Spectral Sequences, Complex-Oriented Cohomology Theories, and Formal Group Laws

Christian Carrick

These are notes and background for a talk I gave in the Chromatic Homotopy Theory seminar at UCLA on October 20th, 2017.

1 SPECTRAL SEQUENCES

1.1 INTRODUCTION One of the motivating principles for singular homology is that it can be computed combinatorially given a decomposition of a space. This principle takes several forms - for example, if you have a ∆- complex, you can compute with ∆-homology using only the data of the simplices and their boundaries, or if you have a CW complex, you can compute with cellular homology using only the data of the cells and the degrees of the attaching maps. The guiding principal in cellular homology, is that a CW complex has a filtration via its skeleta, and the successive quotients of the filtration are wedges of spheres, which we know the homology of. In fact, the result is that we have the cellular filtration

∅ ⊂ X0 ⊂ X1 ⊂ · · · ⊂ X

p p−1 p p−1 of a CW complex X, and the data of the groups H∗(X , X ) along with the maps H∗(X , X ) → p−1 p−1 p−2 H∗−1(X ) → H∗−1(X , X ) recovers the data H∗(X), where the ﬁrst map comes from the long exact sequence of the pair (Xp, Xp−1) and the second from that of (Xp−1, Xp−2). The idea of the proof is that we have a long exact sequence in homology for each of the pairs (Xp, Xp−1), and by stringing p ∼ them together in the above way and using the fact that H∗(X ) = H∗(X) for ∗ < p, we can compute H∗(X). Another way to say that these sequences can be strung together is to say that if we take the direct sum of all the maps in these long exact sequences, they ﬁt into a single triangle

L p L p Hn(X ) Hn(X ) n,p n,p

k j L p p−1 Hn(X , X ) n,p

and the result is that H∗(X) can be computed as ker(j ◦ k)/im(j ◦ k). We can ask more generally if we have a ﬁltered chain complex C∗, say

0 1 C∗ ⊂ C∗ ⊂ · · · ⊂ C∗

can we compute the homology groups of C∗ with the knowledge of the homology of the quotient p p−1 chain complexes C∗ /C∗ . It turns out that it is not always as nice as the above example, but p p−1 in most cases, we can compute the quotients H∗(C∗ )/H∗(C∗ ). In other words, from the data p p−1 p p−1 H∗(C∗ /C∗ ), we can compute H∗(C∗ )/H∗(C∗ ). If we therefore could identify which extension

1 p p p−1 p−1 p H∗(C∗ ) of H∗(C∗ )/H∗(C∗ ) by H∗(C∗ ) was, that is, if we could compute H∗(C∗ ) from the knowl- p p−1 p−1 edge of H∗(C∗ )/H∗(C∗ ), H∗(C∗ ) and the short exact sequence

p−1 p p p−1 0 → H∗(C∗ ) → H∗(C∗ ) → H∗(C∗ )/H∗(C∗ ) → 0

p p p−1 we could inductively compute H∗(C∗ ) and therefore H∗(C∗). We therefore say that the data H∗(C∗ , C∗ ) determines H∗(C∗) modulo extension problems. We generalize our triangle above in the following way:

Deﬁnition 1.1. An exact couple is a diagram of abelian groups

D1 i D1

k j E1 that is exact at each node.

By analogy with the exact couple used for cellular homology, we know that since the diagram is exact at each node, (E1, j ◦ k) forms a chain complex, and in fact we see something stronger:

Proposition 1.2. For an exact couple, if we set D2 = im(i) and E2 = ker(j ◦ k)/im(j ◦ k), then

i|im(i) D2 D2

k [j◦i−1] E2 is an exact couple.

Proof : (j ◦ k)2 = 0 because k ◦ j = 0 by exactness. By [j ◦ i−1] we mean taking a preimage under i, applying j and then taking homology. By k we mean that the restriction of the map k : E1 → D1 to ker(j ◦ k) descends to the quotient ker(j ◦ k)/im(j ◦ k) and lands in im(i) - this is clear from the fact that im(j ◦ k) ⊂ ker(k) since im(j) = ker(k), and from the fact that im(i) = ker(j) and k(ker(j ◦ k)) ⊂ ker(j). i is clearly well-deﬁned, we checked that k is, and [j ◦ i−1] is well deﬁned because if x0, x1 are two preimages of z then j(x0) − j(x1) = j(x0 − x1) ∈ im(j ◦ k) =⇒ [j(x0)] = [j(x1)], since x0 − x1 ∈ ker(i) = im(k). The only thing left to check is that the diagram is exact at each node. We leave this as an exercise.

1.2 SPECTRAL SEQUENCES Given an exact couple, we saw that we could iteratively form new exact couples by taking homology with respect to the maps j ◦ k, and we get a sequence of groups E1, E2,... with each being the homology of the previous. We therefore deﬁne:

Deﬁnition 1.3. A homological bigraded spectral sequence in the category of R-modules is a r sequence of bigraded R-modules Ep,q, where r ≥ 1 and p, q ∈ Z, such that there are diﬀerentials r r d r Ep,q −→ Ep−r,q+r−1 that make

r r M r d M r d M r · · · → Ei,j −→ Ei,j −→ Ei,j → · · · i+j=n+1 i+j=n i+j=n−1

into a chain complex over R-mod, and the homology of this chain complex is Er+1, i.e.

dr ker(Er −→ Er ) p,q p−r,q+r−1 r+1 r = Ep,q r d r im(Ep+r,q−r+1 −→ Ep,q)

2 r A spectral sequence is said to be bounded if only ﬁnitely many of the R-modules Ep,q are nonzero. r r We call the bigraded R-module Ep,q the E page of the spectral sequence - the analogy being that we r can think of each Ep,q as a grid of R-modules.

