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 D- complex, you can compute with D-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 Æ ⊂ 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 ijim(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) 2 im(j ◦ k) =) [j(x0)] = [j(x1)], since x0 − x1 2 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 ho- mology 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 2 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 fE , drg 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 fE , d g is bounded, we take r = ¥ r r maxfr(p, q)g, 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 2 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] 2 E2 has a (usually not unique) representative x 2 E1. In terms of the exact couple, this says that we compute d2([x]) by taking a representative x 2 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.