Examples of Spectral Sequences

Examples of spectral sequences

Man Cheung, Yao-Rui April 16, 2019

Spectral sequences are a collection of useful tools that helps compute the cohomology of various algebraic or topological objects. In this note we will only give examples of ﬁrst-quadrant cohomological spectral sequences. Dual notions exists for homology and for other quadrants, and one can see [7, Chapter 5] and [3] for deﬁnitions and examples.

1 Introduction

We brieﬂy introduce the notion of a spectral sequence in this section. Since the construction of spectral sequences is not very illuminating, we have chosen to spend more time on examples; see the next few sections for some of them. Here is an abstract deﬁnition of a spectral sequence.

Definition 1. Let R be a ring, and let r0 be a fixed nonnegative integer. A (first-quadrant cohomological) spectral sequence E consists of the following data: p,q • a collection of R-modules Er , with integers p, q ≥ 0 and r ≥ r0, p,q p,q p+r,q−r+1 • and a collection of differentials dr = dr : Er −→ Er such that p,q 2 p,q p,q (dr ) = 0 and Er+1 is the homology at Er , i.e.

ker d : Ep,q −→ Ep+r,q−r+1 Ep,q = r r r . r+1 p−r,q+r−1 p,q im dr : Er −→ Er

p,q th The collection Er = {(Er , dr): p, q ≥ 0} for a ﬁxed r is called the r -page. The spectral sequence converges if Es = Es+1 = Es+2 = ··· for a large enough s, and we denote this page by E∞.

Remark. We may sometimes assign an algebra structure to a page Er of a spectral sequence. In this case, the algebra structure ^ needs to respect the p,q p0,q0 p+p0,q+q0 grading, i.e. ^: Er × Er −→ Er . The reason why spectral sequences are useful is because of the following deﬁnition, which characterizes results that are related to spectral sequences. The examples in this note will make clear of this deﬁnition.

1 Definition 2. A filtered chain complex is a chain complex C = (Ci, δi) of R- modules together with a differential-preserving filtration on each Ci, i.e. for each Ci, there is an increasing sequence of R-modules

F0Ci ⊂ F1Ci ⊂ · · · ⊂ Fp−1Ci ⊂ FpCi ⊂ Fp+1Ci ⊂ · · · such that [ \ FpCi = Ci, FpCi = 0, δi(FpCi) ⊂ FpCi+1. p p

∗ A filtration on the chain complex induces a filtration FpH (C) on the cohomology groups H∗(C). The associated graded pieces are defined by

n n n GpH = FpH (C)/Fp+1H (C).

A spectral sequence E converges to H∗(C), written Ep,q ⇒ Hp+q(C), if E con- r0 verges and p,q ∼ p+q E∞ = GpH . As the deﬁnition implies, a convergent spectral sequence does not compute the cohomology groups we want, but rather a ﬁltration of it. Hence we have the following question, called the lifting problem.

n M p,q How is H (C) related to E∞ ? p+q=n

There are many instances where the two terms are equal. An instance is when the diagonal p + q = n contains exactly one nontrivial term, which we will see a lot of in the next section. If a nontrivial algebra structure is involved, the following easy proposition will guarantee equality.

Proposition 3. If the E∞-page is a free graded commutative algebra, then n T ot(E∞) equals H (C). We end the introduction by stating a consequence of the deﬁnition of spectral sequences, which follows by diagram chasing.

Proposition 4. Given a spectral sequence starting from the E2-page and con- p,q p+q verging to the cohomology of C (so E2 ⇒ H (C) in the notation of Deﬁnition 2), there is an associated ﬁve-term exact sequence

1,0 1 0,1 d2 2,0 2 0 −→ E2 −→ H (C) −→ E2 −−→ E2 −→ H (C).

2 Three examples of spectral sequences

In this section we state three examples of spectral sequences, and give examples for each of them. One can refer to the references for proofs of these spectral sequences.

2 Double complexes Theorem 5 (Double complex spectral sequence [6, Section 1.7]). Let C = Cp,q be a double complex over an abelian category. Then there exists spectral sequences p,q p,q p,q q,p E0 = C ,E0 = C , p+q p,q both converging to H (T ot(C)). The diﬀerentials in the respective E0 -pages are induced from the maps in the double complex C. Example 6. Let us use the above theorem to prove the Snake Lemma, which says that the following commutative diagram with exact rows 0 / A0 / B0 / C0 / 0 O O O f g h

0 / A / B / C / 0 gives rise to the exact sequence

δ 0 / ker f / ker g / ker h / coker f / coker g / coker h / 0.