r Definition 1.4. (Convergence): A homological bigraded spectral sequence {E , dr} is said to converge r r(p,q) if for all p, q, there exists r(p, q) < ∞ such that Ep,q = Ep,q for all r ≥ r(p, q). It is said to converge to a filtered graded R-module C∗ if, denoting the filtration on C∗ by −1 0 1 · · · ⊂ C∗ ⊂ C∗ ⊂ C∗ ⊂ · · · r(p,q) ∼ p p−1 ∞ r(p,q) we have Ep,q = Cp+q/Cp+q for all p, q. We define Ep,q := Ep,q . The notation r Ep,q =⇒ Cp+q

means the spectral sequence E converges to C∗ in the above sense.

r(p,q) Remark 1.5. If we can choose a single r(p, q) for all p, q, we say that the bigraded R-module Ep,q = ∞ ∞ r r Ep,q is the E -page of the spectral sequence. For instance, when {E , d } is bounded, we take r = ∞ r r max{r(p, q)}, and E = E . In this case, when Ep,q =⇒ Cp+q we have an isomorphism of bigraded ∼ ∞ R-modules grC∗ = E∗,∗, where grC∗ is the associated graded R-module of C∗ with respect to the given ﬁltration - which gets one grading from the ﬁltration and the other from C∗. ∼ ∞ We also remark that in practice, the isomorphism grC∗ = E∗,∗ is not some abstract isomorphism, but comes from maps built into the construction of the spectral sequence. As we will see below, in the case of a spectral sequence coming from an exact couple, this isomorphism is usually canonical.

Remark 1.6. Proposition 1.2 tells us that given an exact couple, we can iteratively form new exact couples from it by taking homology with respect to the diﬀerential j ◦ k. In fact, we have shown that 1 1 1 1 if D , E are bigraded R-modules, and i : Dp−1,q+1 → Dp,q (i.e. we say i has bidegree (1, −1)), j has bidegree (0, 0), and k has bidegree (−1, 0)), then (Er, dr) is a homological bigraded spectral sequence of R-modules, where dr = jr ◦ kr, where j and k are the corresponding maps in the r-th exact couple.

Remark 1.7. There are of course more general notions of a spectral sequence than the one deﬁned here, but since we are using ﬁltered chain complexes as the motivating example, we stay near this context.

Computing the differentials in spectral sequences can be very hard - indeed it is common to work only with spectral sequences which have no nonzero differentials after the E2 page to avoid this problem. However, we can describe the differentials coming from an exact couple explicitly after taking representatives in the E1 page. This doesn’t necessarily make computing differentials any easier, but it does shed light on the convergence picture. Note that by definition of homology, Er is always a subquotient of E1. Recall that we defined d2([x]) for x ∈ E2 by [j ◦ i−1](k([x])) = [j ◦ i−1](k(x)) = [j(i−1(k(x)))], using that E2 is a subquotient of E1, and so the class [x] ∈ E2 has a (usually not unique) representative x ∈ E1. In terms of the exact couple, this says that we compute d2([x]) by taking a representative x ∈ E1 of [x], and moving right to left in the following diagram

D1 i D1

j k E1 E1 and then taking the equivalence class of the result in the quotient forming E2. In fact, since the spectral sequence is constructed iteratively from the exact couples, if [x] ∈ Er, and x ∈ E1 is a representative for Er, a subquotient of E1, we have dr([x]) = [j(i−(r−1)(k(x)))], or we move right to left in the diagram

D1 i ··· i D1

j k E1 E1

3 where the map i appears r − 1 times in the top row, and then we take the equivalence class in Er. We prove this statement in the following form.

Proposition 1.8. Let (Er, dr) be the spectral sequence as above from the exact couple D1 → D1 → E1 → D1. We set Zr := ker(dr), Br := im(dr), then by deﬁnition Er+1 = Zr/Br. Let Z˜ r = k−1(im(ir)), B˜r = j(ker(ir)), then Z˜ r/B˜r =∼ Zr/Br = Er+1.

Proof : We have that

x ∈ ker(j ◦ k) ⇐⇒ k(x) ∈ ker(j) ⇐⇒ k(x) ∈ im(i) by exactness, and therefore Z1 = k−1(i(D)) = Z˜ r. Similarly

x ∈ im(j ◦ k) ⇐⇒ x = j(k(y)) ⇐⇒ x = j(y) for y ∈ ker(i) and therefore B1 = j(ker(i)) = B˜1. Since we only used the fact that we had an exact couple to compute this, we can repeat it using our derived exact couple, and we have

−1 Z2 = k (i2(D)) = [k−1(im(i2)) ∩ ker(j ◦ k)] = [k−1(im(i2))]

2 −1 −1 2 B = [j ◦ i ](ker(i|im(i))) = [j(i (ker(i|im(i))))] = [j(ker(i ))] where ker(j ◦ k) [−] : ker(j ◦ k) im(j ◦ k) is the quotient map. We have the ﬁltration

im(j ◦ k) ⊂ j(ker(i2)) ⊂ k−1(im(i2)) ⊂ ker(j ◦ k)

and therefore k−1(im(i2))/im(j ◦ k) k−1(im(i2)) E3 = Z2/B2 = =∼ j(ker(i2))/im(j ◦ k) j(ker(i2)) Since each exact couple is derived from the previous in exactly the same way, we can repeat this ar- −1 r gument inductively, and we see that r+1 = r r =∼ k (im(i )) , and r and r are the image of this E Z /B = j(ker(ir)) Z B numerator and denominator respectively under a quotient map. We therefore deﬁne Z˜ r = k−1(im(ir)), and B˜r = j(ker(ir)). We thus can compute Er for all r directly from the ﬁrst exact couple by forming a single subquotient.

