Beta families arising from a $v_2^9$ self map on $S/(3,v_1^8)$

We show that $v_2^9$ is a permanent cycle in the 3-primary Adams-Novikov spectral sequence computing $\pi_*(S/(3,v_1^8))$, and use this to conclude that the families $\beta_{9t+3/i}$ for $i=1,2$, $\beta_{9t+6/i}$ for $i=1,2,3$, $\beta_{9t+9/i}$ for $i=1,\dots,8$, $\alpha_1\beta_{9t+3/3}$, and $\alpha_1\beta_{9t+7}$ are permanent cycles in the 3-primary Adams-Novikov spectral sequence for the sphere for all $t\geq 0$. We use a computer program by Wang to determine the additive and partial multiplicative structure of the Adams-Novikov $E_2$ page for the sphere in relevant degrees. The $i=1$ cases recover previously known results of Behrens-Pemmaraju and the second author. The results about $\beta_{9t+3/3}$, $\beta_{9t+6/3}$ and $\beta_{9t+8/9}$ were previously claimed by the second author; the computer calculations allow us to give a more direct proof. As an application, we determine the image of the Hurewicz map $\pi_*S \to \pi_*tmf$ at $p=3$.

For i ≤ 7 and β 7 + cβ 9/9 , these assertions can be read off the exhaustive calculation [Rav86, Table A3.4] of π n S ∧ 3 in stems n ≤ 108; see also [Oka81] for many of the survival results and [Shi97] for the non-survival results.The element β 7 + cβ 9/9 is an Arf invariant class (an odd-primary analogue of the p = 2 Kervaire invariant classes), discussed in [Rav78,p.439];the survival of the Arf invariant classes is not known in general at p = 3.The survival of β 9/j for j ≤ 8 is a consequence of Theorem 5.1, which does not depend on prior knowledge about this element, but we do not claim originality for this result.
These results arise from exhaustive calculations in tractable stems, but it is possible to prove results about β elements outside the range of feasible computation.One strategy is as follows.Suppose β i/j is a permanent cycle.It is v 1 -power-torsion; that is, there exists a type 2 complex V such that β i/j factors as S → S (we omit degree shifts for clarity of notation).If v t 2 is a v 2 self map on V , then we may construct elements of π * (S) as follows: For this family to be of interest, one must also show that the elements are nonzero, for example by identifying their Adams-Novikov representatives.
Following the strategy outlined above, much of the work involves showing that v 9 2 is a permanent cycle in the Adams-Novikov spectral sequence computing π * (S/(3, v 8 1 )) (Theorem 4.6).All of our explicit calculations are in the Adams-Novikov spectral sequence for S/3.To relate this to S/(3, v 8 1 ) we use a lemma due to the second author (see Lemma 4.1) that relates v m 1 -extensions in the Adams-Novikov spectral sequence for S/3 to differentials in the Adams-Novikov spectral sequence for S/(3, v m 1 ).Combined with Oka's result [Oka79] that S/(3, v m 1 ) is a ring spectrum for m ≥ 2, this implies the existence of a v 9 2 -self-map.
There is also a similar result for m = 9, but correction terms for v 9 2 are needed; see Remark 4.8.

Our proof that v 9
2 is a permanent cycle in the Adams-Novikov spectral sequence computing π * (S/(3, v 8 1 )) relies on analysis of the 143-stem in the Adams-Novikov spectral sequence for the sphere.This is greatly aided by software written by Wang [Wan;Wan21], which computes the E 2 page of the Adams-Novikov spectral sequence for the sphere using the algebraic Novikov spectral sequence.In addition, the software computes multiplication by p, α 1 , and arbitrary β i/j elements.Wang's software was originally written for use at p = 2; the minor modifications we used to change the prime are available at [BWb] and data, charts, and more documentation are available at [BWa].The calculations that make use of computer data occur solely in Section 3.
We now comment on the overlap between this work and the preprint [Shi06] by the second author: both works construct v 9 2 and the families β 9t+3/3 , β 9t+6/3 , and β 9t+9/8 , but we find the methods here to be more straightforward.The earlier preprint uses the machinery of infinite descent to control the complexity of the Adams-Novikov spectral sequence, while we opt to work directly with the Adams-Novikov E 2 page, controlling the complexity using Wang's program.In particular, our analysis in Section 3, which is the crucial input for the construction of v 9 2 , follows from the β 1 -multiplication structure given by the computer calculations, as most of the elements in play are highly β 1 -divisible.
