by -modules, H R H ⊗ H → H r+1 r r−1 s+1 s n 0 = A ⊆ A ⊆ A ⊆ ... ⊆ A ⊆ A = H , that makes H into an algebra. A spectral sequences of algebras is a spectral sequence such that together with short exact sequences of R-modules: {Er, dr}r≥2 Er is a dierential bigraded algebra, i.e., there is a s+1 n s,n−s (4.1) product, A → H → E∞ Er ⊗ Er → Er, As+2 → As+1 → Es+1,n−s−1 ∞ and the product in E is that induced by the prod- ... r+1 uct in Er after taking cohomology with respect to r r−1 r−1,n−r+1 A → A → E dr. A spectral sequence of algebras converges as an ∞ algebra to the algebra if it converges as a spectral 0 → Ar → Er,n−r. H ∞ sequence and there is a decreasing ltration of H∗, Remark 4.1. Pay attention to the dierence between (3.1) and (4.1): the indexing is upside down. ... ⊆ F n+1 ⊆ F n ⊆ F n−1 ⊆ ... ⊆ H∗, Example 4.2. Topological complex K-theory is a which is compatible with the product in H, generalized cohomology theory. For any compact topological space X, K0(X) is the Grothendieck group F n · F m ⊆ F n+m, of isomorphism classes of complex vector bundles over n ∼ n+2 and which satises the conditions below. When one X, and K (X) = K (X) for all n ∈ Z. For a point, we have 0 ∼ and 1 . Let n be the restricts this ltration to each dimension, one gets K (∗) = Z K (∗) = 0 CP the ltration explained in (4.1). More precisely, if complex projective space of dimension n and consider the bration we dene F pHp+q = F p ∩ Hp+q, we have short exact sequences n id n ∗ → CP −→ CP . p+1 p+q p p+q p,q F H → F H → E∞ . For a bration we have the Atiyah-Hirzebruch spectral sequence 10.5. In this case, the second page is Then there are two products in the page E∞: given by the cohomology of P n with coecients in C (c) The one induced by E∞ being the limit of the the K-theory of a point, and it converges to the K- algebras , n E2,E3,... theory of CP : (d) The one induced by the ltration as follows: For p,q ∼ p n q p+q n E2 = H (CP ; K (∗)) ⇒ K (CP ). elements a and b belonging to n The integral cohomology groups of CP are given p,q p p+q p+1 p+q p n ∼ E∞ = F H /F H by H (CP ; Z) = Z for 0 ≤ p ≤ n, p even, and p,q otherwise. So ∼ if and are even and 0 0 0 0 0 0 0 0 0 E2 = Z p q and Ep ,q = F p Hp +q /F p +1Hp +q respectively, 0 ≤ p ≤ 2n, and it is 0 otherwise. This is the picture we set:∞ for 3: CP a · b = a · b Z 0 Z 0 Z 0 Z in p+p0,q+q0 p+p0 p+p0+q+q0 p+p0+1 p+p0+q+q0 0 0 0 0 0 0 0 E∞ = F H /F H . Z 0 Z 0 Z 0 Z We say that the spectral sequence converges as an 0 0 0 0 0 0 0 algebra if these two products are equal. Z 0 Z 0 Z 0 Z 0 0 0 0 0 0 0 5 Additional structure: algebra 16

2i+1 j 2i j+2 Remark 5.1. Note that, by denition, the graded Analogously, one deduces that d2(y x ) = y x and bigraded products satisfy n m n+m and 2i j H · H ⊆ H and that d2(y x ) = 0. Hence d2 kills elements as p,q p0,q0 p+p0,q+q0 for . All algebras follows, Er ·Er ⊆ Er r ∈ N∪{∞} we consider are graded (or bigraded) commutative, S3 3T 3 2 3 3 y SSyS xTTTy x y x i.e., SSS TTTT 2 2 SS2S 2TTT2T 3 ab = (−1)|a||b|ba, (5.1) y y x y x) y x* 2 3 where and are the degrees (total degrees) of the yVVVyxWWW yx yx |a| |b| VVVVWWWWW homogeneous elements a and b respectively. More- 1 xVV xV*2 WWWxW+3 over, every dierential d is a derivation, i.e., and is d(ab) = d(a)b + (−1)|a|ad(b). E3 0 0 0 0 ∗,∗ Finally, if C is a bigraded module or a double y2 y2x 0 0 cochain complex, by Total(C) we denote the graded algebra or the cochain complex such that 0 0 0 0 1 x 0 0 M Total(C)n = Cp,q. p+q=n By degree reasons, there cannot be any other non- trivial dierentials and hence the spectral sequence Example 5.2. Consider the short exact sequence of collapses at . Moreover, the bigraded alge- cyclic groups: E3 = E∞ bra E∞ is given by C2 → C4 → C2. 0 2 0 The cohomology ring of C2 with coecients in the E∞ = F2[x, x ]/(x ) = Λ(x) ⊗ F2[x ], eld of two elements is given by ∗ H (C2; F2) = F2[x] with |x| = 1. For the extension above there is the where x0 = y2, |x| = (1, 0) and |x0| = (0, 2). It turns out that ∗ 0 with Lyndon-Hochschild-Serre spectral sequence of alge- H (C4; F2) = Λ(z) ⊗ F2[z ] |z| = bras converging as an algebra 10.4, 1, |z0| = 2. We cannot deduce this directly from 2 Theorem 5.5 below as the relation x = 0 in E∞ is p,q p q p+q not one of the free relations 5.1 for the eld . But E2 = H (C2,H (C2; F2)) ⇒ H (C4; F2). F2 it is easy to deduce it from Theorem 6.2 below in a As the extension is central, we have that similar way to that of Example 6.4. p q ∼ p q Exercise 5.3. Compare last example and Example H (C2,H (C2; F2)) = H (C2; F2) ⊗ H (C2; F2) ∼ 3.6. = F2[x] ⊗ F2[y] (see the comments after the statement of Theorem Exercise 5.4. Do the analogous computation for the extension given that ∗ 10.4). Hence, the corner of the page ∼ C3 → C9 → C3 H (C3; F3) = E2 = F2[x, y] with , . Solution in 12.2. has the following generators: Λ(y) ⊗ F3[x] |y| = 1 |x| = 2 In general we cannot recover the graded algebra y3 y3x y3x2 y3x3 structure in H∗ from the bigraded algebra structure 2 2 2 2 2 3 of E∞. This problem is known as the lifting problem. y y x y x y x The next theorem gives some special conditions un- 2 3 der which we can reconstruct ∗ from ∗,∗. They y yx yx yx H E∞ 1 x x2 x3 apply to Exercise 5.4.

Theorem 5.5 ([17, Example 1.K, p. 25]). If E∞ is a free, graded-commutative, bigraded algebra, then H∗ 2 Because the extension is non-split, d2(y) = x . An- is a free, graded commutative algebra isomorphic to other way of deducing this is to use the fact Total(E∞). 1 ∼ ∼ A free graded (or bigraded) commutative algebra H (C4; F2) = Hom(C4, F2) = F2. is the quotient of the free algebra on some graded Then, as the terms 1,0 and 0,1 contribute (bigraded) symbols modulo the relations (5.1). The- E∞ = hxi E∞ to 1 , the must die, and 2 is its orem 5.5 translates as that, if are the free H (C4; F2) y d2(y) = x x1, . . . , xr only chance. From here, we can deduce the rest of generators of E∞ with bidegrees (p1, q1),..., (pr, qr), then ∗ is a free graded commutative algebra on gen- the dierentials as d2 is a derivation, for instance: H erators z1, . . . , zr of degrees p1 + q1, . . . , pr + qr. The d (yx) = d (y)x + yd (x) = x3, as d (x) = 0, next example is the integral version of the earlier Ex- 2 2 2 2 ample 5.2 and here the lifting problem becomes ap- 2 over . d2(y ) = d2(y)y + yd2(y) = 2d2(y)y = 0 , F2 parent. 5 Additional structure: algebra 17

Example 5.6. . Consider the short exact sequence at (n, 0). Then zx at (n, n − 1) must die killing of cyclic groups: n−1 2 dn(zx) = dn(z)x + (−1) zdn(x) = x .

