The mod 2 Adams Spectral Sequence for tmf

Transcription

1 The mod 2 Adams Spectral Sequence for tmf Robert Bruner Wayne State University, and Universitetet i Oslo Isaac Newton Institute 11 September 2018 Robert Bruner (WSU and UiO) tmf at p = 2 INI 1 / 65

2 Outline 1 Introduction 2 tmf 3 Duality 4 The Torsion-free quotient 5 The Davis-Mahowald spectral sequence 6 Ext A(2) (F 2, F 2 ) 7 Key Differentials Robert Bruner (WSU and UiO) tmf at p = 2 INI 2 / 65

3 Introduction Acknowledgements A report on joint work in progress with John Rognes. Thanks are due to the Simons Foundation for travel funds and to the Universitetet i Oslo for giving me excellent working conditions and colleagues during these last three years. Thanks also the the Newton Institute, the organizers of the HHH programme, and the Institute staff. Robert Bruner (WSU and UiO) tmf at p = 2 INI 4 / 65

4 Introduction Introduction tmf -modules whose ordinary homology we can readily calculate, but whose BP homology is much harder to get. The ordinary mod 2 Adams spectral sequence is thus a reasonable tool to understand these. We first needed to understand the ordinary Adams spectral sequence for tmf itself in greater detail. Have tmf, tmf C2, tmf Cη, and tmf Cν in gory detail. Tools: DMSS, Gröbner bases, ways to make calculations finite. Oddity: no Toda brackets needed except as a heuristic. Side benefit: independent verification of earlier results, with thorough documentation. Robert Bruner (WSU and UiO) tmf at p = 2 INI 5 / 65

5 tmf tmf Robert Bruner (WSU and UiO) tmf at p = 2 INI 7 / 65

6 tmf We may think of ko as made up of two HZ s with a bit of 2-torsion tagging along. In the same way, tmf is made up of eight ko s with some B-torsion tagging along. Here, B π 8 (tmf ) maps to the Bott class in π 8 (ko) under a natural map tmf ko. We will call this lift to tmf the Bott class as well. Robert Bruner (WSU and UiO) tmf at p = 2 INI 8 / 65

7 tmf Generators of tmf First, there is the periodicity element M π 192 (tmf ), not a zero-divisor. Group remaining generators by Ann Z[B] : for 0 k 7, (0) D k π 24k (tmf ), B k π 24k+8 (tmf ), C k π 24k+12 (tmf ) (2) η k π 24k+1 (tmf ), and (2, B) ν k π 24k+3 (tmf ), ɛ k π 24k+8 (tmf ), κ k π 24k+14 (tmf ), and κ k π 24k+20 (tmf ). D k, B k, C k and ν k are defined for all k. The rest are defined only for some values of k. Robert Bruner (WSU and UiO) tmf at p = 2 INI 9 / 65

8 tmf When k = 0, we have familiar elements: The unit i : S tmf sends 1, η, ν, ɛ, κ and κ to D 0 = 1, η 0, ν 0, ɛ 0, κ 0 and κ 0. Omit the subscript 0 accordingly. The map tmf ko sends D 0 = 1, B 0 = B, and C 0 = C to generators of ko in degrees 0, 8 and 12, respectively. The relation between B and κ is the same as that between 2 and η, or between η and ν, a kind of Frobenius, detected by Sq 0 in Ext. κ 2 = B κ. Robert Bruner (WSU and UiO) tmf at p = 2 INI 10 / 65

9 tmf 20 tmf BD1 8 Bκ=ɛκ B κ=ɛ κ B1 ɛ1 κ κ D1 C ν1 4 B κ κ η1 ɛ η ν Robert Bruner (WSU and UiO) tmf at p = 2 INI 11 / 65