We will see below that this description of Er allows us to say that a wide variety of spectral sequences converge.

1.3 FILTERED CHAIN COMPLEXES

Definition 1.9. A filtered chain complex C∗ is a filtered graded abelian group as above such that restricting the differential to each piece of the filtration gives a chain complex. In other words C∗ is a filtered object in the category of chain complexes.

Proposition 1.10. A ﬁltered chain complex determines an exact couple and thus a spectral sequence.

1 L p 1 L p p−1 Proof : Set D = Hp+q(C∗ ), E = Hp+q(C∗ /C∗ ), then since a short exact sequence of chain p,q p,q complexes determines a long exact sequence in homology, these form an exact couple.

4 Proposition 1.11. Let C∗ be a ﬁltered chain complex with bounded ﬁltration, i.e.:

0 1 n C∗ ⊂ C∗ ⊂ · · · ⊂ C∗ = C∗

p p−1 We have a spectral sequence Hp+q(C∗ /C∗ ) =⇒ Hp+q(C∗), where Hp+q(C∗) is ﬁltered by

p im(Hp+q(C∗ ) → Hp+q(C∗))

Proof : First note that since the filtration is bounded, the spectral sequence collapses at the En+1 page. By this we mean that all the differentials on the En+1 page are zero, and hence En+1 = En+2 = ··· and therefore En+1 = E∞. This is because the differentials dn+1 are of the form

n+1 n+1 → n+1 d : Ep,q Ep−(n+1),q+r−1 and since the ﬁltration has length n, either the domain or the codomain of this map is zero. Now we have a morphism of short exact sequences

p−1 p−1 p−1 0 ker(Hn(C∗ ) → Hn(C∗)) Hn(C∗ ) im(Hn(C∗ ) → Hn(C∗)) 0

p p p 0 ker(Hn(C∗ ) → Hn(C∗)) Hn(C∗ ) im(Hn(C∗ ) → Hn(C∗)) 0

By the snake lemma and the fact that the righthand vertical map is an inclusion, we therefore have a short exact sequence

p p p ker(H (C ) → H (C )) H (C ) im(H (C ) → H (C )) → n ∗ n ∗ → n ∗ → n ∗ n ∗ → 0 p−1 p−1 p−1 0 ker(Hn(C∗ ) → Hn(C∗)) Hn(C∗ ) im(Hn(C∗ ) → Hn(C∗)) Using exactness (in the exact couple) and the first isomorphism theorem, we can canonically identify the p p lefthand term with j(ker(Hn(C∗ ) → Hn(C∗))) and the middle term with j(Hn(C∗ )), and the righthand p S r term is the filtration quotient on Hn(C∗). But ker(Hn(C∗ ) → Hn(C∗)) = ker(i ) since Hn(C∗) = r S p S r n+1 p Hn(C∗ ), hence the lefthand term is j( ker(i )) = B˜ . Similarly, j(Hn(C∗ )) = im(j) = ker(k), but r since our filtration is bounded below, Z˜ n+1 = k−1(T im(ir)) = k−1(0) = ker(k), hence our short exact r sequence yields ˜ n+1 p Zp,q im(Hp+q(C∗ ) → Hp+q(C∗)) Ep,q =∼ =∼ = grp,q(H (C )) ∞ ˜ n+1 p−1 ∗ ∗ Bp,q im(Hn(C∗ ) → Hn(C∗))

Our hypothesis that the filtration was finite is rather restrictive. Indeed, in the cellular homology case for instance, we need to deal with infinite CW complexes. Before we can relax this hypothesis, however, we need to define a weaker notion of convergence that can support unbounded spectral sequences with limiting behavior.

∞ T −1 r r ∞ S r r Definition 1.12. Define Z˜ = k (im(i )) = lim Z˜ , and B˜ = j(ker(i )) = colimr B˜ . Then r ←−r r ˜ ∞ ˜ ∞ ∼ we say the spectral sequence weakly converges to a filtered graded abelian group H∗ if Zp,q/Bp,q = Fp Hp+q/Fp−1 Hp+q. Note that in practice, most of our spectral sequences will just collapse on an early page, and then since Z˜ r/B˜r stabilizes in r iff Zr/Br does, this definition of convergence will look the same as the one for a bounded spectral sequence.

Theorem 1.13. Let D1 → D1 → E1 be a bigraded exact couple with the right bidegrees so that we can associate a homological spectral sequence. Let 1 1 1 Hn := colim · · · → D0,n → D1,n−1 → D2,n−2 ···

5 1 r r Filter Hn by Fp Hn = im(Dp,n−p → colimk(Dk,n−k)). Then {E , d } weakly converges to H∗ if and only if j(D) = k−1(0).

