The 2-primary Hurewicz image of tmf

We determine the image of the 2-primary tmf-Hurewicz homomorphism, where tmf is the spectrum of topological modular forms. We do this by lifting elements of tmf_* to the homotopy groups of the generalized Moore spectrum M(8,v_1^8) using a modified form of the Adams spectral sequence and the tmf-resolution, and then proving the existence of a v_2^32-self map on M(8,v_1^8) to generate 192-periodic families in the stable homotopy groups of spheres.


Introduction
The Hurewicz theorem implies that the Hurewicz homomorphism h : π * (S n ) → H * (S n ; Z) is an isomorphism for * = n, implying the well known result that the 0th stable stem is given by π s 0 ∼ = Z.

1
The computation of the real K-theory of a point (the homotopy groups of the spectrum KO representing real K-theory) is a consequence of the Bott periodicity theorem [Bot59]: these groups are given by the following 8-fold periodic pattern.
2-locally, the homotopy groups of tmf are merely 192-periodic. These homotopy groups were originally computed by Hopkins and Mahowald [DFHH14] (see also [Bau08]). These homotopy groups are displayed in Figure 1.1. In this figure: • A series of i black dots joined by vertical lines corresponds to a factor of Z/2 i which is annihilated by some power of c 4 = v 4 1 . • An open circle corresponds to a factor of Z/2 which is not annihilated by a power of c 4 . • A box indicates a factor of Z (2) which is not annihilated by a power of c 4 . • The non-vertical lines indicate multiplication by η and ν.
• A pattern with a dotted box around it and an arrow emanating from the right face indicates this pattern continues indefinitely to the right by c 4multiplication (i.e. tensor the pattern with Z (2) [c 4 ]). 1 Here, tmf denotes connective topological modular forms. 2 The 3-primary Hurewicz image has also not been resolved, but would follow from the results in a recent preprint of Shimomura [Shi]. Since π * tmf (p) has no torsion for p ≥ 5, the p-primary tmf-Hurewicz image is trivial in positive degrees for these primes. After localization at the prime 2, the element ∆ 8 = v 32 2 is a permanent cycle in the descent spectral sequence, and π * tmf is given by tensoring the pattern depicted in Figure 1.1 with Z[∆ 8 ]. Our choice of names for generators in Figure 1.1 is motivated by the fact that the elements η, ν, , κ,κ, q, u, w in the stable stems map to the corresponding elements in π * tmf under the tmf-Hurewicz homomorphism.
The main theorem of this paper is the following.
Besides representing an advance in our understanding of v 2 -periodic homotopy at the prime 2, Theorem 1.2 also has applications to smooth structures on spheres, as explained in [BHHM20]. Specifically, Hill, Hopkins, and the first two authors consider the following question. Question 1.3. In which dimensions n do there exist exotic smooth structures on the n-sphere?
Such spheres with exotic smooth structures are called exotic spheres. The work of Kervaire and Milnor [KM63] relates the existence of exotic spheres to the triviality of the Kervaire homomorphism π s 4k+2 → Z/2 and the non-triviality of the cokernel of the J-homomorphism J : π n SO → π s n . Specifically, they prove that exotic spheres exist in dimensions n for which n = 4k: n ≥ 8 and there exists a non-trivial element of coker J, n = 4k + 1: there exists a non-trivial element of coker J, or there does not exist an element of Kervaire invariant 1 in dimension n + 1, n = 4k + 2: there exists a non-trivial element of coker J with Kervaire invariant 0, n = 4k + 3: n ≥ 7.
In the case of n = 8k + 2 ≥ 10, Adams' elements µ 8k+2 with non-trivial KO-Hurewicz image are not in the image of J and have trivial Kervaire invariant. It thus follows that: there exist exotic spheres in all dimensions n = 8k + 2 ≥ 10.
As is explained in [BHHM20], many of the 192-periodic families of elements of Theorem 1.2 also are not in the image of J and have trivial Kervaire invariant. Theorem 1.2 therefore has the following corollary. 3 Corollary 1.4. There exist exotic spheres in the following congruence classes of even dimensions n ≥ 8 modulo 192: (This accounts for over half of the even dimensions.) We will prove Theorem 1.2 by first showing (Theorem 6.1) that the elements of π * tmf not in the subgroup described by Theorem 1.2 are not in the Hurewicz image. This will be a relatively straightforward consequence of some v 1 -periodic computations. The elements of Theorem 1.2(1) are already established to be in the Hurewicz image by the preceding discussion, and the elements (2) are in the Hurewicz image because they are the images of the elements µ 8i+j . We are left to show that the elements of type (3) lift to π s * . This is the main task of this paper.
In [BR], Bruner and Rognes give a systematic and careful study of the Adams spectral sequence for tmf, and in particular they have independently established the Hurewicz image in many low-dimensional cases. Specifically, they prove Theorem 1.2 for degrees * ≤ 101 and also show that wκ 3 , w 2κ , wκ 4 , 2∆ 4 κκ, and 4∆ 6 ν 2 (in dimensions 105, 110, 125, 130, and 150) are in the Hurewicz image. Also, they use a different technique (Anderson duality) to prove that the Hurewicz image is contained in the subgroup of tmf * described in Theorem 1.2.
Our strategy to lift elements from π * tmf to π s * is to use the methods of [BHHM20]. We summarize that strategy here. We recall the following from [BHHM20, Prop. 6.1].
Let M (2 i ) denote the cofiber of 2 i , and let M (2 i , v j 1 ) denote the cofiber of a v 1 -self map v j 1 : Σ 2j M (2 i ) → M (2 i ).
3 In fact, the v 32 2 -self map of Theorem 1.7 which is used to construct the periodic families of Theorem 1.2 also immediately implies the existence of some elements not in the image of the J-homomorphism which are in the kernel of the tmf-Hurewicz homomorphism, such as the beta elements β 32k/8 . However, we will not concern ourselves here with the few additional dimensions such considerations add to the list of Corollary 1.4. Corollary 1.6. Every v 1 -torsion element x ∈ π * (tmf) lifts to an element x ∈ tmf * M (8, v 8 1 ) so that the projection to the top cell maps x to x.