10 tmf All the generators except M: Start B C η ν ɛ κ κ 24 D 1 B 1 C 1 η 1 ν 1 ɛ 1 48 D 2 B 2 C 2 ν 2 72 D 3 B 3 C 3 (ν 3 ) 96 D 4 B 4 C 4 η 4 ν 4 ɛ 4 κ 4 ( κ 4 ) 120 D 5 B 5 C 5 ν 5 ɛ D 6 B 6 C 6 ν D 7 B 7 C 7 (ν 7 ) Here, ν 3 = η 3 1, κ 4 = κd 4 and ν 7 = 0 are not needed to generate π (tmf ), but are convenient in general formulas. Robert Bruner (WSU and UiO) tmf at p = 2 INI 12 / 65

11 tmf Families of elements in tmf Consider the natural map tmf MF /2 to the ring of modular forms MF = Z[c 4, c 6, ]/(1728 = c4 3 c6 2 ) = Z[c 4, ][ c ] The discriminant is not in the image of the map to MF /2, but it does exist in the spectral sequences leading to tmf Differential on kills ν κ, detected by h 2 g in the Adams spectral sequence, giving Massey products at E 2 of the Adams spectral sequence (x) = h 2, g, x (x) = g, h 2, x Robert Bruner (WSU and UiO) tmf at p = 2 INI 13 / 65

12 tmf Theorem Repeated application of gives classes detecting the following sequences of elements of π (tmf ): 8D D 1 2D 2 D 3 4D 4 D 5 2D 6 D 7 8M. C C 1 C 2 C 3 C 4 C 5 C 6 C 7 CM. B +ɛ B 1 +ɛ 1 B 2 B 3 B 4 +ɛ 4 B 5 +ɛ 5 B 6 B 7 BM. 8B 8B 1 8B 2 8B 3 8B 4 8B 5 8B 6 8B 7 8BM. Robert Bruner (WSU and UiO) tmf at p = 2 INI 14 / 65

13 o Figure 9.1. on h 1 and h 2 in Ext tmf 9.3. THE MASSEY PRODUCTS AND on classes detecting the i and i family h1 h1 h0 / / o h0 h1 h 2 1 h 3 1 = h 2 0h2 h0h2 h2 h1 h 2 1 = h0 h1w2 {h 2 1w2, 2 } h1 2 = h 2 0h2w2 h0h2w2 h2w2 w2 h1 w2 {h 2 1 w2 = h0 w2, 3 } w2 h1w 2 2 {h 2 1w 2 2, 2 w2} h1 2 w2 = h 2 0h2w 2 2 h0h2w 2 2 h2w 2 2 w 2 2 h1 w 2 2 {h 2 1 w 2 2 = h0 w 2 2, 3 w2} w 2 2 h1w 3 2 {h 2 1w 3 2, 2 w 2 2} h1 2 w 2 2 = h 2 0h2w 3 2 h0h2w 3 2 h2w 3 2 w 3 2 h1 w 3 2 {h 2 1 w 3 2 = h0 w 3 2, 3 w 2 2} w 3 2 h1w 4 2 {h 2 1w 4 2, 2 w 3 2} h1 2 w 3 2 = h 2 0h2w 3 2 h0h2w 3 2 h2w 3 2 Robert Bruner (WSU and UiO) tmf at p = 2 INI 15 / 65

14 / THE tmfhomotopy GROUPS OF tmf Effect on η i and ν i Figure 9.2. survive to E 1 0 on classes detecting the i and i family which / o 2 o = = = = = = Robert Bruner (WSU and UiO) tmf at p = 2 INI 16 / 65

15 tmf Let Γ B (tmf ) be the B-power torsion, those x tmf such that B i x = 0 for i >> 0. Theorem ν k, ɛ k, κ k, κ Γ B (tmf ). Theorem If x Γ B (tmf ) then Bx = ɛx. Robert Bruner (WSU and UiO) tmf at p = 2 INI 17 / 65

16 tmf General principle: if x, y {η, ν, ɛ, κ} then x i y j depends only on x, y, and i + j. Exceptions stem from the varying 2-divisibility of the 24k + 3 stem. For example, { 2ννk+1 k = 1, 5 ν1 ν k = 0 other k 7 ν2 ν 2 = νν 4, ν 2 ν 4 = νν 6, and ν 2 ν 6 = νν 8 = ν 2 M. (In general, let x k+8 = x k M.) ν4 ν 6 = ±νν 2 M. Robert Bruner (WSU and UiO) tmf at p = 2 INI 18 / 65