C2 → C4 → C2. 0 0 0 0 0 0 0 0 0 0 The integral cohomology ring of the cyclic group z 0 ... 0 zx 0 ... 0 zx2 0 C2 @ C is given by ∗ , with . The @@ CC H (C2; Z) = Z[x]/(2x) |x| = 2 0 0 @... 0 0 0 CC... 0 0 0 @ dn C dn Lyndon-Hochschild-Serre spectral 10.4 is ...@@ CC @@ CC 0 0 ... @0@ 0 0 ... 0CC 0 0 Ep,q = Hp(C ,Hq(C ; )) ⇒ Hp+q(C ; ). @@ C! 2 2 2 Z 4 Z 1 0 ... 0 x 0 ... 0 x2 0 As the extension is central, the corner of the page E = [x, y] has the following generators: 2 F2 Pushing this argument further, one may conclude ∗ 2 2 2 2 2 3 that H (K(Z, n); Q) = Q[x] with |x| = n. Now as- y 0 y x 0 y x 0 y x ∗ sume that n is odd. Then H (K(Z, n − 1); Q) = Q[z] 0 0 0 0 0 0 0 with |z| = n − 1 and again z at (0, n − 1) must die 2 3 y 0 yx 0 yx 0 yx killing dn(z) = x at (n, 0). Next, we deduce that 2 n−1 and hence 0 0 0 0 0 0 0 dn(z ) = dn(z)z + (−1) zdn(z) = 2zx 1 2 3 is killed by 2: 1 0 x 0 x 0 x zx 2 z 0 0 ... 0 0 0 Recall that |x| = (2, 0) and |y| = (0, 2). As in Exam- 1 2 2 2 z 0 ... 0 z x 0 ple 4.2, parity implies that the spectral sequence col- AA lapses at and hence . 0 0AA... 0 0 0 E2 = E∞ E∞ = Z[x, y]/(2x, 2y) AA dn Nevertheless, ∗ with . ...A H (C4; Z) = Z[z]/(4z) |z| = 2 AA 0 0 0A 0A 0 0 AA In the next section, we deepen into the lifting z 0 ... 0 zx 0 problem to pin down the relation between the bi- E EE graded algebra and the graded algebra ∗. We 0 0EE... 0 0 0 E∞ H EE dn ...EE nish this section with a topological example. EE 0 0 ... E0E 0 0 EE Example 5.7. We are set to determine the ring 1 0 ... 0 " x 0 ∗ H (K(Z, n); Q), where K(Z, n) is by denition the Eilenberg-MacLane space whose homotopy groups satisfy Following these arguments, one may conclude that , if , ∗ Z i = n H (K(Z, n); Q) = Λ(x) with |x| = n. πi(K(Z, n)) = , otherwise. 0 The next three exercises are meant to highlight the So, we know that K( , 1) = S1 and hence role of the base ring R. The computations are similar Z to those of Example 5.7 but with integral coecients ∗ H (K(Z, 1); Q) = Λ(z) with |z| = 1. instead of rational coecients. We prove by induction that Exercise 5.8. Determine the cohomology ring ∗ H (K(Z, 2); Z) using the pathspace bration [z], if n is even, H∗(K( , n); ) = Q ΩK( , 2) → PK( , 2) → K( , 2), Z Q Λ(z), if n is odd, Z Z Z where we have ΩK(Z, 2) ' K(Z, 1) and PK(Z, 2) ' where |z| = n. We consider the pathspace bration ∗. Solution in 12.3. ΩK(Z, n) → PK(Z, n) → K(Z, n), Exercise 5.9. Determine the cohomology ring ∗ H (K(Z, 3); Z) using the pathspace bration where ΩK(Z, n) ' K(Z, n − 1) and PK(Z, n) ' ∗, and its Serre spectral sequence 10.2. Notice that ΩK( , 3) → PK( , 3) → K( , 3) is simply connected as . Then p,q Z Z Z K(Z, n) n ≥ 2 E2 = p q . As ∗,∗ and the previous exercise, where we have ΩK( , 3) ' H (K(Z, n); Q) ⊗ H (K(Z, n − 1); Q) E2 ⇒ Z ∗ and . Solution in 12.4. H (∗; Q) = Q, all terms but the Z in (0, 0) must dis- K(Z, 2) PK(Z, 3) ' ∗ ∗ appear. If n is even then H (K(Z, n − 1); Q) = Λ(z) with . Moreover, the only chance of dying Exercise 5.10. Determine the cohomology ring |z| = n − 1 ∗ n using the pathspace bration for z at position (0, n − 1) is by killing d (z) = x H (ΩS ; Z) n ΩSn → PSn → Sn, where PSn ' ∗. Solution in 12.5. 6 Lifting problem 18

6 Lifting problem Theorem 6.2 ([6, Theorem 2.1]). If E∞ is a nitely generated bigraded commutative algebra, then H∗ is a Recall the general setting for the lifting problem: we nitely generated graded commutative algebra. More- ∗ ∗ have a graded algebra H with a ltration F and over, given a presentation of E∞ by double homoge- then we consider the associated bigraded algebra E∞ neous generators and double homogeneous relations: with the induced product as in (d). So, for F pHp+q = 1. Generators for H∗ are given by lifting the given F p ∩ Hp+q, we have Ep,q ∼ F pHp+q/F p+1Hp+q and ∞ = generators of E , and they have the same total the product p,q p0,q0 p+p0,q+q0 sends ∞ E∞ ⊗ E∞ → E∞ degree. (a + F p+1Hp+q) · (b + F p0+1Hp0+q0 ) 2. Relations for H∗ are given by lifting the given 0 0 0 relations of E∞, and they have the same total = (a · b) + F p+p +1Hp+q+p +q . degree. Example 6.1 ([17, Example 1.J, p. 23]). Consider In Example 6.1, we have the following two algebras: ∗,∗ 2 2 2 E∞,1 = Q[¯x, y,¯ z¯]/(¯x , y¯ , z¯ , x¯y,¯ x¯z,¯ y¯z¯) H∗ = [x, y, z]/(x2, y2, z2, xy, xz, yz) 1 Q and these generators and relations lift to give ∗ 2 2 2 . For with |x| = 7, |y| = 8, |z| = 15, and H1 = Q[x, y, z]/(x , y , z , xy, xz, yz) ∗ 2 2 2 E∗,∗ = [¯u, v,¯ w¯]/(¯u2, v¯2, w¯2, u¯v,¯ u¯w,¯ v¯w¯), H2 = Q[u, v, w]/(u , v , w , uv − w, uw, vw) ∞,2 Q with |u| = 7, |v| = 8, |w| = 15. The following de- the relation u¯v¯ = 0 lifts as uv = w. This theorem has the following immediate consequence. creasing ltrations F1 and F2 of H1 and H2 (respectively) are compatible with the respective products: Corollary 6.3 ([6, Theorem 2.1]). If R is a nite ∗ 8 7 6 5 4 3 eld, the graded algebra structure of H is determined, 0 = F1 ⊂ F1 = F1 = F1 = hzi ⊂ F1 = F1 = hy, zi up to a nite number of possibilities, by the bigraded 2 1 algebra structure of . ⊂ F1 = F1 = H1, E∞ 8 7 6 5 4 3 0 = F2 ⊂ F2 = F2 = F2 = hwi ⊂ F2 = F2 = hv, wi At this point, the original picture of the bridge 2 1 and the tree should be clearer. That there is a bridge ⊂ F2 ⊂ F2 = H2. means that the spectral sequence converges, and you The next pictures depict the corresponding bigraded can cross it if you are able to gure out enough dier- algebras E and E : entials. Climbing the tree implies solving extension ∞,1 ∞,2 and lifting problems. This may require extra information. 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0z ¯ 0 8 0 0 0 0 0 0w ¯ 0 Convergence H x 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 −→  extension and d4... ··  lifting problems 6 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 d3 ···················· ··················· E∞ 5 0x ¯ 0 0 0 0 0 0 5 0u ¯ 0 0 0 0 0 0 d2 ····· E2 4 0 0 0y ¯ 0 0 0 0 4 0 0 0v ¯ 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 We exemplify Theorem 6.2 by means of the next 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 example. 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Example 6.4. Consider the dihedral group D8 described as a wreath product: ∼ We have E∞,1 = E∞,2 as bigraded algebras because all products of generators are zero in both cases. In D8 = C2 o C2 = C2 × C2 o C2 = ha, bi o hci, the former case, this is a consequence of the products being zero in the graded algebra ∗. In the latter where c interchanges a and b: ca = b, cb = a. The H1 case, the product u¯v¯ is zero because |u¯| = (2, 5), |v¯| = Lyndon-Hochschild-Serre spectral sequence 10.4 with coecients in the eld of two elements is , and 6,9 . What is happening (4, 4) |u¯v¯| = (6, 9) E∞,2 = 0 is that 6,9 6 15 7 15 implies that p,q p q p+q u¯v¯ = 0 ∈ E∞,2 = F2 H2 /F2 H2 E2 = H (C2; H (C2 × C2; F2)) ⇒ H (D8; F2). 7 15 . So we may deduce that uv ∈ F2 H2 = hwi uv = λw The extension is not central and it is not imme- for some λ ∈ , but we cannot know whether λ = 0 diate to describe . Recall that ∗ Q E2 H (C2 × C2; F2) = or not. ∗ ∗ , with H (C2; F2) ⊗ H (C2; F2) = F2[x, y] |x| = |y| = The following result claries the general situation. 1. Here, x = a∗ and y = b∗ are the corresponding du- als. This ring can be decomposed as a C2-module into 7 Edge morphisms 19