To do this, we ﬁrst use the second spectral sequence, with E0-page as follows. 0 0 0 AA0 0 BB0 0 CC0 0 0 0 0

By exactness, this spectral sequence converges to zero. Let us now look at the ﬁrst spectral sequence, with the following E0-page. 0 0 0 0 0 0 A0 B0 C0 0 0 ABC 0

The E1-page will look like 0 0 0 0 0 0 coker f coker g coker h 0 0 ker f ker g ker h 0

and the E2-page is as follow. 0 0 0 0 0 0 ker(coker f −→ coker g) 0 0 0 0 0 0 (ker h)/(ker g) 0

3 Since the spectral sequence converges to zero, we have an isomorphism

d : ker(coker f −→ coker g) −→(ker h)/(ker g).

Therefore δ := d−1 induces the connecting homomorphism in the Snake Lemma, while the other maps are the natural ones (which also corresponds to the diﬀer- entials in the E1-page above).

Fibrations The next spectral sequence we will look at is associated to a fibration, and is usually given as the first example of a spectral sequence. Theorem 7 (Serre spectral sequence, [3, Theorem 5.1]). Let F −→ E −→ B be a fibration with B simply connected, and let R be a ring. Then there is a first quadrant spectral sequence of algebras

p,q p q p+q E2 = H (B; H (F ; R)) ⇒ H (E; R). The left hand side is the cohomology of B with local coeﬃcients in the cohomology of F with respect to R. Furthermore, the multiplication structure on E2 is compatible with the cup product structure on cohomology.

Example 8. Let K(Z, n) be the Eilenberg-Maclane space with πn(K(Z, n)) = Z and πi(K(Z, n)) = 0 otherwise. We now show by induction that the cohomology ring ( ∗ Q[z], if n is even, H (K(Z, n), Q) = Q[z]/z2 if n is odd, where |z| = n. (This is still true if we replace Q by a ﬁeld of characteristic 0.) For n = 1 one has K(Z, 1) = S1, so this is clearly true. For n ≥ 2 we consider ΩK(Z, n) −→ PK(Z, n) −→ K(Z, n), where ΩK(Z, n) ' K(Z, n − 1) is the loop space and PK(Z, n) ' ∗ is the path space of K(Z, n). Note that π1(K(Z, n)) = 1, so by the Serre spectral sequence

p,q p q p+q E2 = H (K(Z, n); Q) ⊗Q H (K(Z, n − 1); Q) ⇒ H (∗, Q).

This implies that the E∞ page of the spectral sequence is Z at coordinate (0, 0) and zero everywhere. 0,n−1 Let us ﬁrst consider the case where n ≥ 2 is even. Then E2 is generated by some ﬁxed 2-torsion element z ∈ Hn−1(K(Z, n − 1)), and for the page to converge to 0 along this diagonal we require dn(z) = x with dn an isomorphism. n,n−1 Thus E2 is generated by xz, and by the Leibniz rule

n 2 dn(xz) = dn(x)z + (−1) xdn(z) = x ,

4 since dn(x) = 0. Inductively, we get the following diagram on the E2-page.

0 0 0 0 0 0 0 0 0 0 z 0 ... 0 xz 0 ... 0 x2z 0 0 0 ... 0 0 0 ... 0 0 0 ...... dn...... dn...... 0 0 ... 0 0 0 ... 0 0 0 1 0 ... 0 x 0 ... 0 x2 0

This gives us what we want for n ≥ 2 even. Next we consider the case where 0,k(n−1) n,(k−1)(n−1) n ≥ 2 is odd. Then E2 must be isomorphic to E2 for any positive integer n, and the rest of the E2 page is zero except at (0, 0). Similar 0,n−1 n−1 to before, E2 is generated by z ∈ H (K(Z, n − 1)), and dn(z) = x with 0,n−1 n,0 dn : E2 −→ E2 an isomorphism. Now

2 n−1 dn(z ) = dn(z)z + (−1) zdn(z) = 2xz,

n,n−1 and so E2 is generated by xz (since Q has characteristic zero).