Given a v 1 -torsion element x ∈ π <192 (tmf), Proposition 1.5 implies it lifts to an element x ∈ tmf * M (8, v 8 1 ) so that the projection to the top cell maps x to x. We will then show that x lifts to an element y ∈ π * M (8, v 8 1 ). Then the image y ∈ π s * given by projecting y to the top cell is an element whose image under the tmf-Hurewicz homomorphism is x.
Every v 1 -torsion element x ∈ π ≥192 tmf is of the form v 32k 2 x for x ∈ π <192 tmf. We will prove the following theorem. If x ∈ tmf * M (8, v 8 1 ) is a lift of x, and y ∈ π * M (8, v 8 1 ) is a lift of x, as in the discussion above, then the resulting element v 32k 2 y ∈ π * M (8, v 8 1 ), obtained by composing with the k-fold iterate of the v 32 2 -self map, projects to an element y ∈ π s * which maps to x under the tmf-Hurewicz homomorphism.
As in [BHHM20], the analysis above rests on a systematic analysis of the homotopy groups π * M (8, v 8 1 ). This will be based on computations using the modified Adams spectral sequence (MASS). The E 2 -term of the modified Adams spectral sequence will be analyzed in a region near its vanishing line by means of another spectral sequence, the algebraic tmf resolution.
The work of [BHHM20] was hampered by the fact that all of the algebraic tmf resolution computations were performed on the level of the E 1 -term of the algebraic tmf resolution. In this paper, we will show that the weight spectral sequence, used in the context of bo-resolutions by [LM87] and [BBB + 20], can be used to analyze the E 2 -term of the algebraic tmf resolution, greatly simplifying the computations.

Conventions.
• Homology will be implicitly taken with mod 2 coefficients.
• We let A * denote the dual Steenrod algebra, A / / A(2) * denote the dual of the Hopf algebra quotient A / / A(2), and for an A * -comodule M (or more generally an object of the stable homotopy category of A * -comodules [Hov04]) we let Ext s,t A * (M ) denote the group Ext s,t A * (F 2 , M ). • Given a Hopf algebroid (B, Γ), and a comodule M , we will let C * Γ (M ) denote the associated normalized cobar complex.
• For a spectrum E, we let E * denote its homotopy groups π * E.
Outline of paper. In Section 2, we recall the modified Adams spectral sequence (MASS), which takes the form mass E * , * 2 = Ext A * (H * X ⊗ H(8, v 8 1 )) ⇒ π * (X ∧ M (8, v 8 1 )) for a certain object H(8, v 8 1 ) in the stable homotopy category of A * -comodules. We recall how the E 2 -term of the MASS can be studied using the algebraic tmf resolution, which is a spectral sequence that takes the form tmf alg E 1 (M ) * , * , * ⇒ Ext * , * A * (M ) for any M in the stable category of A * -comodules. We then recall how the E 1 -term of the algebraic tmf resolution decomposes as a sum of Ext groups involving tensor powers of bo-Brown-Gitler comodules, and also summarize an inductive method to compute these Ext groups.
In Section 3, we study the d 1 differential in the algebraic tmf resolution for F 2 , and introduce a tool, the weight spectral sequence (WSS) tmf alg E 1 = wss E 0 ⇒ tmf alg E 2 , which serves as an analog of the May spectral sequence, and converges to the E 2term of the algebraic tmf resolution. The E 0 -page of the v 0 -localized weight spectral sequence is identified with the cobar complex of a primitively generated Hopf algebra, and this allows us to give "names" to the v 0 -torsion-free classes of tmf alg E 1 . We include many charts of summands of tmf alg E 1 (F 2 ) corresponding to tensor powers of bo-Brown-Gitler comodules which illustrate this naming convention, and provide the essential data for the rest of the computations in this paper. Finally, we study the g-local WSS 4 using recent work of Bhattacharya-Bobkova-Thomas [BBT18], and show that many classes are killed in the g-local WSS by d 1 -differentials. This is the key fact we will use to systematically remove obstructions for lifting classes from tmf * X to π * X.
In Section 4 we study the structure of the MASS for M (8, v 8 1 ). We recall the structure of the MASS for tmf * M (8, v 8 1 ), and we explain how to adapt the Ext charts of Section 3 to give the corresponding computations of tmf alg E 1 (H(8, v 8 1 )). We then explain how to translate the computations of the g-localized algebraic tmf resolution of Section 3 to the case of H(8, v 8 1 ).
Section 5 is dedicated to the proof of Theorem 1.7. We recall the work of Davis, Mahowald, and Rezk, who discovered topological attaching maps between the first two bo-Brown-Gitler spectra which comprise tmf∧tmf, which give extra differentials in the Adams spectral sequence of tmf ∧tmf that kill some g-torsion-free classes. We then prove a technical lemma (Lemma 5.5) which lifts differentials from the MASS for tmf s ∧ M (8, v 8 1 ) to the MASS for M (8, v 8 1 ). We prove Theorem 1.7 by listing all elements in tmf alg E 1 (H(8, v 8 1 )) which could detect a non-trivial differential d r (v 32 2 ) in the MASS for M (8, v 8 1 ), and then we systematically eliminate these possibilities. Most of these classes are g-torsion-free, and are eliminated in the WSS, or by using Lemma 5.5.
In Section 6, we explain how v 1 -periodic computations give an upper bound on the Hurewicz image.
Section 7 is devoted to showing this upper bound is sharp, by producing lifts of the remaining elements of π * tmf to the sphere. We begin by identifying multiplicative generators of the Hurewicz image in dimensions less than 192, so that it suffices for us to lift these. We then lift these elements by producing elements in the MASS for M (8, v 8 1 ) which we show are permanent cycles, and detect elements of π * M (8, v 8 1 ) which project to the desired elements on the top cell. These elements are then propagated to v 32 2 -periodic families using the self-map, thus proving Theorem 1.2 in all dimensions. ously sharing their results on their study of the Adams spectral sequence of tmf, and also to Rognes for pointing out a redundancy in Section 7. This project would have not been possible without the Ext computational software developed by Bob Bruner and Amelia Perry, and the detailed computations of the Adams spectral sequence of the sphere by Isaksen, Wang, and Xu. The authors are especially grateful to Bob Bruner for providing them with a module definition file for A/ /A(2). The first author would also like to express his appreciation to Agnès Beaudry, Prasit Bhattacharya, Dominic Culver, Kyle Ormsby, Nat Stapleton, Vesna Stojanoska, and Zhouli Xu, whose previous collaborative work on the tmf resolution was essential for the results of this paper, as well as to Mike Hill and Mike Hopkins, whose collaboration with the first two authors was the genesis of this paper. The third author also wishes to thank the first two authors for the opportunity to contribute to this project. The first author was supported by NSF grants DMS-1050466, DMS-1452111, DMS-1547292, DMS-1611786, and DMS-2005476 over the course of this work. The third author was partially supported by NSF grant DMS-1547292.