a direct sum of two type of C2-modules: either the 7 Edge morphisms trivial C2-module or the free transitive C2-module: Consider a cohomological type spectral sequence E2 ⇒ Ù which is rst quadrant, i.e., such that p,q × Ö 2 2 Ô 3 3 2 2 2 2 H E2 = 0 1 x y x y xy x y x y xy x y ... whenever p < 0 or q < 0. It is clear that there are monomorphisms and epimorphisms for all : Now, the cohomology of the trivial module has been p, q ≥ 0 already mentioned ∗ with . H (C2; F2) = F2[z] |z| = 1 E0,q ,→ ...,→ E0,q ,→ E0,q ,→ ...,→ E0,q ,→ E0,q, The cohomology of the free transitive module M is ∞ r r−1 3 2 given by the xed points in degree , 0 p,0 p,0 p,0 p,0 p,0 0 H (C2; M) = E2 E3 ... Er−1 Er ... E∞ . id C2 n M , and H (C2; M) = 0 for n > 0 (as 0 → M → 0,q is a free -resolution of ). To sum up, So E0,q is an R-submodule of E and Ep,0 is a quo- M → 0 F2C2 M ∞ 2 ∞ -generators for the corner of are as follows: tient -module of p,0. Now, if we rewrite the exten- F2 E2 R E2 sions (4.1) together with a,b ∼ a a+b a+1 a+b E∞ = F H /F H x3y + xy3, x4 + y4, x2y2 x2y2z x2y2z2 x2y2z3 x2y2z4 x2y2z5 x2y2z6 we get x3 + y3, x2y + xy2 0 0 0 0 0 0 F 1Ha+b → Ha+b → E0,a+b x2 + y2, xy xyz xyz2 xyz3 xyz4 xyz5 xyz6 ∞ 2 a+b 1 a+b 1,a+b−1 x + y 0 0 0 0 0 0 F H → F H → E∞ 1 z z2 z3 z4 z5 z6 ... F a+bHa+b → F a+b−1Ha+b → Ea+b−1,1 A similar description of and the general fact that ∞ E2 a+b a+b a+b,0 the spectral sequence of a wreath product collapses in 0 → F H → E∞ . E may be found in [3, IV, Theorem 1.7, p. 122]. So 2 We deduce that 0,q is a quotient module of q we have E = E = [z, σ , σ ]/(zσ ) where σ = E∞ H = ∞ 2 F2 1 2 1 1 0 q and that p p p,0 is a submodule of p. x + y and σ2 = xy are the elementary symmetric F H F H = E∞ H polynomials. So according to Theorem 6.2, we have Summing up, we have morphisms, called edge mor- ∗ , where , phisms: H (D8; F2) = F2[w, τ1, τ2]/(R) |w| = 1 |τ1| = 1, |τ | = 2 and R is the lift of the relation zσ = 0. 2 1 q 0,q 0,q Recall that |z| = (1, 0) and that |σ1| = (0, 1). So H E∞ ,→ E2 , 1,1 1 2 2 2, where ∗ is some p,0 p,0 p zσ1 = 0 ∈ E∞ = F H /F H F E E ,→ H . unknown ltration of ∗ ∗ . So 2 ∞ H = H (D8; F2) wτ1 ∈ F 2H2 and we need to know what is F 2H2. For total For several spectral sequences, the edge morphisms degree 2, we have extension problems (4.1): may be explicitly described. For the Lyndon-Hochschild-Serre spectral sequence 1 2 2 0,2 2 2 ι π F H →H → E∞ = hx + y , xyi = F2 ⊕ F2 10.4 of a short exact sequence of groups, N → G → 2 2 1 2 1,1 , we have 0,q 0 q q Q F H →F H → E∞ = 0 Q E2 = H (Q; H (N; R)) = H (N; R) 2 2 2,0 2 and the edge morphism, 0 →F H → E∞ = hz i = F2. So we deduce that 2 2 1 2 2 2 . q 0,q 0,q q Q F H = F H = hz i ⊂ H (D8; F2) H (G; R) E∞ ,→ E2 = H (N; R) , So we may take (see Remark 7.1) and then w = z coincides with the restriction in cohomology 2 for some . We cannot deduce the wτ1 = λw λ ∈ F2 value of from the spectral sequence. A computa- λ Hq(ι): Hq(G; R) → Hq(N; R)Q. tion with the bar resolution shows that λ = 0, and 2 hence H (D ; ) = [w, τ , τ ]/(wτ ). p,0 8 F2 F2 1 2 1 We also have E = Hp(Q; H0(N; R)) = Hp(Q; RN ) Exercise 6.5. Calculate the Poincaré series and the edge morphism,2 ∞ X i i Hp(Q; RN ) = Ep,0 Ep,0 ,→ Hp(G; R), P (t) = dim H (D8; F2)t , 2 ∞ i=0 is exactly the ination from the bigraded module E∞ of Example 6.4. Solu- tion is in 12.6. Hp(π): Hp(Q; RN ) → Hp(G; R). Exercise 6.6. Fully describe the Lyndon-Hochschild- Remark 7.1. We have seen that p,0 p Serre spectral sequence with coecients 10.4 of the E∞ ⊆ H (G; R) F2 as an R-module for all p ≥ 0 but in fact E∗,0 ⊆ dihedral group D described as a semi-direct product: ∞ 8 ∗ as a subalgebra because p,0 p0,0 p+p0,0. H (G; R) E∞ ·E∞ ⊆ E∞ D = C C = hai hbi, 8 4 o 2 o Exercise 7.2. Prove that in Example 5.2 we can with b −1. Recall that by Example 6.4 deduce the algebra structure of ∗ from the a = a H (C4; F2) ∗ . Solution is in 12.7. -page. Solution in 12.8. H (D8; F2) = F2[w, τ1, τ2]/(wτ1) E∞ 8 Dierent spectral sequences with same target 20