Proof : The argument is almost exactly as before. Fix n = p + q, and we want to identify the quotients 1 1 in the ﬁltration of Hn, recall Fp Hn = im(Dp,n−p → colimk(Dk,n−k)), hence we ﬁrst look at ker(Dp,n−p → 1 r colimk(Dk,n−k)). We have x ∈ ker(Dp,n−p → colimk(Dk,n−k)) =: Kp,n−p ⇐⇒ i (x) = 0 for some r, S r hence Kp,n−p = ker(i ). We thus have a morphism of SES’s r

1 0 Kp−1,n+1−p Dp−1,n+1−p Fp−1 Hn 0

1 0 Kp,n−p Dp,n−p Fp Hn 0

The rightmost downward arrow is of course an inclusion (it’s a ﬁltration), so by the snake lemma we have a short exact sequence of cokernels of the downward maps, which by exactness arguments is

1 Kp,n−p Dp,n−p Fp Hn 0 → → → → 0 ker(j|Kp,n−p ) ker(j) Fp−1 Hn which by ﬁrst iso theorem is

1 Fp Hn 0 → j(Kp,n−p) → j(Dp,n−p) → → 0 Fp−1 Hn S r S r ˜ ∞ We have j(Kp,n−p) = j ker(i ) = j(ker(i )) = Bp,n−p, and r r

F H 0 B˜ ∞ j(D1 ) p n 0 p,q p,n−p Fp−1 Hn

= =

F H 0 B˜ ∞ Z˜ ∞ p n 0 p,q p,q Fp−1 Hn

−1 T −1 r ∞ The spectral sequence therefore converges to H∗ iﬀ j(D) = k (0) is equal to k (im(i )) = Z˜ . r

Corollary 1.14. If C∗ is a filtered chain complex with filtration bounded below, and the filtration is p p p−1 exhaustive in the sense that C∗ = colimp C∗ , then there is a spectral sequence Hp+q(C∗ , C∗ ) =⇒ Hp+q(C∗), where here we mean weak convergence.

− Proof : The condition j(D) = k 1(0) follows immediately from the ﬁltration being bounded below.

Corollary 1.15. Cellular homology coincides with singular homology.

Proof : Let X be CW complex with a skeletal ﬁltration

∅ ⊂ X0 ⊂ X1 ⊂ · · · ⊂ X

i.e. Xi is the i-skeleton of X. Then we ﬁlter the singular chain complex C∗(X) by setting FpC∗(X) = C∗(Xp). Then we get an exact couple

L L Hp+q(Xp) Hp+q(Xp) p,q p,q

L Hp+q(Xp, Xp−1) p,q

6 and we recall that ( cell 1 Cp (X) q = 0 Ep q = Hp+q(Xp, Xp−1) = , 0 q 6= 0

since Xp/Xp−1 is a wedge of spheres - one for each p-cell of X. Then since the filtration is bounded 1 below, the spectral sequence weakly converges to H∗(X). The differential on the E page is by definition the cellular boundary map, and there are no nonzero differentials on the E2 page because the E1 page is nonzero only on the line q = 0. So our spectral sequence collapses and we can upgrade weakly converges to converges, and we have that

( cell ∞ 2 Hp (X) q = 0 Ep q = Ep q = , , 0 q 6= 0

The fact that this spectral sequences converges to the graded object H∗(X) then says that in the ﬁltration 0 ⊂ im(Hp(X0) → Hp(X)) ⊂ · · · ⊂ im(Hp(Xn) → Hp(X)) = Hp(X) ∞ ∼ ∞ if we set Fk = im(Hp(Xn) → Hp(X)), then Ek,p−k = Fk/Fk−1. Then since Ek,p−k is nonvanishing only if k = p, the above ﬁltration collapses to

0 = im(Hp(Xp−1) → Hp(X)) ⊂ im(Hp(Xp) → Hp(X)) = Hp(X) cell ∼ and we thus have Hp (X) = Hp(X).

1.4 THE ATIYAH-HIRZEBRUCH SPECTRAL SEQUENCE (AHSS) Let X be a CW complex, and E a generalized homology theory, then by the exactness axiom we have an exact couple

L L Ep+q(Xp) Ep+q(Xp) p,q p,q

L Ep+q(Xp, Xp−1) p,q

To check this weakly converges to the obvious ﬁltration of Ep+q(X), we just need to check that \ k−1(im(ir)) = ker(k) r

This follows easily from the fact that X−r = ∅ for r > 0, then the trick is to identify the diﬀerentials as ∼ cell being the cellular ones since Ep+q(Xp, Xp−1) = Cp (X) ⊗ Ep+q(∗), by the wedge and excision axioms. Leaving this as an exercise or referring the reader to Adams’ blue book, we have a spectral sequence

2 Ep,q = Hp(X; Eq(∗)) =⇒ Ep+q(X) One might notice that we should be able to do all the same arguments in cohomology. To that end, we deﬁne

Deﬁnition 1.16. A cohomological spectral sequence is the same data as in (1.3) except the indices are switched, and the diﬀerentials switch direction - i.e.

r p,q p+r,q−r+1 d : Er → Er

We get a cohomological spectral sequence from an exact couple if we require the bidegree of i to be (−1, 1), that of j to be (1, 0), and that of k to be (0, 0).

7 Theorem 1.17. If E is a cohomology theory, and X a CW complex, there is a spectral sequence

p,q p q p+q E2 = H (X; E (∗)) =⇒ E (X) Proof : In cohomology, it’s not too diﬀerent. We get an exact couple from the exactness axiom all the same, and we have Z˜ ∞ = T k−1(im(ir)) and B˜ ∞ = S j(ker(ir)). This time the fact that X−r = ∅ r r tells us that B˜ ∞ = im(j) since, for r large enough, everything is in ker(ir). We ﬁlter E∗(X) by φp FpEp+q(X) = ker(Ep+q(X) −→ Ep+q(Xp−1)). We have a morphism of SES’s

φp+1 0 Fp+1En(X) En(X) im(φp+1) 0

φp 0 FpEn(X) En(X) im(φp) 0

By the ﬁve lemma, therefore

i Fp/Fp+1 =∼ ker(im(φp+1) −→ im(φp)) = {x ∈ En(Xp+1) : x = φp+1(y), i(x) = 0}

On the other hand, setting n = p + q

{x ∈ En(Xp, Xp−1) : k(x) ∈ im(ir)∀r} Z˜ ∞ /B˜ ∞ = p,q p,q im(j) {x ∈ En(Xp, Xp−1) : k(x) ∈ im(lim (En(Xl)) → En(Xp))} = 0 im(j) {x ∈ En(Xp, Xp−1) : k(x) = φp+1(x)} = im(j) {x ∈ En(Xp, Xp−1) : k(x) = φp+1(x)} = ker(k) =∼ {x ∈ En(Xp) : x = φp+1(y), i(x) = 0}

Moving from the second line to the third, we use the fact that the milnor sequence shows that the map En(X) → lim En(Xp) is a surjection, and φp+1 factors through this surjection. Hence we have weak ←−p convergence.