17 tmf 0 to Bκ=ɛκ D 1 C 4 B κ κ η ɛ η ν Robert Bruner (WSU and UiO) tmf at p = 2 INI 19 / 65

18 tmf 24 to BD 1 ɛɛ 1 ɛκ κ ɛ 1κ ν 1κ C 1 D 2 ɛκ 8 B κ=ɛ κ B 1 ɛ1 κ κ η 1κ κ 2 η 1 κ D 1 ν 1 η Robert Bruner (WSU and UiO) tmf at p = 2 INI 20 / 65

19 tmf 48 to ɛ 1κ κ ɛɛ 1 κ κd 2 BD 2 η 1ɛ κ C 2 η 1 κ 2 κ κ 2 ν 2ɛ ν2κ η 1ɛ 1 κ 3 ɛ 1 κ B 2 η 2 1 κ D 3 D 2 η 2 1 ν Robert Bruner (WSU and UiO) tmf at p = 2 INI 21 / 65

20 tmf 72 to BD 3 16 C η κ2 κ 4 η η1 κ3 D4 D 3 ν 3=η 3 1 B Robert Bruner (WSU and UiO) tmf at p = 2 INI 22 / 65

21 tmf 96 to ɛκ 4 η 2 1 κ3 D 5 20 η 4 1 = κ5 C 4 BD η 1 κ 4 4 κ 4 B 4 κ 4 η 4 κ ɛ 4 D 4 η 4 ν 4 ν 2 ν 4=ηɛ 4+η 1 κ Robert Bruner (WSU and UiO) tmf at p = 2 INI 23 / 65

22 tmf 120 to κ 2 κ 4 η 5 1 BD 5 ɛν 5 ɛɛ 5 C 5 η 1κ 4 ν 5κ ɛ 5κ D 6 24 κκ 4 κ 4 κ D 5 B 5 ɛ5 η 1η 4 ν Robert Bruner (WSU and UiO) tmf at p = 2 INI 24 / 65

23 tmf 144 to η 6 1 η 2 1 B4 ɛɛ 5 κ ɛ 5κ κ 28 η 1 κb 4 ɛ 5 κ BD 6 κd 6 D 7 C 6 ɛν 6 ν 6κ κ 4 κ 2 η 1ɛ 5 B 6 D 6 ν Robert Bruner (WSU and UiO) tmf at p = 2 INI 25 / 65

24 tmf 168 to BD 7 C 7 32 M D 7 B Robert Bruner (WSU and UiO) tmf at p = 2 INI 26 / 65

25 Duality Definition If x π d R and M is an R-module, let ( M[1/x] = hocolim M x Σ d M x Σ 2d M and let M/x be the homotopy cofiber of M M[1/x]. ) x We can iterate this to get M/(x, y ) = M/x R M/y, etcetera. We get 0 π (M)/x π (M/x ) Γ x π 1 (M) 0 Theorem Σ 20 tmf I (tmf /(2, B, M )) Robert Bruner (WSU and UiO) tmf at p = 2 INI 28 / 65

26 Duality Sketch proof: Let N be the Z[B] submodule of π (tmf ) generated in degrees less than 192 (equivalently, 180). Γ B N is zero outside N /B is zero above dimension 172 and is Z, generated by C 7 /B, in degree 172. Multiplication N Z[M] π (tmf ) is a Z[B, M]-isomorphism. Γ M π (tmf ) = 0 N Z[M]/M = π (tmf )/M = π (tmf /M ) Short exact sequence 0 N /B Z[M]/M π (tmf /(B, M )) Γ B N 1 Z[M]/M 0 Robert Bruner (WSU and UiO) tmf at p = 2 INI 29 / 65