Consider the Serre spectral sequence of a bration In particular, we deduce the oriented bordism of X F →ι E →π B 10.2 with B simply connected and F in low dimensions: connected. The edge morphism H (X; ), for q = 0, 1, 2, 3, ΩSO(X) = q Z q 0,q 0,q q q ⊕ H (X; ), for q = 0, 4. H (E; R) E∞ ,→ E2 = H (F ; R), Z 4 Z is Example 7.6 ([10, Proposition 7.3.2]). Consider an q q q H (ι): H (E; R) → H (F ; R). extension of groups N →ι G →π Q which is split, The edge morphism i.e., such that G is the semi-direct product G = . We consider the Lyndon-Hochschild-Serre Hp(B; R) = Ep,0 Ep,0 ,→ Hp(E; R), N o Q 2 ∞ spectral sequence 10.4 with coecients in a ring R coincides with with trivial G-action: Hp(π): Hp(B; R) → Hp(E; R). p,q p q p+q E2 = H (Q; H (N; R)) ⇒ H (G; R). Remark 7.3. We have seen that p,0 p E∞ ⊆ H (E; R) We know that the edge morphism as an R-module for all p ≥ 0 but in fact E∗,0 ⊆ ∞ p p,0 p,0 p ∗ as a subalgebra because 0,q 0,q0 0,q+q0 . H (Q; R) = E E ,→ H (G; R), H (E; R) E∞ ·E∞ ⊆ E∞ 2 ∞ Exercise 7.4. Figure out what are the edge mor- is exactly the ination phisms for the Serre and the Lyndon-Hochschild-Serre p p p homological spectral sequences. H (π): H (Q; R) → H (G; R). The next two examples show two particular situa- Because the extension is split, there is a homomor- p tions where the edge morphisms give much informa- phism s: Q → G with π ◦ s = 1Q. Hence, H (s) ◦ p p tion. H (π) = 1Hp(Q), the ination H (π) is injective and we deduce that p,0 p,0. So all dierentials ar- Example 7.5 ([8, p. 246]). We consider oriented E2 = E∞ bordism ΩSO, which is a generalized cohomology the- riving to the horizontal axis must be zero and we ∗ have an inclusion of algebras ∗ ∗ , ory. For a space X, ΩSO(X) is, roughly speaking, H (Q; R) ⊆ H (G; R) the abelian group of oriented-bordismn classes of - see Remark 7.1. This situation already occurred in n Example 6.4 and appears again in Exercise 6.6. dimensional closed manifolds mapping to X. Then we have the Atiyah-Hirzebruch spectral sequence 10.5 for the bration 8 Dierent spectral sequences with same ι id target ∗ → X → X, where X is any topological space. It states that In Examples 6.4 and Exercise 6.6 we have seen two SO SO dierent spectral sequences converging to the same Hp(X;Ωq (∗)) ⇒ Ωp+q(X). target ∗ . In this section, we present an- H (D8; F2) Also we have that other two examples of this phenomenon, the rst with target ∗ 1+2 and the second with target 0, q < 0 or q = 1, 2, 3, H (3+ ; F3) ΩSO(∗) = H (S3; ). From these examples, one sees that the q Z, for q = 0, 4. ∗ Z price one pays for having an easily described E2-page In this case, the edge morphism, is more complicated dierentials. SO SO 2 ∞ SO Example 8.1. We denote by 1+2 the extraspe- Ωq (∗) H0(X;Ωq (∗)) = E0,q E0,q ,→ Ωq (X), S = 3+ cial group of order 27 and exponent 3. It has the is given by SO . Note that the constant map Ωq (ι) following presentation satises . Hence, SO c: X → ∗ c ◦ ι = 1∗ Ωq (∗) ◦ S = hA, B, C|A3 = B3 = C3 = [A, C] = [B,C] = 1, [A, B] = Ci SO SO Ω (ι) = 1 SO and the edge morphism Ω (ι) is in- q Ωq (∗) q and it ts in the central extension jective. In particular, ΩSO(∗) = H (X;ΩSO(∗)) and q 0 q 1+2 ¯ ¯ 2 ∞ for all . This last condition implies that hCi = C3 → 3+ → C3 × C3 = hA, Bi. E0,q = E0,q q all dierentials arriving or emanating from the verti- Leary describes in [15] the Lyndon-Hochschild-Serre cal axis must be zero: spectral sequence of this extension. Its second page SO is given by 4 Ω4 (∗) H1(X; Z) H2(X; Z) H3(X; Z) H4(X; Z) ∗,∗ ∗ ∗ 3 0 0 0 0 0 E2 = H (C3; F3) ⊗ H (C3 × C3; F3) 2 0 0 0 0 0 = Λ(u) ⊗ F3[t] ⊗ Λ(y1, y2) ⊗ F3[x1, x2], 1 0 0 0 0 0 with |u| = |y | = |y | = 1 and |t| = |x | = |x | = SO 1 2 1 2 0 Ω0 (∗) H1(X; Z) H2(X; Z) H3(X; Z) H4(X; Z) 2. The dierentials in E∗ are the following, and the 0 1 2 3 4 spectral sequence collapses at E6. 9 A glimpse into the black box 21