We conclude this section by giving an outline of the rest of the paper.In Section 2 we state notational conventions for the rest of the paper and write down some easy facts about the Adams-Novikov spectral sequence that will be used extensively in the remaining sections.Most of the work for proving Theorem 4.6 occurs in Section 3, which makes use of computer calculations to determine the Adams-Novikov spectral sequence for S/3 near the 143 stem.Theorem 4.6, which constructs v 9 2 , is proved in Section 4. In Section 5 we prove Theorem 5.1, which constructs the promised β families.This involves explicit calculations in a tractable range of stems to prove that v 2 1 v 3 2 , v 1 v 6 2 , and )) are permanent cycles.In Section 6 we determine the 3-primary Hurewicz image of tmf (Theorem 6.5).

Acknowledgements:
The first author would like to thank Paul Goerss for suggesting this project, and for many helpful conversations along the way.She would also like to thank Guozhen Wang for explaining some aspects of his program, and for some code changes.

Notation and preliminaries
At a fixed prime p, the Brown-Peterson spectrum BP has coefficient ring converges.Henceforth everything will be implicitly localized at the prime p = 3.The E 2 page of (2.1) can be calculated as the cohomology of the normalized cobar complex though this is not an efficient means of computation.See [Goe07], [Rav86, §4.3, §4.4] for further background on the Adams-Novikov spectral sequence.
Let E s,f r (X) denote the E r page of (2.1), restricted to stem s and Adams-Novikov filtration f .We say that an element in π s X is detected in filtration f if it is represented by a nonzero class in E s,f ∞ (X).Throughout, any equality of homotopy or E 2 page elements should be understood to be true up to units (that is, up to signs).
We will make frequent use of the cofiber sequence We will also consider the cofiber sequences for m ≥ 1, eventually focusing primarily on m = 8.Henceforth degree shifts in cofiber and long exact sequences will usually not be shown.The maps i, j, i m , and j m induce maps of Adams-Novikov spectral sequences, which we will denote with the same letters.We note the effect on degrees: given x ∈ E s,f 2 (S/3), we have j(x) (S/3).The maps i and i m preserve degrees.Sometimes we omit applications of i or i m in the notation for brevity; for example, we write β 1 ∈ E 10,2 2 (S/3) to refer to i(β 1 ).This is justified by regarding E 2 (S/3) as a module over E 2 (S).
By the Geometric Boundary Theorem (see [Rav86,  Notation 2.2. (1) If x ∈ E s,f 2 (S) is 3-torsion, we will let x ∈ E s+1,f −1 2 (S/3) denote a class such that j(x) = x.Note that x may not always be uniquely determined.
In the rest of this section we present some preliminaries that are important for working with the Adams-Novikov spectral sequences for S, S/3, and S/(3, v m 1 ).All of these facts are well-known, and the rest of this section can be skipped by a knowledgable reader.First we recall some frequently-encountered permanent cycles in the 3-primary Adams-Novikov spectral sequence for the sphere.The comparisons below to the Adams spectral sequence are not needed in the rest of the paper, and are just presented for those readers who are more familiar with the Adams elements; a reference for computational facts about the Adams spectral sequence E 2 page is [Rav86, §3.4], and for the corresponding Adams-Novikov elements is [MRW77] or [Goe07,§6] for the Greek letter construction and [Rav86, Theorem 4.4.20]for low stems.
2 (S) is called k = k 0 = h 0 , h 1 , h 1 in the Adams spectral sequence.(This does not correspond to an Adams-Novikov Massey product since h 1 does not exist in the Adams-Novikov E 2 page.) The 0-line (generated by just 1 ∈ E * ,0 2 (S)) and the 1-line E * ,1 2 (S) consist of the image of the J homomorphism.These classes are all permanent cycles; the image of the 1-line under i is Proof.See [Rav86, Proposition 4.4.2]for the statement about X = S.This sparseness for the sphere also implies the first statement for X = S/3 and S/(3, v m 1 ), as can be seen by looking at the degrees of the long exact sequences in Ext groups corresponding to the short . The second statement follows from the first.
Most of our calculations in the Adams-Novikov spectral sequence for S/(3, v m 1 ) for m ≥ 2 implicitly use the following fact.
It is also well-known that S/3 is a ring spectrum.