Preliminaries
The techniques and methods of this paper closely follow those of [BHHM20]. In this section we recall some spectral sequences used in that paper.
The modified Adams spectral sequence. Our computations of π * M (8, v 8 1 ) and tmf * M (8, v 8 1 ) will be performed using the modified Adams spectral sequence (MASS). We refer the reader to [BHHM20, Sec. 6] for a complete account of the construction of the MASS and summarize the form it takes here.  It follows that if X is a ring spectrum, the MASS above is a spectral sequence of (non-associative) algebras.

Let St
We recall the following key theorem of Mathew.
Taking X = tmf ∧ Y for some Y , and applying a change of rings theorem, the MASS takes the form The algebraic tmf-resolution. The E 2 -page of the MASS for M (8, v 8 1 ) will be analyzed using an algebraic analog of the tmf-resolution (as in [BHHM20, Sec. 6]).
The (topological) tmf-resolution of a space X is the Adams spectral sequence based on the spectrum tmf: Here, tmf is the cofiber of the unit: The algebraic tmf-resolution is an algebraic analog. Namely, let M be an object of the stable homotopy category of A * -comodules, and let A/ /A(2) * denote the cokernel of the unit 0 → F 2 → A/ /A(2) * → A/ /A(2) * → 0 5 By this, we mean a spectrum with a possibly non-associative product and a two sided unit in the stable homotopy category.
(note that H * tmf = A/ /A(2) * ). The algebraic tmf-resolution of M is a spectral sequence of the form bo-Brown-Gitler comodules. We recall some material on bo-Brown-Gitler comodules. These are A * -comodules which are the homology of the bo-Brown-Gitler spectra constructed by [GJM86]. Mahowald used integral Brown-Gitler spectra to analyze the bo resolution [Mah81]. The bo-Brown-Gitler comodules play a similar role in the algebraic tmf resolution [BHHM08], [MR09], [DM10], [BOSS19], Endow the mod 2 homology of the connective real K-theory spectrum . .] with a multiplicative grading by declaring the weight of ζ i to be The ith bo-Brown-Gitler comodule is the subcomodule spanned by monomials of weight less than or equal to 4i. It is isomorphic as an A * -comodule to the homology of the ith bo-Brown-Gitler spectrum bo i .
For any M , the computation of can be inductively determined from Ext A(2) * (bo ⊗k 1 ⊗ M ) by means of a set of exact sequences of A(2) * -comodules which relate the bo i 's [BHHM08, Sec. 7] (see also [BOSS19]): Here, tmf j is the jth tmf-Brown-Gitler comodule -it is the subcomodule of spanned by monomials of weight less than or equal to 8j. 6 The exact sequences (2.7) and (2.8) can be re-expressed as resolutions in the stable homotopy category of A(2) * -comodules: which give rise to spectral sequences E n,s,t (2.9) These spectral sequences have been observed to collapse in low degrees (see [BOSS19]) but it is not known if they collapse in general. They inductively build Ext A(2) * (bo i ⊗ M ) out of Ext A(2) * (bo ⊗k 1 ⊗ M ) and Ext A(1) * (tmf j ⊗ M ).

Analysis of the algebraic tmf resolution
In this section we will compute the d 1 -differential in the algebraic tmf resolution, and will introduce a tool, the weight spectral sequence (WSS), which is a variant of the May spectral sequence that converges to the E 2 -page of the algebraic tmf resolution.
The d 1 differential in the algebraic tmf resolution. Our approach to understanding the d 1 -differential in the algebraic tmf-resolution will be to compute it on v 0 -torsion-free classes, and then infer its effect on v 0 -torsion classes by means of linearity over Ext A * (F 2 ).
Consider the algebraic BP 2 and algebraic BP -resolutions.
6 Technically speaking, as is addressed in [BHHM08, Sec. 7], the comodules A(2)/ /A(1) * ⊗tmf j−1 in the above exact sequences have to be given a slightly different A(2) * -comodule structure from the standard one arising from the tensor product. However, this different comodule structure ends up being Ext-isomorphic to the standard one. As we are only interested in Ext groups, the reader can safely ignore this subtlety.
The d 1 -differential in the algebraic tmf-resolution may be studied by means of the zig-zag where F 2 [ζ 2 1 , ζ 2 2 , · · · ] denotes the cokernel of the unit F 2 → F 2 [ζ 2 1 , ζ 2 2 , · · · ]. The Adams spectral sequences BP alg E n, * , * 1 = ass * , * E 2 (BP ∧ BP n ) ⇒ C n BP * BP (BP * ) collapse, where C * BP * BP is the normalized cobar complex for BP * BP , and ζ 2 i ∈ A/ /E * detects t i ∈ BP * BP . We conclude: Lemma 3.2. The d 1 differential in the algebraic BP resolution is the associated graded of the differential in the cobar complex for BP * BP with respect to Adams filtration.
Applying Ext A * (F 2 , −) to the resulting filtered A * -comodule produces a variant of the May spectral sequence which we will call the modified May spectral sequence The map tmf → H induces an inclusion Under this inclusion, the weight filtration restricts to a decreasing filtration on ⊗n * by A * -subcomodules. Because the weights of all of the generators of A/ /A(2) * are divisible by 8, we actually work with weights divided by 8. Applying Ext A(2) * (F 2 , −) and taking cohomology, we get the weight spectral sequence (WSS): 7 The authors of [LSWX19] construct a similar modified May spectral sequence, but with a slightly different filtration.
The WSS serves as an analog of the May spectral sequence for the algebraic tmfresolution.
The map Φ above induces a map of spectral sequences The v 0 -localized algebraic tmf resolution. Observe that we have and that there is an isomorphism We will now compute the localized E 1 -page v −1 0 wss E 1 . The following is immediate from the computation of the cobar differential (modulo terms of higher Adams filtration) on the elements ζ 8 1 and ζ 4 2 , using (3.6), (3.7), and (3.1). Proposition 3.8. There is an isomorphism of differential graded algebras v −1 0 wss E * ,n, * , * where F 2 [ζ 8 1 , ζ 4 2 ] is regarded as a primitively generated Hopf algebra. Corollary 3.9. There is an isomorphism Charts. For the convenience of the reader we include some charts of Ext A(2) * (bo k 1 ) for 0 ≤ k ≤ 3 as well as Ext A(2) * (bo 2 ).