(i) , , where acts by the antipodal map and 3 is the d2(u) = y1y2 d2(t) = 0 C2 RP projective space of dimension . Because the ber is (ii) , 3 d3(t) = x1y2 − x2y1 discrete, the associated Serre spectral sequence 10.1: (iii) i i−1 2 2 , d4(t u(x1y2 − x2y1)) = it (x1x2y2 − x1x2y1) 2 3 3 2 Ep,q = Hp(RP ; Hq(C2; Z)) ⇒ Hp+q(S ; Z) d4(t yi) = u(x1y2 − x2y1)xi, collapses in the horizontal axis of 2: We have (iv) 2 3 3, 2 E d5(t (x1y2 −x2y1)) = x1x2 −x1x2 d5(ut y1y2) = with 3 interchanging 3 3 , . H0(C2; Z) = Z ⊕ Z C2 = π1(RP ) ku(x1y2 − x2y1) k 6= 0 the two 's, and for . Hence, 2 Z Hq(C2; Z) = 0 q > 0 Ep,0 A long and intricate computation leads from E2 to is the twisted homology 3 and 2 Hp(RP ; Z ⊕ Z) Ep,q = 0 E6 [9]. The following table contains representatives for . We deduce that the twisted homology n,m q > 0 of classes that form an 3-basis of E for 0 ≤ n ≤ 6 3 must be for and otherwise. F 6 Hp(RP ; Z⊕Z) Z p = 0, 3 0 and 0 ≤ m ≤ 5: Let us directly compute the twisted homology 3 via the universal cover 3 of 3 Hp(RP ; Z ⊕ Z) S RP 5 3 [11, 3.H]. The sphere S has a structure of free C2- 4 complex with cells 3 uty1y2 2 2 2 ty1, ty2 ty1x1, ty1x2 tx y1, tx y2 3 3 3 2 2 1 1 0 0 1 1 S = e+ ∪ e− ∪ e+ ∪ e− ∪ e+ ∪ e− ∪ e+ ∪ e−, 2 2 ty2x2 tx2y1, tx2y2 where C2 interchanges each pair of cells on every di- 1 uy , uy uy y uy x , uy x ux2y , ux2y 1 2 1 2 1 1 1 2 1 1 1 2 mension. This gives rise to the following chain com- 2 2 uy2x1, uy2x2 ux y1, ux y2 plex over : 2 2 ZC2 0 1 y , y x , x y x , y x x2, x2 x2y , x2y x3, x3 1 2 1 2 1 1 1 2 1 2 1 1 1 2 1 2 0 → ⊕ → ⊕ → ⊕ → ⊕ → → 0, 2 2 2 2 Z Z Z Z Z Z Z Z Z y2x2 x1x2 x2y1, x2y2 x1x2, x1x2 0 1 2 3 4 5 6 with d(1, 0) = (1, −1), d(0, 1) = (−1, 1) in dimensions 1 and 3, d(1, 0) = d(0, 1) = (1, 1) in dimension It turns out that this description of the corner of 2 and d(1, 0) = d(0, 1) = 1 in dimension 0. Set M to E determines the rest of E as there are both vertical be equal to the -module . Then we ten- 6 6 ZC2 H0(C2; Z) and horizontal periodicities. More precisely, there sor every term of the chain complex above by ⊗ M are -isomorphisms n,m ∼ n,m+6 for C2 F3 E6 = E6 n, m ≥ 0 and we compute the resulting homology. and En,m ∼= En+2,m for n ≥ 5 and m ≥ 0. The Setting a = (1, 0) ⊗ (1, 0) = (0, 1) ⊗ (0, 1) and b = 6 6 on every dimension, it extraspecial group S = 31+2 also ts in an extension (1, 0) ⊗ (0, 1) = (0, 1) ⊗ (1, 0) + turns out that d(a) = a−b, d(b) = b−a in dimensions 1+2 ¯ and , in dimension and hB,Ci = C3 × C3 → 3 → C3 = hAi, 1 3 d(a) = d(b) = a + b 2 + d(a) = d(b) = 0 in dimension 0. Thus we obtain the where the action is given by the matrix 1 0 . In desired homology 3 . ( 1 1 ) H∗(S ; Z) this case, the page ∗,∗ ∗ ∗ E2 = H (C3; H (C3 × C3; F3)) Exercise 8.3. Do similar computations to those of is quite complicated as the action of C3 is far from being trivial. It is described in [20], where the au- Example 8.2 for the bration thor also shows that . The page has E2 = E∞ E2 9 I → S3 → X, generators, x1, γ1, x2, y2, γ2, x3, x6, z2, z3, and they lie in the following positions: where I is the binary icosahedral group of order 120 and X is the Poincaré homology sphere. In this case, 6 x you should recover the homology groups 3 6 H∗(S ; Z) 5 as the twisted homology of X with coecients in a 4 module for the group π1(X) = I. 3 x3 2 x2, y2 z3 9 A glimpse into the black box 1 x1 z2 In this section, we show how a ltration gives rise 0 γ1 γ2 to a spectral sequence. An equivalent approach to 0 1 2 3 construct spectral sequences is exact couples [17, p. 37]. An exact couple consist of three module homo- Compare to the E2-page in Equation (8.1), which has morphisms between graded or bigraded modules such 6 generators all of which lie on the axes. that the following diagram is exact at each module: Example 8.2. In Example 3.1, we saw that the Hopf i bration S1 → S3 → S2 gives rise to a spectral se- D / D quence converging to 3 . Consider now the - `AA } H∗(S ) AA }} AA }} bration k AA } j 3 3 ~}} C2 → S → RP , E. 9 A glimpse into the black box 22

The notion of derived couple makes it very natural to (e) The dierential d induces a dierential derive a spectral sequence from a given exact couple. r r r Moreover, sometimes exact couples are the natural dp,q : Ep,q → Ep−r,q+r−1. setting to construct certain spectral sequences, e.g., the Bockstein spectral sequence. Nevertheless, here (f) When taking homology we have we focus on the approach via ltrations. r r ∼ r+1 So let (C, d) be a chain complex and let F∗C be an Hp,q(E∗,∗, d ) = Ep,q . increasing ltration of (C, d), i.e., an ordered family of chain subcomplexes: Exercise 9.1. Compute the modules r 's, r 's Zp,q Bp,q and r 's for Example 2.1. Ep,q ... ⊆ Fn−1C ⊆ FnC ⊆ ... ⊆ C. We may dene a ltration of the homology of via These two facts are consequence of intricate dia- H C gram chasings, and it is here where things become the inclusions: FnH = Im(H(FnC, d) −→ H(C, d)). Thus we have an ordered family of graded modules: delicate. We do not reproduce the details (that may be found in [17, 2.2] or [21, 5.4] for instance), but ... ⊆ Fn−1H ⊆ FnH ⊆ ... ⊆ H. instead we explain the crucial step. The aforementioned chasing arguments reduce to On each total degree p + q, we obtain a ltration of the modules and : the following situation: We are given a 3×3-diagram Cp+q Hp+q of modules and inclusions, ... ⊆ Fp−1Cp+q ⊆ FpCp+q ⊆ ... ⊆ Cp+q A / B / C ... ⊆ Fp−1Hp+q ⊆ FpHp+q ⊆ ... ⊆ Hp+q. We will assume that the ltration is bounded, i.e., d d d that for each total degree , there exist and such n s r D / E / F that FsCn = 0 and FtCn = Cn. Then we have: d d d 0 = FsCp+q ⊆ ... ⊆ Fp−1Cp+q ⊆ FpCp+q G / H / I, ⊆ ... ⊆ FrCp+q = Cp+q (9.1) 0 = FsHp+q ⊆ ... ⊆ Fp−1Hp+q ⊆ FpHp+q where d◦d = 0, and we need to prove (g) and (h) be- ⊆ ... ⊆ F H = H (9.2) low. We dene E1 as taking quotient and homology r p+q p+q (in this order): Because of the boundedness assumption, there exists a spectral sequence r that converges to E {E∗,∗, dr}r≥0 H E = H(B/A → E/D → H/G), in the sense explained in (a) and (b). More precisely, 1 EG = H(D → G → 0), E0 is the associated graded object, i.e., it is obtained 1 ∗,∗ C by taking quotients on (9.1): E1 = H(0 → C/B → F/E), 0 and as taking homology and quotient (in this E = F C /F C , E∞ p,q p p+q p−1 p+q order): and the stable term ∞ at is also obtained as Ep,q (p, q) an asociated graded object, i.e., it is isomorphic to Im(H(B → E → H) → H(C → F → I)) EE = . considering quotients on (9.2): ∞ Im(H(A → D → G) → H(C → F → I)) ∞ ∼ Ep,q = FpHp+q/Fp−1Hp+q. Then the following hold: This correspond to the Equations (3.1). So the prob- (g) The dierential d induces a dierential lem is to dene the rest of pages r and their dif- E∗,∗ ferentials and see how they t together. C d1 E d1 G E1 → E1 → E1 . Dene r −1 and r Zp,q = FpCp+q∩d (Fp−rCp+q−1) Bp,q = FpCp+q ∩ d(Fp+rCp+q+1): (h) When taking homology we have d Fp+rCp+q+1 eee 1 1 reeeeee C d E d G E FpCp+q H(E → E → E ) ∼= E . d eee 1 1 1 ∞ reeeeee Fp−rCp+q−1 Then we set Exercise 9.2. Prove that (g) and (h) hold. Zr The situation for a decreasing ltration of a cochain Er = p,q , p,q Zr−1 + Br−1 complex is similar. Considering a product in the p−1,q+1 p,q cochain complex gives rise to a spectral sequence of and one has to check that: algebras, see [17, 2.3]. 10 Appendix I 23

10 Appendix I Then we have a ltration of E given by J s = π−1(Bs)

There are predictably a large number of interest- ∅ ⊆ J 0 ⊆ J 1 ⊆ J 2 ... ⊆ E. ing spectral sequences in the literature. Classically, the most used and the most well-known are: (i) the Consider the cochain complex of singular cochains on Leray-Serre spectral sequence which manages the in- E with coecients in R, C∗(E; R), given by tertwining relationships between the (co)homology s groups of base, ber and total spaces of a bration, C (E; R) = {functions f : Cs(E) → R}, (ii) the Lyndon-Hochschild spectral sequence which where s continuous are the is used in computing the (co)homology of groups, Cs(E) = {σ : ∆ → E } singular s-chains in E. Now dene a decreasing l- (iii) the Atiyah-Hirzebruch spectral sequence, and ∗ (iv) the Adams spectral sequence for computing the tration of C (E; R) by stable homotopy groups of spheres. We discuss the s ∗ ∗ ∗ s−1 three rst spectral sequences in this appendix, and we F C (E; R) = ker(C (E; R) → C (J ; R)), briey discuss the Adams spectral sequence in Sub- i.e., the singular cochains of that vanish on chains section 11.4. E in J s−1. It turns out that