(1) If there is a nontrivial differential d 5 (x) = y in E 5 (S) where y is not 3-divisible, then there is a nontrivial differential d 5 (i(x)) = i(y) in E 5 (S/3).
We use the following lemma without further mention when working with β elements, applying it to the case Proof.The map of short exact sequences induces a map of long exact sequences after applying Ext BP * BP (BP * , −).In particular, we have a commutative diagram as follows.

Computer-assisted calculations in the 143-stem
In this section we study the Adams-Novikov spectral sequence for S/3 in the 143-stem and nearby stems; this is the main technical input needed for Theorem 4.6.We make use of computer calculations of the Adams-Novikov E 2 page for the sphere; the specific facts from the computer data we use are given in Lemma 3.1.The results from this section that are used 6 later are Lemma 3.2 and Proposition 3.7.The former follows immediately from the F 3 -vector space structure of E * , * 2 (S).The rest of the section is devoted to proving the latter, which says that every permanent cycle in π 143 (S/3) is detected in filtration ≤ 5.This requires more careful analysis using the multiplicative structure of the E 2 page.Lemmas 3.5 and 3.6 give the differentials responsible for killing higher filtration elements in E 143, * 2 (S/3).
(6) Figure 4 displays the vector space structure of E s,f 2 (S) for 140 ≤ s ≤ 144, as well as selected multiplicative structure.
In Figure 4, the names in E 141,15 2 (S) follow from the proof of Lemma 3.3; other names are multiplications computed using Wang's program.
See Lemma 3.1 for element definitions.The second sentence is used implicitly when identifying one of the generators of E 142,14 2 (S) as α 1 β 4 1 x 99 (as seen in Figure 4): Wang's program only shows that there is a nonzero element in E 142,14 2 (S) that is α 1 β 4 1 times an element of E 99,5 2 (S).
Table 2. E s,f 2 (S) in degrees 95 ≤ s ≤ 102, 4 ≤ f ≤ 10. Brown lines represent α 1 -multiplication.Each dot represents a copy of F 3 .The information in this chart used in the proof of Lemma 3.4 is summarized in Lemma 3.1(2)(3).
Proof.We list the elements in E 143,f 2 (S/3) for f > 5.
Filtration f # bottom cell generators # top cell generators 9 1 1 13 0 2 17 1 0 We encourage the reader to refer to Figure 5, which is derived from Figure 4, alongside the rest of the proof.
Filtration 9: We claim that both classes E 143,9 2 (S/3) support d 5 differentials.The bottom cell class does so because of Lemmas 3.6 and 2.6(1), and the top cell class does so because of Lemma 3.5 and Lemma 2.6(2).
β families arising from a v 9 2 self map on S/(3, v 8 1 ) By Lemma 3.6 we have a class t 2 ∈ E 143,5 2 (S) such that d 9 (t 2 ) = β 6 1 β 6/3 .Let t 2 be the top cell class in E 144,4 2 (S/3) associated to the 3-torsion element t 2 .We wish to show that there is a differential d 9 (t 2 ) = β 6 1 β 6/3 .First we check that t 2 survives to the E 9 page.The only possible targets for such a shorter differential are in E 143,9 2 (S/3), and we showed above that these both support nontrivial d 5 differentials.The map induced by j on E 2 pages shows that d 9 (t 2 ) ≡ β 6 1 β 6/3 modulo ker(j).We have E 143,13 9 (S/3) = F 3 {β 6 1 β 6/3 }, and j(β 6 1 β 6/3 ) = β 6 1 β 6/3 which is nonzero in E 9 (S).Thus there is a nonzero d 9 differential as claimed.Remark 3.8.The dependence of Proposition 3.7 on computer calculations would be reduced if we could make precise the observation that much of the Adams-Novikov E 2 -page is β 1periodic, and classes in high filtrations are highly β 1 -divisible.Using [Pal01, Theorem 2.3.1,Remark 2.3.5(c)],one can prove that multiplication by β 1 is an isomorphism on the Adams E 2 page restricted to Adams filtration f A , stem s, and filtration ν in the algebraic Novikov spectral sequence Ext * , * A (F 3 , F 3 ) =⇒ E * , * 2 (S) if f