27 Duality Sketch proof:(cont) π (tmf /(B, M )) is concentrated in degrees 20 and in degree 20 is Z generated by C 7 /BM. It is 0 in degree -21, so the short exact sequence 0 π (tmf /(B, M ))/2 π (tmf /(2, B, M )) Γ 2 π 1 (tmf /(B, M )) 0 implies that π (tmf /(2, B, M )) is concentrated in degrees 20 and in degree 20 is Z/2. π (I (tmf /(2, B, M ))) is concentrated in degrees 20 and is Z 2 in degree 20. Choosing a 2-adic generator, we get a tmf -module map inducing an isomorphism in π 20 between 20-connected spectra Σ 20 tmf I (tmf /(2, B, M )) It is an equivalence by inducing along tmf BP 2 Robert Bruner (WSU and UiO) tmf at p = 2 INI 30 / 65

28 Duality As usual, the equivalence of spectra yields isomorphisms (and pairings) with different shifts on the homotopy. Filter π (tmf ) by where π (tmf ) Γ B π (tmf ) Γ 2 π (tmf ) π (tmf ) is the submodule of Γ B generated by the classes not in degrees 3 (mod 24), and ko[k] is the Z[B] submodule generated by {D k, B k, C k } together with the appropriate η s: 0: η, η 2 1: η 1, ηη 1 2: ηb 2, η1 2 3: ηb 3, η 2 B 3 4: η 4, ηη 4 5: ηb 5, η 1 η 4 6: ηb 6, η 2 B 6 7: ηb 7, η 2 B 7 Robert Bruner (WSU and UiO) tmf at p = 2 INI 31 / 65

29 Duality Proposition As a Z[B, M] module Γ B π (tmf ) π (tmf ) = 7 ν k Z[M] k=0 and π (tmf ) Γ B π (tmf ) = 7 ko[k] Z[M] k=0 Robert Bruner (WSU and UiO) tmf at p = 2 INI 32 / 65

30 Duality Duality in the B-torsion ɛκ ɛ κ κ ν νν 6 ν η 6 1 Figure: Duality between [0] and [6] Robert Bruner (WSU and UiO) tmf at p = 2 INI 33 / 65

31 Duality ɛɛ 1 κν 1 ɛ 1 κ ν 1 ɛ 1 κ κ η 1 κ κ 2 η 1 κ ɛ 5 κ ɛɛ 5 ɛν 5 η 2 1 κ4 η 5 1 κν 5 η 1 κ 4 κ4 κ ɛ 5 ν Figure: Duality between [1] and [5] Robert Bruner (WSU and UiO) tmf at p = 2 INI 34 / 65

32 Duality κ κ 2 η 1 ɛ 1 ɛν 2 κ 3 η 1 κ 2 η 2 1 κ ν 2 κν ɛκ 4 η 4 κ D 4 κ η 2 1 κ3 κ 4 ɛ 4 η 4 1 ν Figure: Duality between [2] and [4] Robert Bruner (WSU and UiO) tmf at p = 2 INI 35 / 65

33 Duality ν 3 κ 4 η 1 κ 3 η 2 1 κ η 2 1 κ2 η 1 κ 3 κ 4 ν Figure: Self-duality of [3] Robert Bruner (WSU and UiO) tmf at p = 2 INI 36 / 65

34 Duality Proposition ν 7 k is Pontrjagin 171-dual to B k /B : 7 k=0 ν k 171 = Hom ( 7 k=0 B k /B, Q/Z (Note however that ν 7 = 0 and 0 = B 0 /B ko[0]/b.) Further, in π (tmf /B ) the class which maps to ν k Γ B π (tmf ) lifts to a class ν k with 2 j ν k = C k /B. ) Robert Bruner (WSU and UiO) tmf at p = 2 INI 37 / 65

