twoside=false

# The $\mathbb{C}$-motivic Adams-Novikov spectral sequence for topological
modular forms

Daniel C. Isaksen  Department of Mathematics, Wayne State University, Detroit,
MI 48009, USA<EMAIL_ADDRESS>,  Hana Jia Kong  School of Mathematics,
Institute for Advanced Study, NJ, USA<EMAIL_ADDRESS>,  Guchuan Li
Department of Mathematics, University of Michigan, MI, USA<EMAIL_ADDRESS>,  Yangyang Ruan  Institute of Mathematics, Academy of Mathematics and Systems
Science, Chinese Academy of Sciences, Beijing, China<EMAIL_ADDRESS>and
Heyi Zhu  Department of Mathematics, University of Illinois, Urbana-Champaign,
IL, USA<EMAIL_ADDRESS>
(Date: 28th August 2024)

###### Abstract.

We analyze the $\mathbb{C}$-motivic (and classical) Adams-Novikov spectral
sequence for the $\mathbb{C}$-motivic modular forms spectrum $\mathit{mmf}$
(and for the classical topological modular forms spectrum $\mathit{tmf}$). We
primarily use purely algebraic techniques, with a few exceptions. Along the
way, we settle a previously unresolved detail about the multiplicative
structure of the homotopy groups of $\mathit{tmf}$.

###### Key words and phrases:

topological modular forms, motivic stable homotopy theory, Adams-Novikov
spectral sequence, Adams spectral sequence, stable homotopy group

###### 2010 Mathematics Subject Classification:

Primary 14F42, 55Q10, 55T15; Secondary 55Q45

The first author was partially supported by National Science Foundation Grant
DMS-2202267. The second author was supported by National Science Foundation
grant DMS-1926686. The third author would like to thank the Max Planck
Institute for Mathematics and the Hausdorff Research Institute for Mathematics
for the hospitality.

##  1\. Introduction

The topological modular forms spectrum $\mathit{tmf}$ plays an essential role
in the study of the stable homotopy groups of spheres [Bau08] [Beh20] [DFHH14]
[Goe10] [Hop95] [Hop02] [HM14] [Rez02]. The unit map
$S\rightarrow\mathit{tmf}$ from the sphere spectrum to $\mathit{tmf}$ detects
much of the structure of the stable homotopy groups of $S$, including the
elements $\eta$ (1-stem), $\nu$ (3-stem), $\epsilon$ (8-stem), $\kappa$
(14-stem), $\overline{\kappa}$ (20-stem), and many additional elements. The
unit map is far from injective (for example, $\sigma$ (7-stem) maps to zero in
$\mathit{tmf}$), so it does not detect all of the stable homotopy groups of
spheres. Moreover, it is also not surjective. The computation of the
$\mathit{tmf}$-Hurewicz image is a difficult problem that leads to the
identification of infinite $v_{2}$-periodic families in the stable homotopy
groups of spheres [BMQ20].

The spectrum $\mathit{tmf}$ serves as an approximation to the sphere spectrum.
This approximation is highly suitable for testing theories and for developing
computational techniques. The structure of $\mathit{tmf}$ is complicated
enough to exhibit the complex phenomena related to the computation of stable
homotopy groups, but it is also simple enough to be computed exhaustively. We
have found that the study of $\mathit{tmf}$ is an indispensable step along the
way to understanding the sphere spectrum.

By comparison, the spectrum $\mathit{ko}$ is arguably too simple to serve as a
test case for computational theories. For example, its Adams spectral sequence
collapses, so its homotopy reduces to an entirely algebraic problem. Neither
the Adams nor the Adams-Novikov spectral sequence collapses for
$\mathit{tmf}$. However, the analysis of $\mathit{tmf}$ does not involve
crossing differentials or crossing extensions in the sense of [IWX20, Section
2.1]. This means that the homotopy of $\mathit{tmf}$ does not share the most
delicate parts of the homotopy groups of spheres.

Bruner and Rognes [BR21] have recently exhaustively studied the Adams spectral
sequence for $\mathit{tmf}$. They completely determine the additive and
(primary) multiplicative structure of the stable homotopy groups of
$\mathit{tmf}$, with one exception.

The goal of this manuscript is to carry out the Adams-Novikov spectral
sequence for $\mathit{tmf}$. In fact, we will work in the more general
$\mathbb{C}$-motivic context and compute the motivic Adams-Novikov spectral
sequence for the $\mathbb{C}$-motivic modular forms spectrum $\mathit{mmf}$.
The classical computation is easily recovered from the motivic computation by
an algebraic localization.

More specifically, there is a certain motivic element $\tau$. Inverting $\tau$
has the effect of collapsing $\mathbb{C}$-motivic computations to classical
computations. In particular, $\tau$-torsion phenomena disappear in the
classical context. Henceforth, we will work in the $\mathbb{C}$-motivic
context. The interested reader can easily recover classical computations from
our work by inverting $\tau$.