A > 1 23 s + 24 23 ν + 159 23 .By keeping track of the effect on the algebraic Novikov spectral sequence, one can derive that β 1 acts injectively (up to higher algebraic Novikov filtration) on the subspace of E s,f 2 (S) in algebraic Novikov filtration ν if (3.1) f > 1 23 s + 1 23 ν + 169 23 .Surjectivity is harder to prove.Even if we knew that β 1 acted isomorphically on the region (3.1) (which is often true), this is not enough to prove the β 1 -divisibility results we need.For example, in Proposition 3.7 we use the fact that the generator x of E 143,17 2 (S) is divisible by β 6 1 .This element has ν = 0, and β −1 1 x and β −2 1 x lie in the region (3.1) but β −3 1 x does not.Improving this bound would also be of use more generally to the study of the 3-primary Adams and Adams-Novikov spectral sequences.In this section, we prove Theorem 4.6, which says that v 9 2 is a permanent cycle in E 2 (S/(3, v 8 1 )).We first explain the choice of exponent of v 1 .Since η R (v 2 ) ≡ v 2 + v 1 t 3 1 − v 3 1 t 1 (mod 3) in the Hopf algebroid (BP * , BP * BP ) (see e.g.[Rav86, (6.4.16)]), we have that v 3 2 is an element of E 2 (S/(3, v m 1 )) for m ≤ 3, and v 9 2 is an element of E 2 (S/(3, v m 1 )) for m ≤ 9. On the other hand, we would like to work with m ≥ 8, since those are the values of m for which β 9/8 is in the image of the composition of Adams-Novikov E 2 page boundary maps E 2 (S/(3, v m 1 )) → E 2 (S/3) → E 2 (S).Trivial modifications to the work in this section show that v 9 2 ± v 8 1 v 7 2 is a self-map on S/(3, v 9 1 ); see Remark 4.8.However, this slight strengthening is not necessary for our purposes, and we write down our results for v 9 2 ∈ π * (S/(3, v 8 1 )) essentially for cosmetic reasons, avoiding the correction term.To obtain the families in Theorem 5.1 other than β 9t+9/j , it suffices to work with S/(3, v 4 1 ).
The main ingredients for proving Theorem 4.6 are Lemma 3.2 and Proposition 3.7 from the previous section, and the following lemma (below, specialized to our setting) due to the second author.It draws a connection between hidden v 8 1 -extensions in π * (S/3), and differentials of the minimum length (i.e., d 5 differentials) in the Adams-Novikov spectral sequence for S/(3, v 8 1 ).
5 (S/3), and let w denote an element in E 5 (S/3) detecting the product v m 1 • {j m (y)} ∈ π * (S/3).Then there is a differential In the next lemma we separate out the general strategy used to prove that v 9 2 and other elements in Section 5 are permanent cycles.
(1) Let x ∈ E s,f 2 (S/3) for f ≤ 3 be an element such that j(x) ∈ E s−1,f +1 2 (S) is a permanent cycle and {j(x)} ∈ π s−1 (S) is an essential element of order 3. Furthermore, suppose that Im(i : Proof.We just prove (1), as (2) is analogous.Consider the exact sequences (For the first long exact sequence, we are using the fact that j induces the zero map in BP -homology.)Suppose that x ∈ E s,f 2 (S/3) is an element such that j(x) is a permanent cycle with 3 • {j(x)} = 0. Then there exists an element ξ ∈ π s (S/3) such that j(ξ) = {j(x)}.Since j : S/3 → ΣS induces a map of Adams-Novikov spectral sequences, the induced map on homotopy j : π * (S/3) → π * (ΣS) = π * −1 (S) respects Adams-Novikov filtration; thus j(x) being detected in filtration f + 1 implies ξ is detected in filtration ≤ f .The assumption f ≤ 3 combined with Fact 2.3 implies that ξ is detected in filtration f .We may write the detecting element as x + y for some y ∈ E s,f 2 (S/3).By the Geometric Boundary Theorem, j(x + y) converges to j(ξ), and we also have that j(x) converges to j(ξ).So j(y) is a boundary.But j(y) has filtration ≤ 4, so Fact 2.3 implies j(y) = 0 in E 2 (S).By (4.1), we have that y is in the image of i.By the assumption about Im(i), y is a permanent cycle, and we have from above that x + y is a permanent cycle.Therefore, x is a permanent cycle.