0 0 ... 0 0 0 ... 1 2 2 2 z 0 ... 0 xz 0 ... 0 0 ... 0 0 0 ...... dn. . . . 0 0 0 0 0 0 ... z 0 ... 0 xz 0 ... 0 0 ... 0 0 0 ...... d ...... n . . . . 0 0 ... 0 0 0 ... 1 0 ... 0 x 0 ...

th Since the 2n column of the E2 page is zero, we conclude that x must be 2-torsion, as desired. Example 9. We show that

∗ n n+1 H (CP ) = Z[x]/(x ), |x| = 2. To do this, we use the Hopf ﬁbration

1 2n+1 n S −→ S −→ CP .

n Explcitly, the map S2n+1 −→ CP is deﬁned by the projection, where we view n n+1 both S2n+1 and CP as the usual subspace and subquotient of C .

5 n Since CP is simply connected, the Serre spectral sequence in this case is

p,q p n q 1 p+q 2n+1 E2 = H (CP ,H (S )) ⇒ H (S ).

The E2-page has nontrivial terms only on the ﬁrst two rows.

0 0 0 0 ... n n n n H0(CP ) H1(CP ) H2(CP ) H3(CP ) ... n n n n H0(CP ) H1(CP ) H2(CP ) H3(CP ) ...

Since the spectral sequence converges after one page to Hp+q(S2n+1), which equals Z when p + q ∈ {0, 2n + 1} and is 0 otherwise, it is easily seen that ( i n Z for i = 0, 2, 4,..., 2n, H (CP ) =∼ 0 otherwise.

As the cup product structure is compatible with the product on E2, the cohomology ring is as claimed.

Groups We now look at the spectral sequence associated to group cohomology. One can either view this purely algebraically, or relate this to topology via the isomorphism Hn(G, A) =∼ Hn(BG, A), where BG is the classifying space of G, and A is a trivial G-module (see [7, Theorem 6.10.5]). For example,

n ⊕n n n H (Z , Z) =∼ H (T ), where Tn is the n-torus. We assume the language of group cohomology; a reference is [2, Chapter 4]. Theorem 10 (Lyndon-Hochschild-Serre spectral sequence, [7, Theorem 6.8.2]). Let 0 −→ N −→ G −→ Q −→ 0 be a short exact sequence of groups, with N normal, and let M be a ring that is also a G-module. Then there is a ﬁrst quadrant spectral sequence of algebras

p,q p q p+q E2 = H (Q, H (N,M)) ⇒ H (G, M). The left hand side is the cohomology of B with local coeﬃcients in the cohomology of F with respect to R. The Lyndon-Hochschild-Serre spectral sequence can be viewed as a special case of the Grothendieck spectral sequence, which states the following: if F : A −→B and G : B −→C are two additive left-exact functors between abelian

6 categories such that F takes acyclic objects to acyclic objects and B has enough injectives, then there is a spectral sequence

p,q p q p+q E2 = (R G ◦ R F )(A) ⇒ R (G ◦ F )(A) for every object A of A. To see this, we view H∗(G, −) as the the derived functor of (−)G, and note that the composition of (−)N and (−)G/N is (−)G. Remark. If we identify Q with G/N, then the associated ﬁve-term exact sequence is the inﬂation-restriction sequence

0 −→ H1(G/N, M N ) −→ H1(G, M) −→ H1(N,M)G/N −→ H2(G/N, M N ) −→ H2(G, M).

0,1 1 G/N 0,1 2 N 0,2 In fact, the diﬀerential d2 from H (N,M) = E2 to H (G/N, M ) = E2 is the transgression map, which we recall here. For the short exact sequence of groups in Theorem 10 one gets an extension

0 −→ N ab −→ G/[N,N] −→ G/N −→ 0, where we identify Q with G/N. Let ξ ∈ H2(G/N, N ab) be the element corresponding to this extension. Now suppose f ∈ H1(N,M)G/N . Then f ∈ Hom(N,M) = Hom(N ab,M) and f is G/N-invariant, so we have an induced homomorphism f ∗ : H2(G/N, N ab) −→ H2(G/N, M). 0,1 0,1 ∗ The map d2 is then deﬁned to be d2 (f) = f (ξ). Example 11. Consider the integral Heisenberg group     1 a c  G = 0 1 b : a, b, c ∈ Z .  0 0 1 

We wish to compute H∗(G, A), where A is a trivial G-module. Note that the Heisenberg group is a semidirect product

G = N o K =∼ (Z ⊕ Z) o Z where        1 0 c   1 a 0  N = 0 1 b : b, c ∈ Z ,K = 0 1 0 : a ∈ Z  0 0 1   0 0 1  and the structure map ϕ : Z −→ Aut(Z ⊕ Z) is given by k 1 0 ϕ(k) = . 1 1 Thus we have a split extension

0 −→ Z −→ G −→ Z ⊕ Z −→ 0.