35 The Torsion-free quotient Maps to MF /2 The elliptic spectral sequence of Hopkins (2002) has edge hom e : π (tmf ) MF /2 = Z[c 4, c 6, ]/(c 3 4 c 2 6 = 1728 ) MF /2 is the ring of integral modular forms, with c 4, c 6 and in weights /2 = 4, 6 and 12, corresponding to topological degrees = 8, 12 and 24. By Hopkins (2002) and Bauer (2008), im(e) is additively Z{a i,j,k c4 i cj 6 k i 0, j {0, 1}, k 0} where 24/ gcd(k, 24) for i = 0 and j = 0, a i,j,k = 1 for i 1 and j = 0, 2 for j = 1. This is an integral result. See also Douglas-Francis-Henriques-Hill (2014) and Konter (2012). Robert Bruner (WSU and UiO) tmf at p = 2 INI 39 / 65

36 The Torsion-free quotient Proposition ker(e) = Γ 2 π (tmf ). B k can be chosen to map to c 4 k, for each 0 k 7. C k can be chosen to map to 2c 6 k, for each 0 k 7. D k can be chosen to map to 2 i k for 1 k 7, where 3 k 1 (mod 2) i = 2 k 2 (mod 4) 1 k = 4 (We complete at 2 here.) The image in MF /2 and the Adams representative in E (tmf ) uniquely determines each of B k, C k and D k in π (tmf ), with the exception of C 2, B 3 and C 6. In each case, the ambiguity is a class of order 2. Robert Bruner (WSU and UiO) tmf at p = 2 INI 40 / 65

37 The Torsion-free quotient C 2 is determined modulo 2 κ 3 = ν 3 ν 2 = ηɛν 2, B 3 is determined modulo κ 4, and C 6 is determined modulo ν 3 ν 6 = ηɛν 6. Robert Bruner (WSU and UiO) tmf at p = 2 INI 41 / 65

38 The Davis-Mahowald spectral sequence General setup The Davis-Mahowald spectral sequence is a substitute for the Cartan-Eilenberg spectral sequence when the sub Hopf algebra is not normal. In Davis and Mahowald (1982) the multiplicative structure is a somewhat ad hoc afterthought. We give precise conditions for it. Γ, Hopf algebra over k Λ Γ, sub Hopf algebra Ω := Γ /Λ = Γ Λ k is a quotient Γ-module coalgebra Robert Bruner (WSU and UiO) tmf at p = 2 INI 43 / 65

39 The Davis-Mahowald spectral sequence DMSS, dual formulation Γ commutative Hopf algebra. Λ quotient Hopf algebra of Γ. Ω = Γ Λ k left Γ -comodule algebra. Require (suitable) Γ -comodule algebra resolution k (Ω R, d). Get multiplicative Davis Mahowald spectral sequence E σ,s, 1 = Ext s, Λ (k, R σ ) = σ Ext s+σ, Γ (k, k). Untwisting Ω R σ = Γ Λ R σ is multiplicative for commutative Γ. Robert Bruner (WSU and UiO) tmf at p = 2 INI 44 / 65

40 The Davis-Mahowald spectral sequence DMSS, dual formulation, cont. Assume a graded Γ -comodule algebra R = σ Rσ and homomorphisms d : Ω R σ Ω R σ+1 such that (Ω R, d) is a differential graded Γ -comodule algebra and the unit k (Ω R, d) is a quasi-isomorphism. Get an algebra spectral sequence E σ,s 1 = Ext s Λ (k, R σ ) = σ Ext s+σ Γ (k, k) Product E σ, 1 E τ, 1 E σ+τ, 1 equals pairing induced by Λ -comodule product R σ R τ R σ+τ. Robert Bruner (WSU and UiO) tmf at p = 2 INI 45 / 65

41 The Davis-Mahowald spectral sequence Main example: A(2) Commutative Hopf algebras Γ = A(2) A(1) = Λ i.e., F 2 [ξ 1, ξ 2, ξ 3 ]/(ξ1, 8 ξ 2, 4 ξ 3) 2 F 2 [ξ 1, ξ 2 ]/(ξ1, 4 ξ 2) 2 The left A(2) -comodule algebra Ω = A(2) A(1) F 2 = E[ξ1, 4 ξ 2, 2 ξ 3 ] is a sub A(2) -comodule algebra, but not a sub coalgebra. Robert Bruner (WSU and UiO) tmf at p = 2 INI 46 / 65