Proof.This will follow from applying Lemma 4.2(2) to v 9 2 .The first two hypotheses of that lemma are satisfied due to Lemma 4.3 and Lemma 4.5.For the last hypothesis, note that E 144,0 2 (S/3) is generated by v 36 1 , which is a permanent cycle.Proof.Naturality of the map S/(3, v 8 1 ) → S/(3, v m 1 ) for m ≤ 8 means that Theorem 4.6 directly implies v 9 2 ∈ E 144,0 2 (S/(3, v m 1 )) is a permanent cycle.By [Oka79, Theorem 6.1], R = S/(3, v m 1 ) is a (homotopy) ring spectrum for m ≥ 2. Thus the desired self map may be obtained as 2 ) is a permanent cycle.The proofs of Lemmas 4.4 and 4.5 go through without modification to show that 2 )} = 0.In the proof of Theorem 4.6, we have This time, v 9 2 and v 8 1 v 7 2 are not permanent cycles since β 9/9 = j(j 9 (v 9 2 )) and β 7 = j(j 9 (v 8 1 v 7 2 )) are not permanent cycles, and v 36 1 is a permanent cycle.Thus v 9 2 ± v 8 1 v 7 2 is a permanent cycle for some choice of sign.
β families arising from a v 9 2 self map on S/(3, v 8 1 ) (see [Bau08,§3]); this is the Y (4)-based Adams spectral sequence for tmf , where Y (4) is the Thom spectrum of ΩU (4) → Z × BU .We will denote this spectral sequence by E ell r (tmf ).We will show (Theorem 6.5) that all classes in filtration ≥ 2 are in the Hurewicz image, and the only classes in filtrations 0 and 1 in the image are the summands generated by 1 and α.Instead of directly mapping to the elliptic spectral sequence, we use the K(2)-local E-based Adams spectral sequence TMF ) where E = E 2 is height 2 Morava E-theory and TMF is the periodic version of tmf .There is a map of spectral sequences E r (S) → E E r (TMF ) induced by the natural maps BP → E and S → TMF .Henn-Karamanov-Mahowald [HKM13, Theorem 1.1] completely determine E E 2 (TMF /3) and provide formulas that we use to compute the map on E 2 pages E 2 (S) → E E 2 (TMF ) in cases of interest (see Lemmas 6.2 and 6.3).For each class in E E 2 (TMF ) in filtration ≥ 2, we identify a preimage in E 2 (S) that is among the classes proved to be permanent cycles in Theorem 5.1 or [BP04] (see Proposition 6.4).As we explain in the proof of Theorem 6.5, it suffices to understand the Hurewicz image in π * (L K(2) TMF ) because there is an injection π * (tmf ) → π * (L K(2) TMF ) (see Lemma 6.6).
First we review some notation and basic facts.We have and there is a natural map BP * → E * that sends v 1 → u 1 u −2 , v 2 → u −8 , and v i → 0 for i > 2. Abusing notation, we will let v i denote its image in E * /3.
By our convention about naming elements in the image of the map BP * → E * , we have Proposition 6.4.For t ≥ 0 the map H : E 2 (S) → E E 2 (TMF ) satisfies: (1) Proof.These statements are all proved the same way; we show (2).First observe that Lemma 6.2 implies v 9 2 ≡ −∆ 6 (mod (3, v 6 1 )).Using Lemma 6.2 and Lemma 6.3 we have: In the last line we are using the fact that j 3 (v 2 1 v 2 ) = α in E E 2 (TMF /3) from Lemma 6.3.
In the next theorem, we show that every element in π * tmf detected in filtration ≥ 2 is in the Hurewicz image.This result is stated without proof in [Hen14, §1], but we do not know of any prior proof in the literature.
Theorem 6.5.The image of the map h : π * S → π * tmf at p = 3 consists of the Z (3) summand generated by 1 and the F 3 summands generated by α, ∆ 3t β i for 1 ≤ i ≤ 4, ∆ 3t αβ, ∆ 3t βb for t ≥ 0.More precisely, we have Proof.Let E ell 2 (tmf ) denote the elliptic spectral sequence for tmf (see [Bau08,§6]); recall this is the Y (4)-based Adams spectral sequence for tmf .There is a map of spectral sequences L : E ell r (tmf ) → E E r (TMF ) that comes from the map on Adams towers induced by the maps Y (4) → M U P → E (where M U P denotes periodic M U ) and tmf → TMF .These maps assemble into a diagram of spectral sequences as follows. (6.1) tmf ) for s = 0 is in the image of h: Lemma 6.6(2) implies x would be detected in filtration 0 of E E ∞ (TMF ), and H : E ∞ (S) → E E ∞ (TMF ) is zero in filtration 0 for nonzero stems.