Ext A(2) * (F 2 ) : (Figure 3.1) All of the elements are c 4 = v 4 1 -periodic, and v 8 2 -periodic. Exactly one v 4 1 multiple of each element is displayed with the • replaced by a •. Observe the wedge pattern beginning in t − s = 35. This pattern is infinite, propagated horizontally by h 2,1multiplication and vertically by v 1 -multiplication. Here, h 2,1 is the name of the generator in the May spectral sequence of bidegree (t − s, s) = (5, 1), and h 4 2,1 = g.

, 3.3, 3.4)
Every element is v 8 2 -periodic. However, unlike Ext A(2) * (F 2 ), not every element of these Ext groups is v 4 1 -periodic. Rather, it is the case that either an element   x ∈ Ext A(2) * (bo ⊗k 1 ) satisfies v 4 1 x = 0, or it is v 4 1 -periodic. Each of the v 4 1 -periodic elements fit into families which look like shifted and truncated copies of Ext A(1) * (F 2 ), and are labeled with a •. We have only included the beginning of these v 4 1 -periodic patterns in the chart. The other generators are labeled with a •. A indicates a polynomial algebra F 2 [h 2,1 ].
Ext A(2) * (bo 2 ) : (Figure 3.5) Via the spectral sequence (2.9), this Ext chart is assembled out of h 2,1 h 2,1 h 2,1 -towers. Our computations of the MASS for M (8, v 8 1 ) will rely on a detailed understanding of this spectral sequence near its vanishing line. Since M (8, v 8 1 ) is a type 2 complex, the Hopkins-Smith Periodicity Theorem [HS98] implies that the E ∞ -page of this MASS has a vanishing line of slope 1/|v 2 | = 1/6. However, g = h 4 2,1 is not nilpotent in the modified Ext groups Ext A * (H(8, v 8 1 )), and h 2,1 -multiplication has slope 1/5. The goal of this subsection is to show that many of the h 2,1 -towers in the E 1 -page of the algebraic tmf resolution actually kill each other off by the E 2 -page of the algebraic tmf resolution. We will then identify specific h 2,1 -periodic elements of Ext A * (F 2 ) that some of these remaining h 2,1 -towers detect.

Consider the quotient Hopf algebra
Proof. Since the element v 8 2 ∈ Ext A(2) * (F 2 ) maps to zero in Ext C * (F 2 ), it follows that there is a factorization in St A(2) * . Explicit computation reveals ,1 ] and it follows that the map (2)/ /C * induces an isomorphism on Ext A(2) * , and is hence an equivalence. The result follows.
Bhattacharya, Bobkova, and Thomas [BBT18] computed the P 1 2 -Margolis homology of the tmf-resolution, and in the process computed the structure of A/ /A(2) ⊗n * as C * -comodules. From this one can read off the Ext groups , which in turn determines the g-local algebraic tmf-resolution by Corollary 3.11 (the spectral sequence in this corollary will collapse in the cases we consider it).
To state the results of [BBT18] we will need to introduce some notation. The

For sets of multi-indices
with I ∩ I = ∅, let x I t I ∈ A/ /A(2) * denote the corresponding monomial. The action of the algebra E[Q 1 , P 1 2 ] on the F 2 -submodule of A/ /A(2) ⊗n * spanned by such monomials is given by For an ordered set J = ((i 1 , j 1 ), . . . , (i k , j k )) of multi-indices, let |J| := k denote the number of pairs of indices it contains. Define linearly independent sets of elements inductively as follows: for J as above with |J| odd, define Let While the set T J depends on the ordering of J, the subspace N J does not.
The following is the main theorem of [BBT18] 8 Theorem 3.12 (Bhattacharya-Bobkova-Thomas). As modules over where J ranges over the non-empty subsets of and v 1 acts trivially on N J for |J| odd. The summand In light of Lemma 3.10 and Corollary 3.11, we may refer to elements of the g-local algebraic tmf resolution as v 8j 2 z, where z is an element of the h 2,1 -localized Ext groups described in the theorem above.
Lemma 3.13. The WSS d 0 -differential on the element Proof. We use the map of spectral sequences wss E 0 → g −1 wss E 0 .
By explicit computation of g −1 Ext A(2) * (bo 2 ), under the map . Again, by explicit computation of g-local Ext groups, under the map 1 x 1,2 ). The result follows. 8 The main theorem of [BBT18] is a computation of P 1 2 -Margolis homology, but the actual content of the paper is a decomposition of A/ /A(2) * in the stable module category of E[Q 1 , P 1 2 ].
Proof. By Lemma 3.10 and Theorem 3.12, the h 2,1 -towers coming from Ext A(2) * (bo ⊗k 1 ) are supported by the elements T {(1,1),...,(1,k)} . By Lemma 3.13, the WSS d 0 induces a surjection for k = 2 For k > 2 even the WSS d 0 gives isomorphisms We shall denote the elements of the Mahowald-Tangora wedge [MT68] in Ext A * (F 2 ) by 9 v i 1 h j 2,1 g 2 , i ≥ 0, j ≥ 0. Recall that the Mahowald operator where B * is the quotient algebra This notation is slightly misleading, as there are a few wedge elements for which the P operator does not take the element we are denoting v i 1 x to the element we are denoting v i+4 1 x, but we justify this notation by the fact that the wedge elements map to elements with such names in Ext A(2) * (F 2 ). of A * [MPT70], [Isa20]. The existence of the element ∆ 2 g 2 ∈ Ext A * (F 2 ) gives elements These elements are all linearly independent, since they project to linearly independent elements of Ext B * (F 2 ).
The following proposition gives the elements of Ext A(2) * that some of the remaining h 2,1 towers in Ext A(2) * detect in the algebraic tmf resolution.