It is not hard to see that the argument we have just presented generalizes Theorem 1.13 replacing colim with and the hypothesis there with ˜ ∞ = im( ). lim←− B j Remark 1.18. The convergence statement of AHSS can also be taken to be an isomorphism of graded E∗(∗)-algebras of the E∞ page with the associated graded ring of E∗(X). We need to deﬁne: a multiplicative structure on a spectral sequence is the structure of a DGA on each page so that the cohomology of (Er, dr) is Er+1 as algebras. In order for this to be true, we essentially just need to check that the leibniz rule holds. The fact that the AHSS is multiplicative is actually quite subtle, and Dan Dugger has a paper on this. We simply take it as given for the rest of this exposition.

2 COMPLEX ORIENTED COHOMOLOGY THEORIES

Deﬁnition 2.1. A complex-oriented cohomology theory is the data (E, x) where E is a multiplicative cohomology theory E, and x ∈ E˜2(CP∞) such that x restricts to 1 in the ring E∗(∗) under the map

E˜2(CP∞) → E˜2(CP1) =∼ E˜2(S2) =∼ E˜0(S0) = E0(∗)

8 E is said to be complex orientable if there exists such a class x.

Example 2.2. HZ, i.e. ordinary cohomology is complex orientable. Indeed we have H∗(CP∞) =∼ Z[x], and the map H˜ 2(CP∞) → H˜ 2(CP1) =∼ H˜ 2(S2) is a surjection sending the class x to the generator of H2(S2), which is sent to 1 by the suspension isomorphism H˜ 2(S2) =∼ H˜ 0(S0) = H0(∗). One can see that H2(CP∞) → H2(CP1) is a surjection by looking at the cellular chain complex and noting that all the boundary maps are zero, then dualizing.

∞ Example 2.3. Let BU(n) be the classifying space for the group U(n) - for instance Grn(C ) is a model for BU(n) - and let En be the universal complex vector bundle of rank n over BU(n). We deﬁne MU(n) to be the Thom space of the bundle En → BU(n) - this is the space D(En)/S(En) for a choice of metric on En. For all n, we have a map

2 S ∧ MU(n) = Thom(C ⊕ En) → Thom(En+1) = MU(n + 1) because the Thom space is functorial under pullbacks of vector bundles. MU is the spectrum deﬁned by MU2n = MU(n) and MU2n+1 = ΣMU(n). We call MU the complex (co)-bordism spectrum. It gets the name because by work of Thom, if X is a smooth manifold, one can identify MU∗(X) with (complex-oriented) bordism classes of smooth maps Z → X such that Z is stably almost complex.

Proposition 2.4. MU is complex orientable.

Proof : We ﬁrst prove MU is a ring spectrum. By the Yoneda lemma, there are maps BU(n) × BU(m) → BU(n + m) for each n, m that classify direct sums. Above this map is the map En × Em → En+m to which we apply the Thom functor to get a map MU(n) ∧ MU(m) → MU(n + m). As in Adams, we may choose a handicrafted smash product model for MU ∧ MU, whereby we let (MU ∧ MU)2n = MUn ∧ MUn and (MU ∧ MU)2n+1 = MUn+1 ∧ MUn, then to check that our direct sum maps deﬁne a µ morphism MU ∧ MU −→ MU, we can reduce to showing that the diagrams

S2 ∧ MU(n) ∧ MU(m) MU(n + 1) ∧ MU(m)

S2 ∧ MU(n + m) MU(n + m + 1)

commute. But this is just the Thom functor applied to the diagram

C ⊕ En × Em En+1 × Em

C ⊕ En+m En+m+1

and everything here is a pullback. Now we need to show there is a unit for this map. The zero section ∞ CP ' BU(1) → MU(1) is a homotopy equivalence because S(E1) is contractible because the unit n 2n+1 n sphere bundle of the bundle Ln → CP is the map S → CP . Hence taking the limit, the sphere bundle is S∞ → CP∞. We therefore know MU(1) is a K(Z, 2). We have a map S2 → MU(1) given by applying the Thom functor to the top row of the pullback diagram

C E1

∗ BU(1)

Then since S2 =∼ CP1 = (CP∞)(2), we may extend the pointed map ι : S2 → MU(1) to a pointed map ι : CP∞ → MU(1) because MU(1) is a K(Z, 2), and CP∞ has no 3-cells. We didn’t just take the zero section CP∞ → MU(1) because this is not a pointed map - although since MU(1) is simply connected,

9 we could have assumed it was without much loss of generality.