Next we turn to elements detected in filtration 1.We have H(α 1 ) = α by Lemma 6.3; since we have H = L • h as maps π * S → π * L K(2) TMF and L is injective by Lemma 6.6, this implies h(α 1 ) = α ∈ π * tmf .The other elements of E ell ∞ (tmf ) in filtration 1 are ∆ 3t α for t ≥ 1 and ∆ 3t b for t ≥ 0; we will show that the permanent cycles they represent are not in the Hurewicz image.By Lemma 6.6(1) they are in the image of h if and only if their images in π * L K(2) TMF are in the image of H.By Lemma 6.6(2) they are also detected in filtration 1 in E E ∞ (TMF ), so if they were in the image of H, they would be the image of a class in E 2 (S) in filtration 0 or 1.We have E s,0 2 (S) = 0 for s > 1, so it suffices to show that the elements in E s,1 2 (S) except for α 1 are in the kernel of h.If x ∈ E s,1 2 (S) with s > 3 then i(x) = α 1 v k 1 for some k ≥ 1.If h denotes the map π * (S/3) → π * (tmf /3) induced by h, we have h (α 1 v 1 ) = 0 since π 7 (tmf /3) = 0. Thus i(h(x)) = h (i(x)) = 0 in π * (S/3), which implies that h(x) is 3-divisible.But Figure 6 shows that there are no 3-divisible nonzero targets in Adams-Novikov filtration 1.
In fact, L is injective on E ∞ pages in all filtrations, but we do not need this fact.(2) Consider an element of ker(L : E ell ∞ (tmf ) → E E ∞ (TMF )) represented by x ∈ E ell 2 (tmf ) in filtration 0 or 1.We claim that x is in ker(L 2 : E ell 2 (tmf ) → E E 2 (TMF )): since L 2 (x) is in filtration 0 or 1, it cannot be the target of a d r differential for r ≥ 2. By comparing the calculations of E ell 2 (tmf ) and E E 2 (TMF /3) in [Bau08, §5] and [HKM13, Theorem 1.1], respectively, it is clear that L 2 : E ell 2 (tmf /3) → E E 2 (TMF /3) is an injection, so the image of x in E ell 2 (tmf /3) is zero, which implies (using exactness of the top row in the diagram) x ∈ E ell 2 (tmf ) is 3-divisible.
E E 2 (TMF ) i / / E E 2 (TMF /3) Since E ell 2 (tmf ) has no 3-divisible classes in filtration 1, we now focus on the filtration 0 case.Let y = x/3 n ∈ E ell 2 (tmf ) be the non-3-divisible generator, which then has nonzero image i(y) in E ell 2 (tmf /3).Since L 2 is an injection, L 2 (i(y)) = i(L 2 (y)) = 0. We claim that the (nonzero) group generated by L 2 (y) is torsion-free: if not, then the corresponding top cell class would be a nonzero class in E E 2 (TMF /3) in filtration −1, contradicting [HKM13, Theorem 1.1].So L 2 (x) = 3 n L 2 (y) = 0, contradicting the fact above that x ∈ ker(L 2 ).Remark 6.7.Our methods are not sufficient to completely determine the image of the map h : π * (S/3) → π * (tmf /3).The remaining nontrivial part of this question is to determine which elements ∆ n α are in the image.Arguments similar to those we have given in this section show that h (β 9t+2 ) = ∆ 6t+1 α and h (β 9t+5 ) = ∆ 6t+3 α.However, the families ∆ 6t α for t ≥ 1 and ∆ 6t+4 α for t ≥ 0 fit into patterns that are not described by our work in this paper.For example, ∆ 4 α ∈ π 99 (tmf /3) is not in the image of h for degree reasons.On the other hand, using the more precise definitions of the β elements in [MRW77, (2.4)] and calculating analogously to Lemma 6.3, we find that the map E 2 (S/3) → E E 2 (TMF /3) sends β 18/11 to ∆ 10 α.As we do not know if β 18/11 is a permanent cycle, we are unable to conclude whether ∆ 10 α is in the image of π * (S/3).

Figure 6 .
Figure 6.The E ∞ page of the elliptic spectral sequence computing π s tmf for 0 ≤ s ≤ 76.Dashed brown lines represent hidden α-multiples.Squares indicate copies of Z (3) and dots indicate copies of F 3 .
Theorem 2.3.4]), the maps induced by j and j m on E 2 pages coincide with the boundary maps in the long exact sequences of Ext groups