M g 2 (Note that the notation Q 2 in the above table refers to the name of the generator of Ext 7,57+7 A * (F 2 ), and not the Milnor generator Q 2 ∈ A.) Proof. The classes corresponding to ∆ 2m v i 1 h k 2,1 are clear, because they are in the image of the map In the case of the classes corresponding to ∆ 2m h k 2,1 n, ∆ 2m h k 2,1 Q 2 , we consider the h j 2,1 multiples of n, Q 2 ∈ Ext A * (F 2 ) for j ≥ 4: gn, gt, rn, mn, g 2 n, · · · , gQ 2 , gC 0 , rQ 2 , mQ 2 , g 2 Q 2 · · · .
It suffices to show that n, t, Q 2 , C 0 are detected in the algebraic tmf resolution by Examination of a computer calculation of Ext A * (A/ /A(2) ⊗2 * ) reveals that none of the elements n, t, Q 2 , C 0 are in the image of the map Since the elements n, t, Q 2 , and C 0 map to zero in Ext A(2) * (F 2 ), they must therefore be detected on the 1-line of the algebraic tmf resolution. Examination of the relevant Ext charts reveals the only possibility is for the elements to be detected by classes of the form (3.18).
If we consider the class M g ∈ Ext A * (F 2 ), one can both check that it is not in the image of (3.19), and that the only class in Ext A(2) * (A/ /A(2) * ) which can detect it is the class . It follows from the multiplicative structure of the wedge, and the fact that In this and following sections, we shall use the notation  The algebraic tmf resolution for H(8, v 8 1 ) The following lemma explains that, in our H(8, v 8 1 ) computations, we may disregard terms coming from Ext A(1) * in the sequence of spectral sequences (2.9).   For n > 0 and i 1 , . . . , i n > 0, the terms that comprise the terms in the algebraic tmf-resolution for H(8, v 8 1 ) are in some sense less complicated than Ext A(2) * (H(8, v 8 1 )).
Most of the features of these computations can already be seen in the computation of Ext A(2) * (bo 1 ⊗ H(8, v 8 1 )), which is displayed in Figure 4.3. This computation was performed by taking the computation of Ext A(2) * (bo 1 ) (see, for example, [BHHM08]) and running the long exact sequences in Ext associated to the cofiber sequences One can similarly compute Ext A(2) * (bo ⊗k 1 ⊗ H(8, v 8 1 )) for larger values of k by applying the same method to the corresponding computations of Ext A(2) * (bo ⊗k 1 ) in [BHHM08]. We do not bother to record the complete results of these computations for small values of k, but will freely use them in what follows. The spectral sequences (2.9) imply these computations control Ext A(2) * (bo I ).
h 2,1 h 2,1 h 2,1 towers in the algebraic tmf resolution for H(8, v 8 1 ) H(8, v 8 1 ) H(8, v 8 1 ). Theorem 3.12 has the following implication for the g-local algebraic tmf-resolution of H(8, v 8 1 ): where J ranges over the non-empty subsets of This leads to the following twist in the analog of Proposition 3.15.
Proof. Everything is identical to the proof of 3.15, except that the differentials now have non-trivial kernel and cokernel.
We now give elements of Ext A * (H(8, v 8 1 )) which these remaining h 2,1 -towers detect in the algebraic tmf resolution. Note that, as pointed out in [MPT70], the Mahowald operator satisfies exist, and we see they are linearly independent by mapping to Ext B * (H(8, v 8 1 )) (where B * is defined in (3.16)).
Proof. The cases of  We now endeavor to prove Theorem 1.7. We first recall the following lemma. It follows from the Leibniz rule that v 32 2 persists to the E 4 -page of the MASS for M (8, v 8 1 ). Our task will then be reduced to showing that d r (v 32 2 ) = 0 for r ≥ 4. We will do this by identifying the potential targets of such a differential, and show that they either the source or target of shorter differentials. This will necessitate lifting certain differentials from the MASS for tmf ∧ tmf n ∧ M (8, v 8 1 ) to the MASS for M (8, v 8 1 ).
As explained in [BOSS19, Sec. 7.4], work of the second author, Davis, and Rezk [MR09], [DM10] implies that the algebraic map realizes to a map where tmf ∧ tmf 2 is a spectrum built out of tmf ∧ Σ 8 bo 1 and tmf ∧ Σ 16 bo 2 . They furthermore show that there is a map which geometrically realizes the inclusion of the direct summand (2.9) The attaching map from tmf ∧ bo 2 to tmf ∧ bo 1 in the spectrum tmf ∧ tmf 2 induces d 3 -differentials from the h 2,1 -towers in bo 2 to the h 2,1 -towers in bo 1 in the ASS for tmf ∧ tmf under the map (5.2). Furthermore, there are differentials in the ASS's for tmf ∧ bo 1 , tmf ∧ bo 2 , and tmf, which induce differentials in the ASS for tmf ∧ tmf under the maps (5.2) and (5.3). We wish to study when these differentials (and more generally differentials in the ASS for tmf ∧ tmf n ) lift via the tmf resolution to differentials in the ASS for the sphere.
To this end we consider the partial totalizations The spectrum T n is a ring spectrum, and in particular has a unit S → T n .
The following lemma will be our key to lifting the desired differentials. (1) The differential d mass in the ASS for M (8, v 8 1 ) is detected by y in the algebraic tmf resolution, or (2) The element y is the target of a differential in the algebraic tmf resolution for H(8, v 8 1 ), or in the algebraic tmf resolution for T n ⊗H(8, v 8 1 ) the element y detects an element of Ext A * (T n ⊗H(8, v 8 1 )) which is zero in mass E r (T n ∧ M (8, v 8 1 )).
Proof. Consider the maps of algebraic tmf resolutions and MASS's induced from the zig-zag . Define x := α * (x) ∈ Ext A * (T n ⊗ H(8, v 8 1 )) Then x is detected by x , regarded as an element of the algebraic tmf resolution for T n ∧ M (8, v 8 1 ). In particular, this means that x = β * (x ) Therefore, the differential d mass r (x ) = y in the MASS for tmf ∧ tmf n ∧ M (8, v 8 1 ) maps to a differential d mass r (x) = y := β * (y ) in the MASS for T n ∧ M (8, v 8 1 ). In particular, either (Case 1) y is nonzero in mass E r (T n ∧ M (8, v 8 1 ) and is detected by y in the algebraic tmf resolution for T n ⊗ H(8, v 8 1 ), or (Case 2) either y = 0 in mass E r (T n ∧ M (8, v 8 1 )) or y is killed in the algebraic tmf resolution for T n ⊗ H(8, v 8 1 ). If the latter is true, then y is killed in the algebraic tmf resolution for H(8, v 8 1 ), since the algebraic tmf resolution for T n ⊗ H(8, v 8 1 ) is a truncation of the algebraic tmf resolution for H(8, v 8 1 ).