7 The associated E2-page looks like

0 0 0 0 AA ⊕ AA 0 AA ⊕ AA 0

0,1 0,1 with one nontrivial diﬀerential d2 : A −→ A. Let us show that d2 = 1. By deﬁnition (see above remark)

0,1 ∗ d2 (f) = f (ξ) where ξ ∈ H2(Z ⊕ Z, Z) is the class corresponding to the extension above. We 0,1 have seen that the extension is split, so ξ = 0 and d2 is the identity. This implies that E3 looks like the diagram below, and the spectral sequence degen- erates after one step.

0 0 0 0 0 A ⊕ AA 0 AA ⊕ A 0 0

By Proposition3 one sees that  A if n = 0, 3,  Hn(G, A) = A ⊕ A if n = 1, 2, 0 otherwise.

3 A sketch of the lifting problem

As we mentioned in the introduction, the lifting problem is the problem where the total cohomology of the E∞-page of a spectral sequence is not equal to the cohomology of the object one is trying to compute. This is because the terms defining the spectral sequence are quotients of predetermined filtrations of cohomology groups, and in general a group is not isomorphic to the direct sum of its filtration.

Example 12. Let Cn be the cyclic group of order n. It is well-known (see Chapter 4.8 of [2]) that

∗ H (Cn,Z) = Z[z]/(nz) where |z| = 2. We now demonstrate the lifting problem via the short exact sequence 0 −→ C2 −→ C4 −→ C2 −→ 0.,

8 which is a central extension. Using the Lyndon-Hoschschild-Serre spectral sequence the corner of the E2-page is

y2 0 y2x 0 y2x2 0 y2x3 0 0 0 0 0 0 0 y 0 yx 0 yx2 0 yx3 0 0 0 0 0 0 0 1 0 x 0 x2 0 x3

It is clear that E2 = E∞, and T ot(E∞) = Z[x, y]/(2x, 2y). However, this is not the cohomology of C4. Although the spectral sequence may not tell us much about the cohomology groups, a little more can be said if we assume ﬁnite generation.

Theorem 13 ([1, Theorem 2.1]). If T ot(E∞) is a finitely generated bigraded commutative algebra, then so is the cohomology ring H∗ that we are trying to compute. Moreover, if T ot(E∞) is presented by (double) generators and homogeneous relations, then the generators and relations of H∗ are lifts of those of T ot(E∞), and they have the same total degree. If we work over finite fields, the above theorem tells us that knowledge ∗ of T ot(E∞) gives us only finitely many possibilities for H . See [5] for more examples of the lifting problem.

9 4 Application: homotopy groups of spheres1

n Our goal now is to prove that πi(S ) is finite if n is odd and i 6= n. The idea is to be more flexible and only work with groups modulo an insignificant n “Serre class” of groups. For example, we’ll see that πi(S ) looks like πi(K(Z, n)) n modulo their torsion. Since πi(K(Z, n)) = 0 for i 6= n, we deduce that πi(S ) is a torsion group whenever i 6= n. Definition 14. A Serre class of abelian groups is a nonempty collection C of abelian groups satisfying: 1. If 0 → A → B → C → 0 is a short exact sequence, then B ∈ C if and only if A, C ∈ C. The Serre class may also satisfy the following optional axioms:

2A. If A, B ∈ C, then A ⊗ B ∈ C and Tori(A, B) ∈ C for all i ≥ 1. 2B. If A ∈ C, then A ⊗ B ∈ C for any abelian group B.2

3. If A ∈ C, then Hi(A; Z) := Hi(K(A, 1); Z) ∈ C for all i ≥ 1. Example 15. The following Serre classes satisfy axioms 1, 2A, 2B, and 3:

(a) The trivial class Ctriv of abelian groups.

