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Mat 9580 – Spring 2015 Introduction to the Adams Spectral Sequence

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Mat 9580 – Spring 2015 Introduction to the Adams Spectral Sequence ALGEBRAIC TOPOLOGY III { MAT 9580 { SPRING 2015 INTRODUCTION TO THE ADAMS SPECTRAL SEQUENCE JOHN ROGNES Contents 1. The long exact sequence of a pair 1 1.1. Er-terms and dr-differentials 2 1.2. Adams indexing 6 2. Spectral sequences 7 2.1. E1-terms 9 2.2. Filtrations 9 3. The spectral sequence of a triple 11 4. Cohomological spectral sequences 15 5. Example: The Serre spectral sequence 16 5.1. Serre fibrations 16 5.2. The homological Serre spectral sequence 16 5.3. The cohomological Serre spectral sequence 16 5.4. Killing homotopy groups 17 5.5. The 3-connected cover of S3 17 5.6. The first differential 17 5.7. The cohomological version 19 5.8. The remaining differentials 19 5.9. Conclusions about homotopy groups 20 5.10. Stable homotopy groups 21 6. Example: The Adams spectral sequence 22 6.1. Eilenberg{Mac Lane spectra and the Steenrod algebra 22 6.2. The d-invariant 22 6.3. Wedge sums of suspended Eilenberg{Mac Lane spectra 23 6.4. Two-stage extensions 23 6.5. The mod p Adams spectral sequence 24 6.6. Endomorphism ring spectra and their modules 25 6.7. The mod 2 Adams spectral sequence for the sphere 25 6.8. Multiplicative structure 28 6.9. The first 13 stems 29 6.10. The first Adams differential 30 7. Exact couples 31 7.1. The spectral sequence associated to an unrolled exact couple 31 7.2. E1-terms and target groups 32 7.3. Conditional convergence 34 8. Examples of exact couples 36 8.1. Homology of sequences of cofibrations 36 8.2. Cohomology of sequences of cofibrations 37 8.3. The Atiyah{Hirzebruch spectral sequence 37 8.4. The Serre spectral sequence 39 8.5. Homotopy of towers of fibrations 40 8.6. Homotopy of towers of spectra 41 9. The Steenrod algebra 42 9.1. Steenrod's reduced squares and powers 42 9.2. The Steenrod algebra 44 Date: April 24th 2015. 1 9.3. Indecomposables and subalgebras 45 9.4. Eilenberg{Mac Lane spectra 47 10. The Adams spectral sequence 49 10.1. Adams resolutions 49 10.2. The Adams E2-term 51 10.3. A minimal resolution at p = 2 53 10.4. A minimal resolution at p = 3 61 11. Bruner's ext-program 66 11.1. Overview 66 11.2. Installation 66 11.3. The module definition format 67 11.4. The samples directory 68 11.5. Creating a new module 69 11.6. Resolving a module 69 12. Convergence of the Adams spectral sequence 70 12.1. The Hopf{Steenrod invariant 70 12.2. Naturality 72 12.3. Convergence 74 13. Multiplicative structure 80 13.1. Composition and the Yoneda product 80 13.2. Pairings of spectral sequences 84 13.3. Modules over cocommutative Hopf algebras 85 13.4. Smash product and tensor product 88 13.5. The smash product pairing of Adams spectral sequences 89 13.6. The bar resolution 95 13.7. Comparison of pairings 96 13.8. The composition pairing, revisited 100 14. Calculations 101 14.1. The minimal resolution, revisited 101 14.2. The Toda{Mimura range 104 14.3. Adams vanishing 107 14.4. Topological K-theory 110 15. The dual Steenrod algebra 115 15.1. Hopf algebras 115 15.2. Actions and coactions 118 15.3. The coproduct 119 15.4. The Milnor generators 121 15.5. Subalgebras of the Steenrod algebra 124 15.6. Spectral realizations 126 References 127 1. The long exact sequence of a pair Let (X; A) be a pair of spaces. The relationship between the homology groups H∗(A), H∗(X) and H∗(X; A) is expressed by the long exact sequence @ i∗ j∗ @ i∗ ::: −! Hn(A) −! Hn(X) −! Hn(X; A) −! Hn−1(X; A) −! ::: Exactness at Hn(X) amounts to the condition that im(i∗ : Hn(A) ! Hn(X)) = ker(j∗ : Hn(X) ! Hn(X; A)) : The homomorphism i∗ induces a canonical isomorphism from Hn(A) cok(@ : Hn+1(X; A) ! Hn(A)) = im(@ : Hn+1(X; A) ! Hn(A)) H (A) = n ker(i∗ : Hn(A) ! Hn(X)) 2 to im(i∗ : Hn(A) ! Hn(X)). Exactness at Hn(A) and at Hn(X; A) amounts to similar conditions, and @ and j∗ induce similar isomorphisms. We can use the long exact sequence to get information about H∗(X) from information about H∗(A) ∼ ∼ and H∗(X; A), if we can compute the kernel ker(@) = im(j∗) = cok(i∗) and the cokernel cok(@) = im(i∗) of the boundary homomorphism @, and determine the extension 0 ! im(i∗) −! H∗(X) −! cok(i∗) ! 