42 The Davis-Mahowald spectral sequence Main example, cont. Resolve by A(2) -comodule algebra R = F 2 [x 4, x 6, x 7 ] with coaction ν(x 4 ) =1 x 4 ν(x 6 ) =1 x 6 + ξ 2 1 x 4 ν(x 7 ) =1 x 7 + ξ 1 x 6 + ξ 2 x 4. Resolution F 2 Ω R has differential d(ξ 4 1) =x 4 d( ξ 2 2) =x 6 d( ξ 3 ) =x 7 Robert Bruner (WSU and UiO) tmf at p = 2 INI 47 / 65

43 The Davis-Mahowald spectral sequence σ = 0 R 0 = F 2 E 0,, 1 = Ext, A(1) (F 2, F 2 ) = ko,. s 8 w 1 v 4 v w t s Robert Bruner (WSU and UiO) tmf at p = 2 INI 48 / 65

44 The Davis-Mahowald spectral sequence σ = 1 R 1 = F 2 {x 4, x 6, x 7 } = Σ 4 H (S η e 2 2 e 3 ). E 1,, 1 = Ext, A(1) (F 2, R 1 ) = ksp, {h 2 }. s 8 4 w 1 h 2 w 1 v h 2 v h 2 h t s Robert Bruner (WSU and UiO) tmf at p = 2 INI 49 / 65

45 The Davis-Mahowald spectral sequence σ = 2 R 2 = F 2 {x 2 4, x 4x 6, x 4 x 7, x 2 6, x 6x 7, x 2 7 }. E 2,, 1 = Ext, A(1) (F 2, R 2 ) = G, 2 {h2 2 }. s 8 4 h 2 2 a 2,0 h t s Robert Bruner (WSU and UiO) tmf at p = 2 INI 50 / 65

46 The Davis-Mahowald spectral sequence σ = 3 dim R 3 = 10. E 3,, 1 = Ext, A(1) (F 2, R 3 ) = G, 3 {h3 2 }. s 8 4 h 3 2 a 3,0 h t Robert Bruner (WSU and UiO) tmf at p = 2 INI 51 / 65

47 Ext A(2) (F 2, F 2 ) Ext A(2) Theorem (Shimada and Iwai) The cohomology of A(2) is Ext A(2) (F 2, F 2 ) = F 2 [h 0, h 1, h 2, c 0, d 0, e 0, g, α, β, γ, δ, w 1, w 2 ]/I. The ideal I has 54 generators: h 0 h 1, h 1 h 2, h0 2h 2 h1 3, h 0h2 2, h c 0 γ h 1 δ, βγ g 2, d0 2 gw 1, γδ h 1 c 0 w 2, γ 2 h1 2w 2 gβ 2, α 4 h0 4w 2 w 1 g 2 Robert Bruner (WSU and UiO) tmf at p = 2 INI 53 / 65

48 Ext A(2) (F 2, F 2 ) Free over F 2 [w 1, w 2 ]; here w 1 and w 2 restrict to v 4 1 and v 8 2, resp. A sum of cyclic R = F 2 [g, w 1, w 2 ]-modules isomorphic to R, R/(g) and R/(g 2 ). Four infinite families, h i 0 αj, i 0, 0 j 3. Thirty-two other summands. E 3, E 4 and E 5 = E are then modules over R 1 = F 2 [g, w 1, w2 2] and R 2 = F 2 [g, w 1, w2 4 ] resp. Mostly cyclic. Robert Bruner (WSU and UiO) tmf at p = 2 INI 54 / 65

49 Ext A(2) (F 2, F 2 ) R 0 generators of Ext A(2) (F 2, F 2 ) No circle indicates an R 0, one circle an R 0 /(g), and two circles an R 0 /(g 2 ). 12 α 2 e0 α 3 d0γ e0γ 8 d0e0 αd0 αe0 δ α 2 αβ β 2 γ 4 d0 e0 c0 α β Robert Bruner (WSU and UiO) tmf at p = 2 INI 55 / 65