(b) The class Ctors of torsion abelian groups. Axiom 3 holds by either by arguing as in p271 of [4], or by a group cohomology argument: write the torsion group A as a direct limit of its n-torsion subgroups, and further write each n-torsion subgroup An as a direct limit of its ﬁnite subgroups Anm. Since homology commutes with direct limits,

H (A; ) ∼ lim H (A ; ), i Z = −→ i nm Z n,m

and since the homology of a ﬁnite group is torsion whenever i ≥ 1, we deduce that Hi(A; Z) is also torsion. The following Serre classes only satisfy axioms 1, 2A, and 3:3

(a) The class Cfinite of ﬁnite abelian groups. The second part of Axiom 2A Z follows from the additivity of Tori (A, B) in the ﬁrst argument and the computation ( B[n] if i = 1 Z Tori (Z/nZ,B) = 0 if i > 1 where B[n] is the n-torsion part of B. Axiom 3 holds by:

1This section blatantly plagiarizes §10.1-2 of Davis and Kirk [4]. 2 Axioms 1 and 2B together imply Axiom 2A since Tori(A, B) is computed by taking a free resolution of A, tensoring by B, and taking the resulting homology. 3 Axiom 2B fails in the next two examples since the tensor product of the finite group Z/2Z by the direct sum ⊕NZ/2Z is neither finite nor finitely generated.

10 • K(A × B, 1) =∼ K(A, 1) × K(B, 1), • The K¨unneththeorem and axiom 2A, • K(Z, 1) = S1, • The spectral sequence computation ([4] theorem 9.30)  if i = 0 Z Hi(Z/nZ; Z) = Z/nZ if i = odd . 0 if i > 0 is even

(b) The class CFG of finitely generated abelian groups. The axioms are verified as in the previous example. Definition 16. Given a Serre class C, a homomorphism ϕ : A → B of abelian groups is a C-isomorphism if both ker ϕ and coker ϕ are in C. C-isomorphism

Deﬁnition 17. Two abelian groups A and B are C-isomorphic if there exists C-isomorphic an abelian group C and two C-isomorphisms f : C → A and g : C → B,4 and we denote this by A =∼ B mod C.5 Example 18. For two abelian groups A and B, ∼ ∼ • A = B mod Ctriv if and only if A = B; ∼ ∼ 6 • A = B mod Ctors if and only if A ⊗ Q = B ⊗ Q. Theorem 19.

(1) (mod C Hurewicz theorem) Let X be 1-connected, and suppose C satisﬁes axioms 1, 2A, and 3.

(a) If πiX ∈ C for all i < n, then HiX ∈ C for all 0 < i < n and the Hurewicz map πnX → HnX is a C-isomorphism.

(b) If HiX ∈ C for all 0 < i < n, then πiX ∈ C for all i < n and the Hurewicz map πnX → HnX is a C-isomorphism.

C 4Such diagram f g is called a “roof”. Looking at the category of abelian

A B groups modulo these roofs is an example of localizing a category. 5This is indeed an equivalence relation. 6In other words, A and B are the same up to torsion (in the new sense) if and only if A and B are the same up to torsion (in the familiar sense). This example illustrates that roofs are Z ∼ id needed for isomorphisms to be reﬂexive: Z = Q mod Ctors since forms a Z Q roof. However there is no Ctors-isomorphism, let alone nonzero group homomorphism, Q → Z.

11 (2) (mod C relative Hurewicz theorem) Let A ⊂ X, A and X be 1-connected, and π2(X,A) = 0. Suppose C satisﬁes axioms 1, 2B, and 3.

(a) If πi(X,A) ∈ C for all i < n, then Hi(X,A) ∈ C for all i < n and the Hurewicz map πn(X,A) → Hn(X,A) is a C-isomorphism.

(b) If Hi(X,A) ∈ C for all i < n, then πi(X,A) ∈ C for all i < n and the Hurewicz map πn(X,A) → Hn(X,A) is a C-isomorphism. (3) (mod C Whitehead theorem) Let f : A → X be a map between 1-connected spaces, and π2(f) an epimorphism. Let C satisfy axioms 1, 2B, and 3. Then the following are equivalent:

(a) πi(f) is a C-isomorphism for i < n and a C-epimorphism for i = n.

(b) Hi(f) is a C-isomorphism for i < n and a C-epimorphism for i = n.

n Application. The homotopy group πi(S ) is ﬁnite if n is odd and i 6= n.