Theorem 10.1 (Serre spectral sequence for homol- p,q p+q p p−1 ogy, [17, Theorem 5.1]). Let M be an abelian group E1 = H (J ,J ; R), and let be a bration with path- F → E → B B the relative cohomology of the pair p p−1 . connected. Then there is a rst quadrant homological (J ,J ) spectral sequence

2 Ep,q = Hp(B; Hq(F ; M)) ⇒ Hp+q(E; M), s s−1 where we are considering homology of B with local E J J F coecients in the homology of the ber F . Theorem 10.2 (Serre spectral sequence for cohomology, [17, Theorem 5.2]). Let R be a ring and let F → E → B be a bration with B path-connected. Then there is a rst quadrant cohomological spectral B Bs Bs−1 sequence of algebras and converging as an algebra

p,q p q p+q E2 = H (B; H (F ; R)) ⇒ H (E; R), The keystep of the proof relies in showing that where we are considering cohomology of with local B p+q p p−1 ∼ p q coecients in the cohomology of the ber F . H (J ,J ; R) = C (B; H (F ; R)), Homology (cohomology) with local coecients is the twisted p-cochains of B with coecients in the q- also known as twisted homology (cohomology). Be- cohomology of the ber, which are dened as follows: ing twisted means that an action of the fundamen- functions S q −1 tal group of the space on the module is taken into { f : Cp(B) → b∈B H (π (b); R) account. It can be understood through the universal q −1 cover, see [11, 3.H], and this is the approach we take such that f(σ) ∈ H (π (σ(v0)); R)}. in Example 8.2. If this action turns out to be trivial Note that, as B is connected, π−1(b) has the same then there is no twist and we recover ordinary ho- homotopy type (F ) for all b ∈ B. Also, we assume mology (cohomology) of the space over the module. that p p is spanned by the vertices . ∆ ⊆ R {v0, . . . , vp+1} For instance, in Theorems 10.1 and 10.2, if π1(B) acts The (twisted) dierential ∗ trivially on H∗(F ; M) or H (F ; R), then the second page is given by ordinary homology of or ordinary B Cp(B; Hq(F ; R)) → Cp+1(B; Hq(F ; R)) cohomology of B respectively. Of course, the action is trivial whenever π1(B) = 1. In particular, in the is given by latter theorem, if π1(B) = 1 and R is a eld we have p+1 p,q p q p+q q X i E2 = H (B; R) ⊗ H (F ; R) ⇒ H (E; R). ∂(f)(σ) = H (Φ; R)(f(σ0)) + (−1) f(σi), Sketch of proof of Theorem 10.2. We follow [17, Chap- i=1 π p ter 5]. Denote by F → E → B the given bration where σi is the restriction of σ to the i-th face of ∆ , −1 and consider the skeletal ltration of B {v0,..., vˆi, . . . , vp+1}, and the map Φ: π (σ(v0)) → π−1(σ(v )) is a lift of the path σ starting in v 0 1 2 1 [v0,v1] 0 ∅ ⊆ B ⊆ B ⊆ B ... ⊆ B. and ending in v1. 10 Appendix I 24

◦ •@ ◦O ◦ ◦ @@ d@v −1 Φ −1 @ dh π (σ(v0)) −→ π (σ(v1)) ◦ ◦@ •@ ◦/ ◦ @@ @@ @@ ◦ ◦ ◦ •@@ ______◦ @@ ·····@@ v1 σ(v1) ∆2 −→σ σ(v0) ····· v0 Filtration by rows σ(v2) v2 ··? ◦ ◦ ◦ Then ∗,∗ is obtained as the cohomology of ?? E2 ?? Cp(B; Hq(F ; R)), and this gives the description in · ·? •? ◦O ◦ ?? the statement of the theorem. ?dv ?? dh · · ◦ •?? ◦/ Theorem 10.3 (Lyndon-Hochschild-Serre spectral ?? ?? sequence for homology, [21, 6.8.2]). Let N → G → Q · · ◦ ◦? • be a short exact sequence of groups and let be a M G-module. Then there is a rst quadrant homological · · ◦ ◦ ◦ spectral sequence Filtration by columns

2 Ep,q = Hp(Q; Hq(N; M)) ⇒ Hp+q(G; M). These two ltrations give rise to spectral sequences Theorem 10.4 (Lyndon-Hochschild-Serre spectral converging to the cohomology of (Total(C), ∂h + ∂v) [21, Section 5.6], sequence for cohomology, [21, 6.8.2]). Let N → G → be a short exact sequence of groups and let be Q R r p,q p q p+q , and a ring. Then there is a rst quadrant cohomological E2 = Hv Hh(C) ⇒ H (Total(C)) spectral sequence of algebras converging as an algebra c p,q p q p+q E2 = HhHv (C) ⇒ H (Total(C)), p,q p q p+q E2 = H (Q; H (N; R)) ⇒ H (G; R). where the subscripts h and v denote taking cohomology with respect to ∂h or ∂v respectively. For the for- In the latter theorem, if N ≤ Z(G) and R is a mer spectral sequence, we obtain r p,q p E2 = H (G; R) eld we have for and p,q for by [16, XI, Lemma q = 0 E2 = 0 q > 0 p,q p q p+q 9.3]. For the latter spectral sequence we get E2 = H (Q; R) ⊗ H (N; R) ⇒ H (G; R). c p,q q Sketch of proof of Theorem 10.4. We follow [16, XI, E1 = HomRQ(Bp(Q),H (N; R)), Theorem 10.1] but consult [12] for a direct ltration as is a free -module and hence commutes on the bar resolution of G. We do not provide de- Bp(Q) RQ tails about the multiplicative structure. Consider the with cohomology, and then double complex c p,q p q E2 = H (Q; H (N; R)). p,q C = HomRQ(Bp(Q), HomRN (Bq(G),R) ∼= Hom (B (Q) ⊗ B (G),R), RG p q Theorem 10.5 (Atiyah-Hirzebruch spectral sequence, [8, Theorem 9.22]). Let be a generalized cohomol- where B∗(Q), B∗(G) denote the corresponding bar h resolutions and RQ, RN and RG denote group rings. ogy theory and let F → E → B be a bration with So B (Q) is the free R-module with basis Qp+1 and B path-connected. Assume hq(F ) = 0 for q small p enough. Then there is a half-plane cohomological with dierential the R-linear extension of ∂(q0, . . . , qp) = Pp i . Then the horizontal and spectral sequence i=0(−1) (q0,..., qˆi, . . . , qp) ∗,∗ vertical dierentials of C are given by p,q p q p+q E2 = H (B; h (F )) ⇒ h (E), 0 p+q+1 0 , and ∂h(f)(b ⊗ b ) = (−1) f(∂(b) ⊗ b ) where we are considering cohomology of B with local 0 q+1 0 ∂v(f)(b ⊗ b ) = (−1) f(b ⊗ ∂(b )). coecients. ∗ There are two ltrations of the graded dierential R- Again, if π1(B) acts trivially on h (F ), then the second page is given by ordinary cohomology of . module (Total(C), ∂h + ∂v) obtained by considering B either all rows above a given row or all columns to the right of a given column: 11 Appendix II 25