If we are in Case (2), we are done. If we are in Case (1), consider the differential in the MASS for M (8, v 8 1 ) (which is defined by hypothesis). We must have α * (y) = y.
Therefore, d mass r (x) is detected by y in the algebraic tmf resolution.
Remark 5.6. We will primarily be applying Lemma 5.5 to the following two cases: ) in the algebraic tmf resolution, and it is proven in [BOSS19] that in the ASS for tmf ∧ tmf there is a differential 1, m ≡ 2 mod 4, 0, otherwise.
[ ] in the algebraic tmf resolution for H(8, v 8 1 ) , and the map (5.3) implies there is a differential . Then Lemma 5.5 implies that either d mass in the algebraic tmf resolution, or v i+3 1 h j+19 2,1 (v −1 0 v 2 2 ζ 8 1 ζ 4 2 )[ ] is killed in the tmf resolution for H(8, v 8 1 ) or it detects an element which is zero in the E 2 -term of the MASS for T 1 ∧ M (8, v 8 1 ). However, the element is non-zero, and is detected by v i+3 [ ] is not killed in the algebraic tmf resolution for H(8, v 8 1 ). Since the algebraic tmf resolution for T 1 ⊗ H(8, v 8 1 ) is a truncation of the algebraic tmf resolution for H(8, v 8 1 ), we conclude that v i+3 is non-trivial in the MASS for M (8, v 8 1 ), and is detected in the algebraic tmf resolution by v i+3 Proof of Theorem 1.7. By Proposition 2.3, it suffices to prove that v 32 2 ∈ Ext A * (H(8, v 8 1 )) is a permanent cycle in the MASS. Furthermore, since v 8 2 ∈ mass E 2 (M (8, v 8 1 )), the Leibniz rule implies that v 32 2 ∈ mass E 4 (M (8, v 8 1 )). We therefore are left with eliminating possible targets of d mass r (v 32 2 ) for r ≥ 4.
Suppose that d r (v 32 2 ) is non-trivial for r ≥ 4. We successively consider terms in the algebraic tmf resolution which could detect d r (v 32 2 ), and then eliminate these possibilities one by one.
Furthermore, bo ⊗s 1 only contributes h 2,1 -towers in this range for s = 5, 6. We list these contributions below, except we do not list elements in h 2,1 -towers coming from bo ⊗s 1 for s ≥ 2 which are zero in the WSS E 1 -term (see Proposition 4.2). Also, since v 32 2 is a permanent cycle in the MASS for tmf ∧M (8, v 8 1 ), we can disregard any terms coming from Ext A(2) * (F 2 ) (the zero-line of the algebraic tmf resolution). Finally, we do not include any terms which can be eliminated through the application of Case 2 of Remark 5.6.
This differential lifts to the top cell of H(8, v 8 1 ) to give d wss [18] detects the element ∆ 4 · M P ∆h 2 0 e 0 [18] in the algebraic tmf resolution for H(8, v 8 1 ). Regarding this element as an element in the MASS for tmf ∧ bo 2 1 , there is a non-trivial differential d mass . By applying (−) ∧ tmf 2 to the map of tmf modules (5.2), we may consider the composite The differential above maps to a non-trivial differential between elements of the same name in the MASS for tmf ∧ tmf 2 . We wish to apply Lemma 5.5.
We now turn our attention to the other potential target coming from bo ⊗2 1 : . This element detects ∆ 2 g 2 v 6 1 h 2,1 M g 3 [0] in the algebraic tmf resolution for M (8, v 8 1 ). However, in the ASS for the sphere, v 6 1 h 2,1 g 3 is a d 2 -cycle, and so there is a differential d ass 2 (∆ 2 g 2 · v 6 1 h 2,1 g 3 ) = d ass 2 (∆ 2 g 2 ) · v 6 1 h 2,1 g 3 = ∆ 2 h 2 2 g 2 e 0 · v 6 1 h 2,1 g 3 = v 7 1 h 22 2,1 g 2 . Applying M (−) = −, 8, g 2 , and mapping under the inclusion of the bottom cell of M (8, v 8 1 ), we get a non-trivial differential d mass h 31 2,1 g(h 2,1 ζ 4 2 ) detects g 8 n ∈ Ext A * (F 2 ) in the algebraic tmf resolution for F 2 (Prop. 3.17). This element can be eliminated by Case (1) of Remark 5.6, but we can also handle it manually using low dimensional calculations in the ASS for the sphere. There is a differential d 3 (mQ 2 ) = g 3 n in the ASS for the sphere [IWX20b], from which it follows that g 8 n is zero on the E 4 -page of the ASS of the sphere, and hence g 8 n[0] is zero on the E 4 -page of the MASS for M (8, v 8 1 ).
Any possible source for such a d 2 -differential would necessarily be detected on the 0-line of the algebraic tmf resolution, and would not support a non-trivial d 2 in the MASS for tmf ∧ M (8, v 8 1  (H(8, v 8  1 )), and we know these all must be permanent cycles in the algebraic tmf resolution because they detect the corresponding wedge elements of Ext A * (H(8, v 8  1 )). The only elements of the algebraic tmf resolution which can detect an element which could support a d 2 -differential killing ) we can eliminate these possibilities on the basis that the elements (5.8) support non-trivial d 2 differentials in the MASS for M (8, v 8 1 ).
We are left with eliminating as possibly detecting d mass

5
(v 32 2 ) in the MASS for M (8, v 8 1 ). This is the trickiest obstruction to eliminate. In the MASS for tmf ∧ tmf ∧ M (8, v 8 1 ) there is a differential The problem is that in the WSS for H(8, v 8 1 ) there is a non-trivial differential d wss Sublemma 5.9. The element v 32 2 is a permanent cycle in the MASS for T 1 ∧ M (8, v 8 1 ).
Proof of sublemma. The elements of the algebraic tmf resolution which could possibly detect the target of a differential d mass r (v 32 2 ), r ≥ 4, in the MASS for T 1 ∧ M (8, v 8 1 ) consist of those terms in Table 5.7 coming from bo 1 and bo 2 .