1 Proof. For n = 1, covering space theory tells us that πi(S ) = πi(R) = 0 for i > 1. Now let n > 1 be odd. The sphere Sn is a ﬁnite CW complex so n n n HiS ∈ CFG for all i > 0. Since S is 1-connected, by (1)(b), πi(S ) ∈ CFG for all i ≥ 0. n n We now show that πiS is a torsion group for i 6= n. Let f : S → K(Z, n) be the map inducing an isomorphism on πn. Then πi(f) is an isomorphism for i ≤ n and an epimorphism for i = n + 1. Applying (3) with the class Ctriv, we see that n Hi(f): Hi(S ; Z) → Hi(K(Z, n); Z) is an isomorphism for i ≤ n and an epimorphism for i = n + 1.The homologies of Sn and K(Z, n) with coeﬃcients in Q vanish in degrees > n,7 so

n Hi(f): Hi(S ; Q) → Hi(K(Z, n); Q) is an isomorphism for all i. By example 18,

n Hi(f): Hi(S ; Z) → Hi(K(Z, n); Z) is a Ctors-isomorphism for all i. Applying (3) with the class Ctors, we deduce that n πi(f): πi(S ) → πi(K(Z, n)) n is a Ctors-isomorphism for all i. Since πi(K(Z, n)) = 0 for i 6= n, πi(S ) ∈ Ctors for i 6= n. n We conclude that πiS is a ﬁnite group for i 6= n.

7 Take the Poincar´edual of our result in example8 to ﬁnd Hi(K(Z, n); Q).

12 5 Proof of theorem 19

In theorem 19, parts (1)(b) and (2)(b) follow from (1)(a) and (2)(a), respectively, and part (3) follows from part (2). It then remains to prove things in the following order: theorem 19(1)(a) and a lemma 20 needed in the proof of 19(1)(a). We refer the reader to [4] p277 for how theorem 19(2)(a) follows from theorem 19(1)(a).

5.1 Proof of 19(1)(a) Strategy. To show that the Hurewicz map is an isomorphism for a space X, f we consider the path ﬁbration ΩX → PX → X and the diagram

f π (X, x ) ∗ π (PX, ΩX) ∂ π (ΩX) n 0 ∼= n ∼= n−1 ρ ρ ρ . (1) f H (X, x ) ∗ H (PX, ΩX) ∂ H (ΩX) n 0 n ∼= n−1

To show the leftmost ρ is a C-isomorphism, it suﬃces to show the lower f∗ and the rightmost ρ are C-isomorphisms, since the ∂ and the top f∗ are already isomorphisms.8 To show the lower f∗ is a C-isomorphism, we need the following spectral sequence lemma, whose proof is postponed to the next subsection:

f Lemma 20. Let F → E → B be a ﬁbration, B be simply-connected, and C a Serre class satisfying axioms 1 and 2A. Suppose further that HpB ∈ C for 0 < p < n and HpF ∈ C for 0 < q < n − 1. Then

f∗ : Hi(E,F ) → Hi(B, b0) is a C-isomorphism for i < n. Actual start of proof. To prove 19(1)(a), we induct on n. For n = 1, we have π1X = 0 = H1X. For n = 2, π1X = 0 = H1X and so lemma 20 tells us the bottom f∗ of diagram1 is an isomorphism. Also π1(ΩX) = π2(X)

8 ∂ is an isomorphism because PX is contractible, and the top f∗ is an isomorphism by applying the ﬁve lemma to the commutative diagram

··· πn(ΩX) πn(PX) πn(PX, ΩX) ···

f∗ . ··· πn(ΩX) πn(PX) πn(X) ···

13 9 is abelian, so by the π1-version of the Hurewicz theorem , the rightmost ρ of diagram1 is an isomorphism. Therefore the leftmost ρ is an isomorphism. Now suppose n > 1 and inductively assume that whenever a simply-connected space Y satisfies πiY ∈ C for i < n − 1, then HiY ∈ C for 0 < i < n − 1 and the Hurewicz map ρ : πn−1Y → Hn−1Y is a C-isomorphism. Let X be a simply-connected space satisfying πiX ∈ C for i < n. Case 1: π2X = 0. Then πi(ΩX) = πi+1X so we apply the inductive hypothesis to ΩX and conclude the rightmost ρ of diagram1 is a C-isomorphism. Lemma 20 shows the bottom f∗ is a C-isomorphism. Case 2: π2X 6= 0. We reduce to case 1 by replacing X by an X2 constructed as follows: after replacing X by a CW complex, let f : X → K(π2X, 2) be the 10 map inducing identity on π2. Let X2 be its homotopy fiber, giving the fibration

K(π2, 1) = ΩK(π2X, 2) → X2 → X.