11 Appendix II Using as coecients the integers localized at p, , i.e., the subring of consisting of fractions with Z(p) Q 11.1 Finitely generated homology. denominator relatively prime to p, yields the next result. Let X be a space. We say that X is of nite type if is a nitely generated abelian group for Theorem 11.5 ([13, Theorem 1.28] 25)[8, Corollary Hi(X; Z) all i ≥ 0. The following result is a nice and easy 10.13]). For n ≥ 3 and p a prime, the p-torsion sub- n application of the Serre spectral sequence 10.1. group of πi(S ) is zero for i < n + 2p − 3 and Cp for i = n + 2p − 3. Theorem 11.1. Let F → E → B be a bration with B simply connected and F path-connected. If two of The Serre spectral sequence may be further ex- the spaces F , E or B are of nite type, then so is the ploited to compute more homotopy group of spheres third. [18, Chapter 12] but it does not give a full answer. The EHP spectral sequence is another tool to com- A proof may be found in [17, Example 5.A] or [13, pute homotopy group of spheres, see for instance [13, Lemma 1.9]. From this, we may derive the following p. 43]. two consequences. Corollary 11.2. The loop space of a simply con- 11.3 Cohomology operations and Steenrod nected nite CW-complex is of nite type. algebra. This corollary is obtained from the previous the- Cohomology operations of type are ex- (n, m, Fp, Fp) orem using the pathspace bration of a space X: actly the natural transformations between the func- tors n and m : ΩX → PX → X. H (−; Fp) H (−; Fp) n m Recall that PX ' ∗ and that, if X is a nite CW - θ : H (−; Fp) ⇒ H (−; Fp). complex, then cellular homology gives directly that Among cohomology operations, we nd stable coho- X is of nite type. By Theorem 11.1, the loop space mology operations, i.e., those families of cohomology of , , is also of nite type. X ΩX operations of xed degree m, Corollary 11.3. If is a nitely generated abelian A n n+m group then K(A, n) is of nite type for all n ≥ 1. {θn : H (−; Fp) → H (−; Fp)}n≥0, Here, K(A, n) is an Eilenberg-MacLane space. The that commute with the suspension isomorphism: proof is as follows: We consider the pathspace bration n θn n+m H (X; Fp) / H (X; Fp)

ΩK(A, n) → PK(A, n) → K(A, n), Σ Σ

θn+1 where PK(A, n) ' ∗ and ΩK(A, n) ' K(A, n − 1). Hn+1(ΣX; ) / Hn+m+1(ΣX; ). Then, by Theorem 11.1 and induction, it is enough Fp Fp to deal with the case , i.e., to compute the ho- n = 1 These stable cohomology operations for mod p co- mology of the group A. Moreover, as A is a product homology are assembled together to form the Steen- of cyclic groups, Theorem 11.1 reduces the problem rod algebra Ap. Moreover, cohomology operations to consider just one of the cyclic factors. Finally, we of a given type are in bijection with know that the homology of either the innite cyclic (n, m, Fp, Fp) the cohomology group m . The co- group or a nite cyclic group is nitely gener- H (K(Fp, n); Fp) Z Zm homology ring ∗ may be determined ated on every degree. H (K(Fp, n); Fp) via the Serre spectral sequence for all n and for all primes p, although this computation is much harder 11.2 Homotopy groups of spheres. than the already seen in Example 5.7. Hence, information about ∗ provides insight into H (K(Fp, n); Fp) The Serre spectral sequence may be used to obtain the Steenrod algebra , which in turn is needed in general results about homotopy groups of spheres. Ap For instance, the cohomology Serre spectral sequence the Adams spectral sequence 11.4. See [13, Theo- rem 1.32], [17, Theorem 6.19], [18, Chapter 9] and [8, with coecients Q is the tool needed for the following result. Section 10.5]. Theorem 11.4 ([13, Theorem 1.21][8, Theorem 10.10]). n 11.4 Stable homotopy groups of spheres. The groups πi(S ) are nite for i > n, except for 2n which is the direct sum of with a nite These groups are dened as follows: π4n−1(S ) Z group. πS = lim (... → π (Sn) → π (Sn+1) → ...), k −→ n+k n+k+1 n→∞ 12 Solutions 26 where the homomorphisms are given by suspension: A(G) → Z. Cohomotopy groups are dened as maps to spheres instead of from spheres: n+k f n n+k n+k+1 Σf n+1 n S −→ S ΣS = S −→ S = ΣS . 0 ∞ ∞ πs (BG) = [Σ BG+, S] = [Σ BG ∨ S, S], In fact, by Freudenthal's suspension theorem, πS = k ∞ 0 π (Sn) for n > k + 1, i.e., all morphisms become where S = Σ S is the sphere spectrum. The map n+k 0 (before completion) takes the tran- isomorphism for n large enough. The Adams spec- A(G) → πs (BG) tral sequence is the tool to compute stable homotopy sitive G-set G/H for H ≤ G to the map groups of spheres: For each prime p, there is a spectral sequence which second page is ∞ trH ∞ ∞ Σ BG+ −→ Σ BH+ → Σ ∗+ = S, s,t s,t where is the transfer map. The conjecture was E2 = ExtA (Fp, Fp) trH p proven by Carlsson [7], who reduced the case of p- and converging to S modulo torsion of order prime groups to the case of -elementary abelian groups. π∗ p to p. Here, A is the Steenrod algebra, see 11.3. The general case had already been reduced to p- p groups by McClure. Ravenel, Adams, Guanawardena The E2-page is so complicated that the May spectral sequence is used to determine it. See [18, Chapter and Miller used the Adams spectral sequence to do 18], [14] and [17, Chapter 9]. the computations for p-elementary abelian groups.

11.5 Hopf invariant one problem 12 Solutions Adams invented and used his spectral sequence to Solution 12.1 (Solution to 3.7). This is [8, p. 243]. solve this problem (although later on he gave another proof via secondary cohomology operations). Solution 12.2 (Solution to 5.4). The Lyndon- Hochschild-Serre spectral sequence of the central ex- The Hopf invariant of a map 2n−1 f n is the only S → S tension is integer H(f) satisfying: C3 → C9 → C3 Ep,q = Hp(C ; ) ⊗ Hq(C , ) x2 = H(f)y, 2 3 F3 3 F3 0 0 p+q = Λ(y) ⊗ F3[x] ⊗ Λ(y ) ⊗ F3[x ] ⇒ H (C9; F3), where x and y are the generators in degrees n and ∗ 2n of the integral cohomology H (X; Z) of certain where |y| = (1, 0), |x| = (2, 0), |y0| = (0, 1) and |x0| = space X. This space obtained by attaching a 2n- (0, 2). So the corner of E2 has the following genera- dimensional cell to Sn via f: tors:

n 2n 02 02 02 02 02 2 X = S ∪f D . x x y x x x yx x x The Hopf invariant one problem consists of determin- y0x0 y0x0y y0x0x y0x0yx y0x0x2 ing for which values of n there exists a map f with x0 x0y x0x x0yx x0x2 H(f) = 1, and its solution is that n ∈ {1, 2, 4, 8}. The problem can also be phrased as determining for y0 y0y y0x y0yx y0x2 n which n does R admit a division algebra structure. 1 y x yx x2 See [11, 4.B] and [17, Theorem 9.38] or Adam's original papers [1], [2]. By elementary group theory, 1 and H (C9; F3) = F3 0 11.6 Segal's conjecture. hence y must die killing d2(y) = x (or −x). Then all odd rows disappear in E as, for instance, This conjecture states that for any nite group G, the 2 natural map from the completion of the Burnside ring d (y0y) = d (y0)y − y0d (y) = xy = yx, of at its augmentation ideal to the stable cohomo- 2 2 2 G 0 0 0 2 topy of its classifying space is an isomorphism: d2(y x) = d2(y )x − y d2(x) = x . Also, we must have 0 because 0 0 , A(G)∧ → π0(BG). d2(x ) = 0 d2(x ) ∈ hy xi s 0 2 d2(y x) = x and d2 ◦ d2 = 0. Then we deduce that, This statement relates the pure algebraic object of for instance, the left hand side to the pure geometric object of 0 0 0 0 the right hand side. The Burnside ring of G is the d2(x y) = d2(x )y + x d2(y) = 0, Grothendieck group of the monoid of isomorphism classes of nite G-sets. The sum is induced by dis- joint union of G-sets and the product by direct product of G-sets with diagonal action. The augmentation ideal is the kernel of the augmenation map 12 Solutions 27