50 Key Differentials First differentials Squaring operations in Ext quickly give us quite a few differentials. d 2 (α) = h 2 w 1 and d 2 (β) = h 0 d 0 d 3 (α 2 ) = h 1 d 0 w 1 and d 3 (β 2 ) = h 1 gw 1 d 3 (w2 2) = Sq9 (d 2 (w 2 )) From these many others follow by the Leibniz rule. Robert Bruner (WSU and UiO) tmf at p = 2 INI 57 / 65

51 Key Differentials Key differentials There are three hard differentials, from which we can deduce everything else using the product structure. They are Theorem d 3 (e 0 ) = c 0 w 1 d 4 (e 0 g) = gw 2 1 d 2 (w 2 ) = αβg Robert Bruner (WSU and UiO) tmf at p = 2 INI 58 / 65

52 Key Differentials d 3 (e 0 ) = c 0 w 1 The Im(J) generator ρ π 15 (S) in Adams filtration 4 must either map to 0 or ηκ in π 15 (tmf ). ηρ π (S) is detected by {Pc 0 } in π (S), which maps to c 0 w 1 in Ext A(2) (F 2, F 2 ). η 2 κ = 0 in π (S) (Toda). c 0 w 1 must be a boundary and d 3 (e 0 ) is the only chance. Robert Bruner (WSU and UiO) tmf at p = 2 INI 59 / 65

53 Key Differentials 0 to c 0w 1 d 0w 1 α 2 4 α β e 0 g Robert Bruner (WSU and UiO) tmf at p = 2 INI 60 / 65

54 Key Differentials d 4 (e 0 g) = gw 2 1 η 2 κ is detected by Pd 0 in π 22 (S) (Barratt-Mahowald-Tangora, Mimura?). This maps to d 0 w 1 in Ext A(2) (F 2, F 2 ). κ η 2 κ = 0 since η 2 κ = 0 κ η 2 κ is detected by d 0 Pd 0 which maps to d 2 0 w 1 = gw 2 1 in Ext A(2) (F 2, F 2 ). d 4 (e 0 g) is the only class which can hit it. Robert Bruner (WSU and UiO) tmf at p = 2 INI 61 / 65

55 Key Differentials 24 to gw e 0g α 2 β 2 Most differentials not shown Robert Bruner (WSU and UiO) tmf at p = 2 INI 62 / 65

56 Key Differentials d 2 (w 2 ) = αβg Corollary d 4 (d 0 e 0 ) = d 0 w1 2 and d 4(β 2 g) = α 2 e 0 w 1 and these are both nonzero. Theorem d 4 (h1 2w 2) = α 2 e 0 w 1, d 2 (w 2 ) = αβg, and d 3 (h 1 w 2 ) = g 2 w 1. γ must live to at least E 6, so d 4 (γ 2 ) = 0 γ 2 = h 2 1 w 2 + β 2 g, so d 4 (h 2 1 w 2) = α 2 e 0 w 1 0 If d 2 (w 2 ) = 0 then d 4 (h 2 1 w 2) = 0, contradiction, and d 2 (w 2 ) = αβg is the only possibility. If d 3 (h 1 w 2 ) = 0 then d 4 (h 2 1 w 2) = 0, contradiction, and d 3 (h 1 w 2 ) = g 2 w 1 is the only possibility. Robert Bruner (WSU and UiO) tmf at p = 2 INI 63 / 65

57 Key Differentials 34 to 58 s d 0e 0g d 0g 2 α 3 g 10 αd 0g αe 0g αg 2 α 2 e 0 α 2 g αβg β 2 g d 0γ e 0γ γg e 0g w2 g 2 Most differentials not shown βg t s Robert Bruner (WSU and UiO) tmf at p = 2 INI 64 / 65

58 Key Differentials Thank you Robert Bruner (WSU and UiO) tmf at p = 2 INI 65 / 65