0 of graded abelian groups. Let us carefully spell this out in a manner that generalizes from long exact sequences to spectral sequences. We are interested in the graded abelian group H∗(X). The map i: A ! X induces the homomorphism i∗ : H∗(A) ! H∗(X), and we may consider the subgroup of H∗(X) given by its image, im(i∗). We get a short increasing filtration 0 ⊂ im(i∗) ⊂ H∗(X) : More elaborately, we can let 8 0 for s ≤ −1 <> Fs = im(i∗) for s = 0 > :H∗(X) for s ≥ 1 for all integers s. We call s the filtration degree. The possibly nontrivial filtration quotients are im(i∗) H∗(X) = im(i∗) and = cok(i∗) : 0 im(i∗) We find 8 >0 for s ≤ −1 > <im(i∗) for s = 0 fracFsFs−1 = >cok(i∗) for s = 1 > :0 for s ≥ 2. The short exact sequence 0 ! im(i∗) −! H∗(X) −! cok(i∗) ! 0 expresses H∗(X) as an extension of two graded abelian groups. This does not in general suffice to determine the group structure of H∗(X), but it is often a tractable problem. More generally we have short exact sequences 0 ! Fs−1 −! Fs −! Fs=Fs−1 ! 0 for each integer s. If we can determine the previous filtration group Fs−1, say by induction on s, and if we know the filtration quotient Fs=Fs−1, then the short exact sequence above determines the next filtration group Fs, up to an extension problem. In the present example F−1 = 0, F0 = im(i∗) and F1 = H∗(X), so there is only one extension problem, from F0 to F1, given the quotient F1=F0 = cok(i∗). We therefore need to understand im(i∗) and cok(i∗). By definition and exactness ∼ ∼ im(i∗) = cok(@) and cok(i∗) = im(j∗) = ker(@) ; so both of these graded abelian groups are determined by the connecting homomorphism @ : H∗+1(X; A) −! H∗(A) : If we assume that we know H∗(A) and H∗+1(X; A), we must therefore determine this homomorphism @, and compute its cokernel cok(@) = H∗(A)= im(@) and its kernel ker(@) ⊂ H∗+1(X; A). In view of the short exact sequences 0 ! im(@) −! H∗(A) −! cok(@) ! 0 and 0 ! ker(@) −! H∗+1(X; A) −! im(@) ! 0 we can say that the original groups H∗(A) and H∗+1(X; A) have been reduced to the subquotient groups cok(@) and ker(@), respectively, and that both groups have been reduced by the same factor, namely by im(@). This makes sense in terms of orders of groups if all of these groups are finite, but must be more 3 carefully interpreted in general. The change between the old groups and the new groups is in each case created by the non-triviality of the homomorphism @. r r 1.1. E -terms and d -differentials. We can present the steps in this approach to calculating H∗(X) using the following chart. First we place the known groups H∗(A) and H∗(X; A) in two columns of the (s; t)-plane: . t = 2 H2(A) H3(X; A) t = 1 H1(A) H2(X; A) t = 0 H0(A) H1(X; A) t = −1 0 H0(X; A) s = 0 s = 1 We call t the internal degree, even if this is not particularly meaningful in this example. The sum s + t is called the total degree, and corresponds to the usual homological grading of H∗(A), H∗(X) and H∗(X; A). 1 1 This first page is called the E -term. It is a bigraded abelian group E∗;∗, with 1 1 E0;t = Ht(A) and E1;t = H1+t(X; A) 1 for all integers t. We extend the notation by setting Es;t = 0 for s ≤ −1 and for s ≥ 2. This appears as follows . 1 1 1 1 E−1;2 E0;2 E1;2 E2;2 1 1 1 1 E−1;1 E0;1 E1;1 E2;1 1 1 1 1 :::E−1;0 E0;0 E1;0 E2;0 ::: 1 1 1 1 E−1;−1 E0;−1 E1;−1 E2;−1 . with nonzero groups only in the two central columns. 4 We next introduce the boundary homomorphism @. In the (s; t)-plane it has bidegree (−1; 0), i.e., maps one unit to the left. We can display it as follows: . @ t = 2 H2(A) o H3(X; A) @ t = 1 H1(A) o H2(X; A) @ t = 0 H0(A) o H1(X; A) t = −1 0 H0(X; A) s = 0 s = 1 In spectral sequence parlance, this homomorphism is called the d1-differential. It extends trivially to a homomorphism 1 1 1 ds;t : Es;t −! Es−1;t for all integers s and t. In all other cases than those displayed above, this homomorphism is zero, since for s ≤ 0 the target is zero, for s ≥ 2 the source is zero, and for s = 1 and t ≤ −1 the target is also zero.
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