5
(v 32 2 ) in the MASS for T 1 ∧ M (8, v 8 1 ). Our previous arguments eliminate all the other possibilities.
Suppose now for the purpose of generating a contradiction that the differential d mass
Proof of sublemma. Let X k denote the kth modified Adams cover of Xso that the MASS for X k is the truncation of the MASS for X obtained by only considering terms in mass E s,t 2 (X) for s ≥ k, and let X k denote the cofiber X k+1 → X → X k Then we have fiber sequences Define M k to be the homotopy pullback Then the algebraic tmf resolution for M k is the truncation of the algebraic tmf resolution for M (8, v 8 1 ) obtained by omitting, for n ≥ 2 all terms of Ext A(2) * (bo i1 ⊗ · · · ⊗ bo in ⊗ H(8, v 8 1 )) of cohomological degree greater than k − n. It follows from the map of algebraic tmf resolutions and MASS's associated to the map in the MASS for M k . This differential is non-trivial in the MASS for M 36 , because it is non-trivial in the MASS for M (8, v 8 1 ), and any intervening differentials killing the target in the algebraic tmf resolution or MASS for M 36 would lift to M (8, v 8 1 ) because the spectral sequences are isomorphic in the relevant range. The same is not true in the case of M 35 , where . Therefore the proof of Sublemma 5.9 goes through with T 1 ∧ M (8, v 8 1 ) replaced with M 35 to show that there exists an element v 2 32 ∈ π 192 M 35 which is detected by v 32 2 in the MASS. Consider the diagram where the rows are cofiber sequences. The element v 32 2 ∈ π 192 M 35 maps to an element v 32 2 ) is non-trivial in the MASS for M 36 , the element v 32 2 ∈ π 192 T 1 ∧ M (8, v 8 1 ) cannot lift to M 36 , and therefore ∂ (v 32 2 ) = 0. It follows that ∂(v 32 2 ) has modified Adams filtration 34. However we have Sublemma 5.11. There are no elements of π 191 Σ −2 tmf 2 ∧ M (8, v 8 1 ) of modified Adams filtration 34.
Proof of sublemma. The only possible elements in the algebraic tmf resolution for tmf 2 ∧M (8, v 8 1 ) which could contribute to modified Adams filtration 34 in this degree are ) and the elements of Table 5.7 of algebraic tmf filtration greater than 1 in the appropriate modified Adams filtration. However, the previous arguments eliminate all of the candidates coming from Table 5.7, so we are left with eliminating (5.12). We wish to lift the differential in the MASS for tmf ∧ tmf 2 ∧ M (8, v 8 1 ) to a differential in the MASS for tmf 2 ∧ M (8, v 8 1 ). We therefore must argue that d mass in the MASS for tmf 2 ∧ M (8, v 8 1 ). We will therefore argue there are no elements in the algebraic tmf resolution for tmf 2 ∧M (8, v 8 1 ) which could detect the target of such a d 2 . Ignoring any possibilities which are eliminated by Proposition 4.2, the only possibilities are . However, these are killed by the respective WSS differentials: Thus we have arrived at a contradiction, as we have produced an element of modified Adams filtration 34, and subsequently showed no such elements exist. We conclude that our supposition, that the differential d mass

Determination of elements not in the tmf Hurewicz image
Theorem 6.1. The elements of tmf * not in the subgroup described in Theorem 1.2 are not in the Hurewicz image.
We first recall some well known K-theory computations. Recall that π * KO is given by the following v 4 1 -periodic pattern: denote the Moore spectrum for Z/2 ∞ . Consider the following diagram of cofiber sequences: The groups KO * M (2) are well-known to be given by the following v 4 1 -periodic pattern: where we denote lifts of elements of KO * along the map p of Diagram (6.2) with a tilde, and the images of the map (·) with a bar. It then follows easily from the map of long exact sequences coming from the above diagram that KO * M (2 ∞ ) is given by the v 4 1 -periodic pattern where again we denote lifts over the map p with a tilde, and images under the map (·) with a bar. The infinite sequences of dots going down represent the elements Proof of Theorem 6.1. Recall that we have an equivalence [Lau04, Cor. 3] where j −1 = ∆/c 3 4 . Applying π 0 to this equivalence, we have a commutative diagram Consider the following diagram Suppose that x ∈ tmf >0 has non-trivial image in L(x) ∈ c −1 4 tmf * , and suppose that x = h(y). Since y is torsion, it lifts over p to an element The commutativity of the diagram, implies that 0 = L(x) ∈ Im(p • i) and this implies that . Now consider elements of the form x = α∆ k ν ∈ tmf * with α ≡ 0 mod 8. Suppose that x = h(y). Lift y to an element y ∈ π * +1 M (2 ∞ ).

Then we have
But the commutativity of the diagram implies that Lh( y) is in the image of i, which implies that k = 0.
7. Lifting the remaining elements of tmf * to π s * .
Multiplicative generators of the Hurewicz image below the 192-stem. In this section, we determine a set of elements which multiplicatively generate the tmf-Hurewicz image the below the 192-stem. The results in this section drastically reduce the number of classes which we must lift in the sequel.
Lemma 7.1. The Hurewicz map S → tmf is a map of E ∞ -ring spectra. In particular, it preserves multiplication and Toda brackets.
This lemma may be applied as follows. Suppose we wish to lift a class α ∈ π * (tmf) to a classα ∈ π * (S).
With this in mind, it suffices to find a subset of the Hurewicz image which generates the entire Hurewicz image up to the 192-stem under products and Toda brackets. Our desired generating subset is given in Corollary 7.18. We will obtain our generating set by listing generators in lemmas and then recording their products in corollaries, until we have exhausted the tmf-Hurewicz image up to stem 192.
Proof. These classes are detected by h 0 , h 1 , and h 2 , respectively, in the ASS for the sphere and for tmf. The Hurewicz map induces a map of spectral sequences which sends h i → h i . The map in homotopy π * (S) → π * (tmf) then sends 2 → 2, η → η, and ν → ν, respectively, since each element survives in the ASS.
Proof. By [Isa19, Table 8], the class q is detected by ∆h 1 h 3 in the ASS for S, u is detected by ∆h 1 d 0 , and w is detected by ∆h 1 g. The same holds in the ASS for tmf by inspection of [DFHH14,Pg. 215]. The Hurewicz map S → tmf induces a map which sends these elements to the element with the same name. Since there are no elements in higher Adams filtration (except for possibly v 1 -periodic classes), we conclude that the same holds in homotopy.