Then X2 satisﬁes: ∼ 11 (a) HiX2 = HiX mod C for all 0 < i ≤ n. ∼ 12 (b) πi(X2) = πi(X) for all i > 2. 13 (c) πi(X2) = 0 for i = 1, 2.

So X2 satisﬁes the same inductive hypotheses as X except now X2 belongs to case 1, so the Hurewicz map for X2 at level n is an isomorphism. Therefore, the same holds for X.14

9 Note: the only inputs needed for this theorem is the π1-version of the Hurewicz theorem and the Serre spectral sequence 10SE39735 11 We assumed π2X ∈ C, so by axiom 3, we deduce Hi(K(π2X, 1)) ∈ C for all i > 0. Therefore:

• Hi(X2) → Hi(X2,K(π2X, 1)) is a C-isomorphism for all i > 0.

• Together with lemma 20, Hi(X2,K(π2X, 1)) → Hi(X, x0) is a C-isomorphism for 0 < i ≤ n. (Typo in [4] on p277 says 0 < i < n.)

The composite Hi(X2) → Hi(X, x0) is then a C-isomorphism for 0 < i ≤ n. 12This follows from the long exact sequence for the ﬁbration. 13 As in (b), note that f induces an isomorphism on π2. 14 Formally, we would draw the diagram and observe the middle ρ is an isomorphism for i > 2: ∼= ··· πi(X2) πi(X) πi(K(π2X, 2)) ···

∼= ρ ρ ρ . ∼= ··· Hi(X2) Hi(X) Hi(K(π2X, 2)) ···

14 5.2 Proof of lemma 20

The spectral sequence for a relative ﬁbration F → (E,F ) → (B, b0) states that

2 ∼ Ep,q = Hp(B, b0; HqF ) ⇒ H•(E,F ).

2 15 ∞ In the range p = 0, 1 or (p, q) ∈ (0, n) × (0, n − 1), Ep,q ∈ C so Ep,q ∈ C as well.16 ∞ Now Hi(E) is ﬁltered by the E terms on the p + q = i diagonal. For ∞ i ≤ n, entries of this diagonal lie in C except possibly Ei,0. This gives a map ∞ Hi(E,F ) Ei,0 that is a C-isomorphism. ∞ 2 k+1 k Also, Ei,0 ⊂ Ei,0 is a C-isomorphism: Ei,0 is the homology at entry Ei,0 where the arrows are slanted so that (for i ≤ n) the kernel and cokernel either lie in the shaded region in the ﬁrst quadrant (hence take values in C), or lie in the other quadrants (so zero). By Theorem 9.12 of [4], we can identify the composite of these two maps with f∗ so f∗ is a C-isomorphism as well.

Figure 1: From [4]

15 2 Since in the latter range for (p, q), Ep,q = (Hp(B, b0) ⊗ HqF ) ⊕ Tor(Hp(B, b0),HqF ) has 2 all parts lying in C by hypothesis. In the range p = 0, 1, the Ep,q terms are zero since B is 1-connected. 16 i i+1 i Suppose that Ep,q ∈ C. We get Ep,q by taking homology at entry Ep,q, so by axiom 1, i+1 Ep,q ∈ C.

15 References

[1] Jon Carlson. Coclass and cohomology. J. Pure Appl. Algebra 200 (2005), 251–266. [2] John Cassels and Albrecht Frohlich. Algebraic Number Theory. Thompson Book Company, 1967. [3] John McCleary. A User’s Guide to Spectral Sequences (2nd Edition). Cam- bridge University Press, 2000. [4] James Davis and Paul Kirk. Lecture Notes in Algebraic Topology. American Mathematical Society, 2001. [5] Antonio Diaz Ramos. Spectral Sequences via Examples. Graduate J. Math. 1 (2017), 10–28. [6] Ravi Vakil. The Rising Sea: Foundations of Algebraic Geometry. [7] Charles Weibel. Introduction to Homological Algebra. Cambridge University Press, 1995.