and in fact d2 is zero on even rows. Summing up, What are the dierentials d2, d3,...? It is straight- forward that the Poincaré series of ∗,∗ is E2 x02 x02y x02x x02yx x02x2 0 0 0 0 0 0 0 0 0 0 2 2 3 1 y xVVVy xVVyV y xXXxX y x yx y x x P (t) = 1 + 2t + 3t + 4t + ... = . VVVV VVVV XXXXX 2 VVVV VVVV XXXXX (t − 1) x0 x0yV xV0* xV xVV0yx+ xXX0x+ 2 0 0 0 0 0 2 This coincides with the Poincaré series of ∗ yUU y UyU y XxX y yx y x H (D8; F2) UUU UUU XXXXX by Solution 12.6. So all terms in the -page must UUUU UUUU XXXXX E2 UUUU UUUU XXX+2 1 y x* yxU* x survive, all dierentials must be zero and E∞ = E2 = . By Theorem 6.2, ∗ Λ(y) ⊗ F2[x] ⊗ F2[z] H (D8; F2) = 0 where 0 are lifts of respec- is as follows, F2[z, τ, τ ]/(R) z, τ, τ z, y, x E3 tively (see Remark 7.1) and R is a lift of the relation 2 . This lift must be of the form x02 x02y 0 0 0 y = 0 0 0 0 0 0 τ 2 = λτz + λ0z2, x0 x0y 0 0 0 and we know that λ = 1 and λ0 = 0. 0 0 0 0 0 1 y 0 0 0 Solution 12.8 (Solution to 7.2). Recall that we have 0 2 0 with E∞ = F2[x, x ]/(x ) = Λ(x) ⊗ F2[x ] |x| = (1, 0) and |x0| = (0, 2). Hence, by Theorem 6.2, we have and the spectral sequence collapses at the free graded ∗ 0 with , 0 and H (C4; F2) = F2(z, z )/(R) |z| = 1 |z | = 2 commutative algebra 0 . Then E3 = E∞ = Λ(y)⊗F3[x ] where R is a lift of the relation x2 = 0. Note that by Theorem 5.5, we have ∗ 0 H (C9; F3) = Λ(z) ⊗ F3[z ] x2 ∈ E2,0 = 0 = F 2H2 ⊂ H2(D ; ) by 4.1. If z is a with and 0 . ∞ 8 F2 |z| = 1 |z | = 2 lift of x, then the lift of the relation x2 = 0 must be Solution 12.3 (Solution to 5.8). This is [13, Exam- z2 = 0 and we are done. ple 1.15] and the arguments are already described in 5.7. References Solution 12.4 (Solution to 5.9). This is [13, Exam- ple 1.19] or [4, p. 245]. [1] J.F. Adams, On the structure and applications of Solution 12.5 (Solution to 5.10). This is [13, Ex- the Steenrod algebra, Comment. Math. Helv. 32 ample 1.16] or [4, p. 204] 1958 180214. Solution 12.6 (Solution to 6.5). The answer is [2] J.F. Adams, On the non-existence of elements of Hopf invariant one, Ann. Math. Vol. 72, No. 1, 2 3 1 72 (1), 2004, (1960). P (t) = 1 + 2t + 3t + 4t + ... = 2 . (t − 1) [3] A. Adem, R.J. Milgram, Cohomology of nite Solution 12.7 (Solution to 6.6). The Lyndon- groups, Second edition. Grundlehren der Math- Hochschild-Serre spectral sequence is: ematischen Wissenschaften [Fundamental Prin- ciples of Mathematical Sciences], 309. Springer- p q p+q H (C2; H (C4; F2)) ⇒ H (D8; F2). Verlag, Berlin, 2004. The cohomology ring ∗ with H (C4; F2) = Λ(y) ⊗ F2[x] [4] R. Bott, L.W. Tu, Dierential forms in alge- |y| = 1, |x| = 2 was described in Example 5.2. On braic topology, Graduate Texts in Mathematics, each degree , the -module q is equal to q F2 H (C4; F2) 82. Springer-Verlag, New York-Berlin, 1982. , and hence it must be trivial as a -module. So, F2 C2 although the given extension is not central, the E2- [5] K.S. Brown,Cohomology of groups, Graduate page is still equal to Texts in Mathematics, vol. 87, Springer-Verlag, ∗,∗ ∗ ∗ New York-Berlin, 1982. E2 = H (C4; F2)⊗H (C2; F2) = Λ(y)⊗F2[x]⊗F2[z], [6] J.F. Carlson, Coclass and cohomology, J. Pure where is a generator of 1 . Generators in z H (C2; F2) Appl. Algebra 200 (2005), no. 3, 251266. the corner of E2 lie as follows: [7] G. Carlsson, Equivariant stable homotopy and Se- x2 x2z x2z2 x2z3 gal's Burnside ring conjecture, Ann. of Math. (2) yx yxz yxz2 yxz3 120 (1984), no. 2, 189-224. x xz xz2 xz3 [8] J.F. Davis, P. Kirk, Lecture notes in algebraic y yz yz2 yz3 topology, Graduate Studies in Mathematics, 35. American Mathematical Society, Providence, RI, 1 z z2 z3 2001. 12 Solutions 28

[9] A. Díaz Ramos, O. Garaialde Ocaña, The cohomology of the sporadic group over , Forum J2 F3 Math. 28 (2016), no. 1, 7787. [10] L. Evens, The cohomology of groups, Oxford Mathematical Monographs. Oxford Science Pub- lications. The Clarendon Press, Oxford University Press, New York, 1991. [11] A. Hatcher, Algebraic Topology, online book https://www.math.cornell.edu/~hatcher/ AT/ATpage.html [12] G. Hochschild, J.P. Serre, Cohomology of group extensions, Trans. Amer. Math. Soc. 74, (1953). 110134. [13] A. Hatcher, The Serre Spectral Sequence, online notes https://www.math.cornell.edu/ ~hatcher/SSAT/SSch1.pdf. [14] A. Hatcher, The Adams Spectral Sequence, online notes https://www.math.cornell.edu/ ~hatcher/SSAT/SSch2.pdf. [15] I. Leary, A dierential in the Lyndon- Hochschild-Serre spectral sequence, J. Pure Appl. Algebra 88 (1993), no. 13, 155168. [16] S. Mac Lane, Homology, Die Grundlehren der mathematischen Wissenschaften, Bd. 114 Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg (1963). [17] J. McCleary, A user's guide to spectral sequences, Second edition. Cambridge Studies in Advanced Mathematics, 58. Cambridge Univer- sity Press, Cambridge, 2001. [18] R.R. Mosher, M.C. Tangora, Cohomology operations and applications in homotopy theory, Harper & Row, Publishers, New York-London 1968. [19] J.J. Rotman, An introduction to homological algebra, Second edition. Universitext. Springer, New York, 2009. [20] S.F. Siegel, The spectral sequence of a split extension and the cohomology of an extraspecial group of order p3 and exponent p, J. Pure Appl. Algebra 106 (1996), no. 2, 185198. [21] C.A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathe- matics, 38. Cambridge University Press, Cam- bridge, 1994.

Antonio Díaz Ramos Departamento de Álgebra, Geometría y Topología Universidad de Málaga, Apdo correos 59, 29080 Málaga, Spain E-mail address: [email protected]