Corollary 7.17. The classes {ν∆ 6 }2ν ∈ π 150 (tmf), {ν∆ 6 }ν 2 ∈ π 153 , {ν∆ 6 }ν 3 ∈ π 156 , {ν∆ 6 }κη ∈ π 162 (tmf) and {ν∆ 6 }κν ∈ π 164 (tmf) are in the Hurewicz image. Lifting generators. We will now describe our method for lifting generators. Given an element x ∈ tmf * , we want to lift it to an element y ∈ π s * . To this end, we consider the diagram of (M)ASS's: First, we identify an element which detects the element x in the ASS for tmf * , and then we identify an element x ∈ Ext A(2) * (H(8, v 8 1 )) which maps to it. This element x can be regarded as an element of the zero line of the algebraic tmf-resolution for Ext A * (H(8, v 8  1 )). We will show that the element x is a permanent cycle in the algebraic tmf-resolution, and thus lifts to an element y ∈ Ext A * (H(8, v 8 1 )). We will then show that the element y is a permanent cycle in the MASS for M (8, v 8 1 ), and hence detects an element y ∈ π * M (8, v 8 1 ). Let y ∈ π s * be the projection of y to the top cell. It then follows that the image of y in tmf * equals x, modulo terms of higher Adams filtration (AF). Furthermore, using the v 32 2 -self map on M (8, v 8 1 ), we deduce that the element v 32k 2 y ∈ π * M (8, v 8 1 ) projects on the top cell to an element v 32k 2 y ∈ π s * whose image in tmf * is ∆ 8k x modulo terms of higher Adams filtration. Finally, Theorem 6.1 eliminates the potential ambiguity caused by elements of higher Adams filtration, since the elements of higher Adams filtration are v 4 1 -periodic.
Proof. We will check that each element lifts using the Atiyah-Hirzebruch spectral sequence (AHSS).
(1) We begin with q ∈ π 32 (tmf), which we will define to be the unique nontrivial c 4 -torsion class detected by the element v 4 2 c 0 ∈ Ext 7,7+32 A(2) * (F 2 ) in the ASS for tmf. The element v 4 2 c 0 does not lift to Ext A * . Nevertheless, we claim that there is an element q ∈ π s 32 10 detected by the element in the ASS for the sphere, which maps to q under the tmf Hurewicz homomorphism. Our strategy will be to argue that q and q lift to  (M (8, v 8 1 )). A similar but easier analysis reveals that the lift q[18] exists. The elements ∆h 1 h 3 ∈ Ext A * (F 2 ) and v 4 2 c 0 ∈ Ext A(2) * (F 2 ) are h 0torsion, and hence lift to elements 10 The element we are calling q ∈ π s 32 is traditionally called q, but we add the tilde to distinguish it from the element we are calling q in π 32 tmf. (2) Since u ∈ π 39 tmf is detected by an element of Ext A(2) * in the image of the map we immediately see that the element u ∈ π 39 (S) maps to it. We are left with lifting u ∈ π s 39 to the top cell of M (8, v 8 1 (3) The element w ∈ π 45 tmf is detected by an element which is in the image of the map (7.22), and thus we deduce that w ∈ π 45 (S) maps to it. A similar argument to the case above shows that w lifts to w[18] ∈ π 63 (M (8, v 8 1 )).
(1) We begin with ∆ 2 ν 2 ∈ π 54 (tmf). This class lifts to an element ∆ 2 ν 2 [1] ∈ tmf 55 (M (8)) 11 We are specifically using case (5) of the Geometric Boundary Theorem since the relevant class (denoted p * (y) in the theorem statement) is a permanent cycle. We will be using this argument repeatedly in subsequent proofs in this section, and for brevity will simply say "by the Geometric Boundary Theorem..." in these subsequent instances.  12 We will say that an element x ∈ Ext A(2) * (H(8, v 8 1 )) has relative position (t − s, s) in Ext A(2) * (bo I ⊗ H(8, v 8 1 )) if the image of a differential supported by x in the algebraic tmf resolution lies in Ext s+1,t A(2) * (bo I ⊗ H(8, v 8 1 )), and the image of a differential supported by x in the MASS could be detected in the algebraic tmf resolution by an element in Ext s+r,t−r+1 H(8, v 8  1 )). In other words, if you were to pretend x were an element in Ext s,t A(2) * (bo I ⊗ H(8, v 8 1 )), then drdifferentials in the algebriac tmf resolution "look" like Adams d 1 's, and dr-differentials in the MASS "look" like Adams dr's.
We now check that v 10 2 v 4 1 d 0 h 0 [1] is a permanent cycle in the algebraic tmf resolution for H(8, v 8 1 ). Its relative position in Ext A(2) * (bo 1 ⊗H(8, v 8 1 )) is t−s = 76 and AF = 20, its relative position in Ext A(2) * (bo ⊗2 1 ⊗H(8, v 8 1 )) is t − s = 69 and AF = 19, and its relative position in Ext A(2) * (bo ⊗3 1 ⊗ H(8, v 8 1 )) is t − s = 62 and AF = 18, the last of which has targets only above the vanishing line. Inspection of the relevant charts shows that v 10 2 v 4 1 d 0 h 0 [1] cannot support a nontrivial d 1 -differential since the target bidegrees are zero. Therefore v 10 2 v 4 1 d 0 h 0 [1] is a permanent cycle in the algebraic tmf-resolution for H(8, v 8 1 ) and detects an element {v 10 (H(8, v 8  1 )). Finally, inspection of the same charts reveals that there are no possible targets for a nontrivial differential supported by {v 10 (3) The class ∆ 2 η 2κ ∈ π 70 (tmf) lifts to an element which is detected by Proof.
The only possibility for this element to support a non-trivial MASS differential is for it to support a d 3 -differential whose target to by detected by the element  )). Therefore the hypotheses of Lemma 5.5 are satisfied, and we deduce that v 1 h 19 2,1 (v −1 0 v 2 2 [ζ 8 1 , ζ 4 2 ])[18] detects an element which is zero in the E 3 -page of the MASS, and hence cannot be the target of a non-trivial d 3 -differential in the MASS. (2) The class ∆ 6 κν ∈ π 161 (tmf) lifts to an element