The map ι : S2 → MU(1) deﬁnes a map ι : S → MU, and it is a unit for µ because for the spectrum S ∧ MU we choose the same handicrafted smash product as for MU ∧ MU so that 2n ι∧1 (S ∧ MU)4n = S ∧ MU(n), then the map S ∧ MU −→ MU ∧ MU is determined by the maps

1∧ι∧1 S2n ∧ MU(n) = S2(n−1) ∧ S2 ∧ MU(n) −−−→ S2(n−1) ∧ MU(1) ∧ MU(n) → MU(n) ∧ MU(n)

Composing this map with the map MU(n) ∧ MU(n) → MU(2n), we have the 2(n − 1) fold suspension ι∧1 µ of the map S2 ∧ MU(n) −→ MU(1) ∧ MU(n) −→ MU(n + 1), so it suﬃces to show this map is the same as the bonding map S2 ∧ MU(n) → MU(n + 1) - recall we are trying to show that the diagram

S ∧ MU MU ∧ MU µ ' MU

commutes, and the map S ∧ MU → MU is the equivalence exhibited by the bonding maps of MU. But ι is the bonding map.

By showing that ι is a unit for the ring structure on MU, we have also shown that the element of 2 MUg (CP∞) given by the extension of ι to CP∞ restricts to 1 ∈ E0(∗), since ι is deﬁned to have this restriction.

Theorem 2.5. If E is a multiplicative cohomology theory, then the following are equivalent: 1. E is complex orientable.

2. For every complex vector bundle V → X over a pointed space X, there is a "Thom class" xV ∈ 2·rkV E˜ (Thom(V)) such that xV restricts to 1 along the maps

2·rk(V) 2·rk(V) 2·rk(V) 2·rk ∼ 0 E˜ (Thom(V)) → E˜ (Thom(V|∗ → ∗)) → E˜ (S (V)) = E (∗)

The classes xV are required to satisfy a naturality condition which says that if f : Y → X is a ∗ map and V → X is a complex vector bundle then x f ∗V = f (xV ), and they are required to satisfy 0 a multiplicativity condition, which says that xV⊕V0 = xV · xV0 , where V, V are complex vector bundles over X. Our ﬁrst condition can be seen as a nontriviality condition.

3. There is a map of ring spectra MU → E.

4. The Atiyah-Hirzebruch Spectral Sequence for E∗(CP∞) collapses at the second page. In 2 and 3 the data given uniquely speciﬁes a complex orientation on E, and in 4 a choice of isomorphism ∗ ∞ of E (CP ) with the E∞ = E2 page of the AHSS speciﬁes a complex orientation on E.

Proof : We prove this in the appendix.

∗ Theorem 2.6. If (E, xE) is a complex-oriented cohomology theory, then as a graded E (∗)-algebra ∗ ∞ ∼ ∗ ∗ ∗ E (CP ) = E (∗)[[xE]] with |x| = 2. We often abbreviate E (∗) by E .

Proof : The Atiyah-Hirzebruch spectral sequence for a multiplicative cohomology theory E is a multiplicative spectral sequence with signature

p,q p q p+q E2 = H (X; E (∗)) =⇒ E (X) where X is a CW complex. Letting X = CPn, we have by the universal coeﬃcient theorem we have that ∗ ∗ n+1 ∗ n ∼ n+1 the E2 page is isomorphic as an E -algebra to E [x]/x since H (CP ; R) = R[x]/x where |x| = 2.

10 r ∗ ∗,∗ Then since d is a graded derivation of the graded E -algebra Er for each r, it follows that every element of the E2 page is a permanent cycle if 1 and x are, and to say that every element is a permanent cycle is to say that the spectral sequence collapses here. 1 is the element corresponding to 0 n 0 0,0 0 0 1 ∈ H (CP ; E (∗)) which we may represent as the unit in the ring E1 = E (CP , ∅). To say that 1 survives the r-th stage is to say that the image of a representative of 1 under the map E0(CP0, ∅) → E0(CP0) (this is the identity map) is in the image of E0(CPr) → E0(CP0), but this is true for all 2,0 r because this map is always a ring map. Similarly, we may represent x by the element of E1 = E2(CP1, ∗) = E˜2(CP1) =∼ E0(∗) corresponding to 1. The map j in the exact couple is just the inclusion E˜2(CP1) ,→ E2(CP1), so looking at the exact couple, we see that to say x is a cycle thru the r-th stage 2 n (r) is to say that there is an element xr ∈ E ((CP ) ) such that xr restricts to 1 under the maps

E2((CPn)(r)) → E˜2(CP1) =∼ E0(∗)

which we have by functoriality and our complex orientation. Therefore, this spectral sequence collapses.

Now we use the convergence of this spectral sequence to calculate E∗(CP∞), and we ﬁrst deduce p n ∞ 0 n ∞ p+1 p the group structure. We have a ﬁltration Fn on E (CP ), where Fn = E (CP ), Fn ⊂ Fn , and p p+1 ∼ p,n−p Fn /Fn = E2 . Collecting these into graded maps, we have a short exact sequence

1 ∗ ∞ 0,∗ ∗ 0 → F∗ → E (CP ) → E2 = E (∗) → 0 Since the Fp are kernels of maps induced by maps of spaces, they are E∗-modules, so this is a short exact ∗ ∗ ∞ ∼ 1 0 1 ∼ ∗ 1 sequence of E -modules, and it therefore splits, so we have E (CP ) = (F /F )∗ ⊕ F∗ = E (∗) ⊕ F∗ , and the splitting sends 1 ∈ E∗(∗) to 1 ∈ E∗(CP∞). The next exact sequence gives us F1 = F2 by evenness. Then we have 3 1 ∗−2 0 → F∗ → F∗ → E (∗) → 0 Again by freeness of E∗−2, we have a splitting, and tracing through where this map comes from in the exact couple, the splitting sends 1 to xE, our orientation class. We proceed inductively, and we have ∗ ∞ ∼ 0 1 2 3 4 5 2n 2n+1 E (CP ) = (F /F )∗ ⊕ (F /F )∗ ⊕ (F /F )∗ ⊕ · · · ⊕ (F /F )∗ ∼ 0,∗ 2,∗−2 4,∗−4 2n,∗−2n = E2 ⊕ E2 ⊕ E2 ⊕ · · · ⊕ E2 =∼ E∗(∗) ⊕ E∗−2(∗) ⊕ E∗−4(∗) ⊕ · · · ⊕ E∗−2n(∗)

p p+1 ∼ p,n−p ∗ and since the isomorphisms Fn /Fn = E2 respect the graded E -algebra structure, at each stage ∗ n ∼ we get a splitting by sending 1 to the appropriate power of xE. We therefore have E (CP ) = ∗ 2 n ∗ n+1 E {1, x, x ,..., x } as an E -module, and this along with the fact that xE = 0 determines an iso- ∗ ∗ n ∼ ∗ n+1 n+1 n morphism of E -algebras E (CP ) = E [xE]/xE . Recall that xE = 0 because CP can be covered by n + 1 contractible subsets, hence using homotopy invariance and naturality of the ring structure, all n + 1-fold products of reduced classes are zero. For more explanation of this last point on reduced classes, see COCTALOS page 3. Note that the Milnor sequence for CP∞ = S CPn implies that n E∗(CP∞) =∼ lim E∗[x ]/xn =∼ E∗[[x ]] because the maps E∗(CPn+1) → E∗(CPn) are surjective. This ←−n E E E is true because there is an induced map of AHSS’s, and since it is a surjection on the E2 page and the ∗ SS collapses, this implies that it is a surjection on E .

3 FORMAL GROUP LAWS

Definition 3.1. If (E, x) is a complex-oriented cohomology theory, and L → X is complex line bundle, E ∗ ˜2 we define the first E-chern class of L, c1 (L), to be the class f (x) ∈ E (X) where f : X → CP∞

is a classifying map for L.

11 In HZ, we know that if L1, L2 are complex line bundles over a space X, then c1(L1 ⊗ L2) = c1(L1) + c1(L2). Quillen asked - for nice cohomology theories, like MU, KU, etc. - does this formula still hold, and if not, how bad can it be? He found out that - in the case of MU - it was literally as bad as it could possibly be.

Proposition 3.2. Let (E, x) be a complex-oriented cohomology theory. There is a power series E ∗ F (x, y) ∈ E [[x, y]] such that if L1, L2 are complex line bundles over a space X, then

E E E E c1 (L1 ⊗ L2) = F (c1 (L1), c1 (L2)) Proof : Using a similar argument with the AHSS along with the Kunneth formula, one has an isomorphism of E∗-algebras E∗(CP∞ × CP∞) =∼ E∗[[x, y]] with |x| = |y| = 2. If L is the tautological line ∞ ∗ ∗ ∞ ∞ bundle over CP , then we can form the line bundle pr1(L) ⊗ pr2(L) over CP × CP , and it therefore has a classifying map CP∞ × CP∞ → CP∞ and therefore an induced map in E∗(−)

E∗[[x]] =∼ E∗(CP∞) → E∗(CP∞ × CP∞) =∼ E∗[[x, y]] and x is therefore sent to an element in E∗[[x, y]], a power series which we denote FE(x, y). But ∗ ∗ pr1(L) ⊗ pr2(L) is the universal case of the tensor product of complex line bundles - indeed, if L1, L2 are complex line bundles over X, then Li has a classifying map fi, and L1 ⊗ L2 is the pullback

∗ ∗ L1 ⊗ L2 pr1(L) ⊗ pr2(L)

f × f X 1 2 CP∞ × CP∞

up to isomorphism. This is because we get the map CP∞ × CP∞ → CP∞ from the yoneda lemma and the natural transformations

⊗ [−, CP∞ × CP∞] =∼ [−, CP∞] × [−, CP∞] −→ [−, CP∞]

by plugging in the identity - and if we do so, we see that the map CP∞ × CP∞ → CP∞ classiﬁes ∗ ∗ pr1(L) ⊗ pr2(L). Then since we have deﬁned Chern classes by pulling back from the universal case, we have

E E ∗ E ∗ ∗ c1 (L1) ⊗ c1 (L2) = ( f1 × f2) (c1 (pr1(L) ⊗ pr2(L))) ∗ E ∗ E ∗ E = ( f1 × f2) F pr1(c1 (L)), pr2(c1 (L)) E ∗ ∗ E ∗ ∗ E = F ( f1 × f2) pr1(c1 (L)), ( f1 × f2) pr1(c1 (L))

E E E = F (c1 (L1), c1 (L2))

Deﬁnition 3.3. A formal group law over a commutative ring R is a bivariate power series F(x, y) ∈ R[[x, y]] satisfying

1. F(x, 0) = x and F(0, y) = y

2. F(x, y) = F(y, x)

3. F(F(x, y), z) = F(x, F(y, z))

Remark 3.4. The power series we found above for E∗ is a fgl because tensor products of line bundles are (up to isomorphism) unital (with unit the trivial line bundle), associative, and commutative.

If F is a formal group law over a ring S, and f : R → S is a ring map, then we can define a formal group law f∗F over S by letting the coefficients be f (aij) where aij are the coefficients of F. Lazard

12 showed:

Theorem 3.5. (Lazard) There is a ring L (the so-called Lazard ring) and a formal group law Funiv over L such that the map

HomCRing(L, R) → {Formal Group Laws over R} sending f 7→ f∗Funiv is an isomorphism. Furthermore L has a grading such that L is isomorphic as a graded ring to Z[x1, x2,...] where |xi| = 2i.

Proof : It is easy to see L exists, we take the ring Z[aij] and mod out by the ideal generated by all the relations: we have ai0 = 1 if i = 1 and 0 otherwise, we have aij = aji, and we have other relations cor- i j responding to the associativity which are not as easy to write down. We then let Funiv(x, y) = ∑ aijx y . i,j We give L a grading by letting |aij| = 2(i + j) and extending out from there multiplicatively. We omit the proof that L is a polynomial ring.

Theorem 3.6. (Quillen) Let f : L → MU∗ be the map classifying the formal group law coming from the canonical complex orientation on MU. f is an isomorphism of graded rings.

Remark 3.7. This tells us that MU carries the universal formal group law with it, and if E is a complex oriented cohomology theory, the ring map MU → E determines a ring map MU∗ → E∗, and this is is the map L → E∗ classifying the formal group law.

4 PROOF OF THEOREM 2.5

Many of the equivalences in 2.5 we could have proven with the tools we have available already, but it is unavoidable that we have to develop the theory of Conner-Floyd Chern classes for some of these statements. We follow the Grothendieck theory of Chern classes. Let p : V → X be a complex vector bundle of rank n, and let P(V) be the corresponding projective bundle - recall if Ui ∩ Uj → GLn(C) are n−1 the transition functions for V, then Ui ∩ Uj → GLn(C) → AutTop(CP ) are the transition functions n−1 for P(V), where the last map comes from the action of GLn(C) on CP , whereby a line through the n n origin in C is sent to its image under a linear automorphism - equivalently GLn(C) acts on C and commutes with the action of C×. There is a tautological line bundle over P(V) - namely we have a trivializing cover {Ui} of X for −1 −1 ∼ n−1 V, and we have a cover of P(V) by open sets p (Ui) with isomorphisms p (Ui) = Ui × CP . Over n−1 each of these we can construct Ui × Ln−1, where Ln−1 is the tautological bundle over CP . We then check that we have gluing data satisfying a cocycle condition, and assemble these together into the line 2 E bundle LV over P(V). We deﬁne t ∈ Ee (P(V)) to be the Chern class c1 (LV ). We have the following:

Proposition 4.1. If E is a COCT, E∗(P(V)) is a free E∗(X) module with basis {1, t,..., tn−1}.

Proof : We first prove the Leray-Hirsch theorem - let CPn → X → B be a fibration with B connected, ∗ ∗ n and suppose that H (X) → H (CP ) is a surjection. Then π1(B) acts trivially on the cohomology of the fiber. This is because we can take γ ∈ π1(B) and the action is given by the isomorphism in the bottom row of H∗(X)

H∗(F ) H∗(X ) H∗(F ) 0 =∼ γ =∼ 1 But since these restrict from H∗(X), one can see from the diagram that the action is trivial. The E2 ∗ 0,2 ∼ 2 n page is isomorphic to H (B)[x] where x ∈ E2 = H (CP ), and then the fact that x is hit by a class in H2(E) tells you that the spectral sequence collapses just as in the COCT argument, and again we

13 use that the rows are free H∗(B) modules to conclude there are no extension problems as modules over H∗(B), so then H∗(E) is a free module with basis {1, x,..., xn} where x is the class restricting to the generator of H2(CPn).

But this argument uses nothing in particular about H∗(−) other than the existence of the SSS, but we can build the same SS using E∗(−) and the action from the base is still trivial, so we just need to check that t restricts to a generator of E2(CPn−1). But the classifying map P(V) → CP∞ takes ∞ n−1 ∞ (x, l) ∈ P(V) where l ∈ P(Vx) to l ∈ CP , so it’s clear that the map CP ,→ P(V) → CP is just the inclusion map, so that the fact that t restricts to a generator is just that the complex orientation − x ∈ Ee2(CP∞) restricts to a generator of Ee2(CPn 1).

2i Deﬁnition 4.2. We therefore have unique classes ci ∈ E (X) such that

n n−1 n−2 n t = c1t − c2t + ··· + (−1) cn

and we deﬁne ci to be the i-th Chern class of the bundle V → X.

Remark 4.3. Here is some motivation for deﬁning the classes in this way. If we form the pullback

L V

P(V) X

We have L = {(x, l, v) : x ∈ X, l ∈ P(Vx), v ∈ Vx}, and hence there is an embedding LV ,→ L sending (x, l, v) 7→ (x, l, v), as LV = {(x, l, v) : x ∈ X, l ∈ P(Vx), l ∈ P}. Since SES’s of vector bundles split, ∼ we have an isomorphism L = LV ⊕ Q for some bundle Q of rank n − 1. If we deﬁne the total chern 2 k class of a bundle E of rank k to be cs(E) = 1 + c1(E)s + c2(E)s + ··· + ck(E)s in an indeterminate s, 0 0 we want the cartan formula cs(E ⊕ E ) = cs(E) · cs(E ) to hold, so we have

cs(L) = cs(LV ⊕ Q) = cs(LV )cs(Q) = (1 + ts)cs(Q)

since this is how we have deﬁned t. We therefore have s = −1/t is a root is a root of cs(L), hence

1 1n 0 = 1 + c (L) − + ··· + c (L) − 1 t n t and clearing denominators, we have the expression we used to deﬁne cn(L). The fact that we went to the projective bundle to deﬁne these can be seen as a sort of universality statement - the universal ∞ ∞ line bundle is the bundle LV where V = C → ∗ since P(V) = CP . We want to build higher Chern classes out of the ones for line bundles using some kind of splitting principle argument (i.e. the Leray-Hirsch theorem), and LV is the natural way to construct a line bundle from any vector bundle, and the universal line bundle is constructed in this way.

Proof of Theorem 2.5 :