From another perspective, we also compute the $\mathbb{C}$-motivic effective
slice spectral sequence for $\mathit{mmf}$, since it agrees with the Adams-
Novikov spectral sequence over $\mathbb{C}$. This identification of spectral
sequences does not appear to be cleanly stated in the literature, but it is a
computational consequence of the weight 0 result of [Lev15, Theorem 1].

Our goal is not merely to record the details of the Adams-Novikov spectral
sequence, which have previously appeared in [Bau08]. More specifically, we
have attempted to give proofs that are as algebraic as possible. Such
algebraic proofs are less likely to contain subtle mistakes, and they are more
easily verifiable by machine. The motivic context provides us with additional
algebraic tools that are not accessible in the strictly classical context. We
also correct a few oversights and minor mistakes in the analysis of [Bau08].

###  1.1. Algebraic philosophy

We do not use any information from the sphere spectrum as input for our
computations. We do, however, assume full knowledge of the algebraic structure
of the motivic Adams and motivic Adams-Novikov $E_{2}$-pages for
$\mathit{mmf}$. This is consistent with our goal of using algebraic techniques
whenever possible. It is also consistent with our philosophy that the role of
$\mathit{tmf}$ is to inform us about the sphere spectrum. By comparison, in
[BR21] it is necessary to import the relation $\eta^{2}\kappa=0$ to
$\mathit{tmf}$ from previous knowledge of the sphere spectrum. Fortunately for
us, we have the relation $h_{1}^{2}d=0$ in the Adams-Novikov $E_{2}$-page for
$\mathit{mmf}$. Because there are no elements in higher filtration, the
relation $\eta^{2}\kappa=0$ therefore has an entirely algebraic proof.

A computation involving the Adams or Adams-Novikov spectral sequence breaks
into two main stages. The first stage is entirely algebraic and involves the
computation of the $E_{2}$-page. In the modern era, this first stage is
typically conducted by machine. The computation of the $E_{2}$-pages for
$\mathit{tmf}$ is not elementary, but it can be done manually with enough
patience [Bae] [Bau08, Section 7] [BR21] [Rez02, Section 18].

The second stage of the process involves the computation of differentials and
hidden extensions. This stage typically requires input from topology, so it
cannot be fully automated because it is not entirely algebraic.

Our contribution is to recognize that much of this topological second stage
actually can be carried out using only algebraic information. The key idea is
to use the additional structure of the motivic context in order to pass back
and forth between the Adams and Adams-Novikov spectral sequences. Each
$E_{2}$-page tells us some things about the homotopy groups of $\mathit{tmf}$.
The information contained in these $E_{2}$-pages does overlap, but not
perfectly. The union of the information in both $E_{2}$-pages is strictly
greater than the information in either one of the $E_{2}$-pages.

We give several concrete examples of information available in only one of the
two $E_{2}$-pages.

  1. (1)

In the classical Adams $E_{2}$-page for $\mathit{tmf}$, we have the relation
$h_{1}^{4}=0$. This implies the relation $\eta^{4}=0$ in homotopy. However, in
the classical Adams-Novikov $E_{2}$-page, the element $h_{1}^{4}$ is non-zero
and is hit by an Adams-Novikov $d_{3}$ differential. Thus, the relation
$\eta^{4}=0$ has an entirely algebraic proof, but only in the Adams spectral
sequence.

  2. (2)

In fact, the relation $h_{1}^{4}=0$ is a consequence of the Massey product
$h_{1}^{2}=\langle h_{0},h_{1},h_{0}\rangle$ in the Adams $E_{2}$-page. In the
classical Adams-Novikov $E_{2}$-page, the corresponding Massey product
$\langle 2,h_{1},2\rangle$ is zero. Consequently, the Toda bracket
$\eta^{2}=\langle 2,\eta,2\rangle$ has an entirely algebraic proof, but only
in the Adams spectral sequence.

  3. (3)

In the classical Adams-Novikov $E_{2}$-page for $\mathit{tmf}$, we have the
relation $h_{2}^{3}=h_{1}c$. This implies the relation $\nu^{3}=\eta\epsilon$.
However, in the classical Adams $E_{2}$-page, we have $h_{2}^{3}=0$. In fact,
there is a hidden $\nu$ extension from $h_{2}^{2}$ to $h_{1}c$ in the Adams
spectral sequence. Thus, the relation $\nu^{3}=\eta\epsilon$ has an entirely
algebraic proof, but only in the Adams-Novikov spectral sequence.

  4. (4)

In fact, the relation $h_{2}^{3}=h_{1}c$ is a consequence of the Massey
product $c=\langle h_{2},h_{1},h_{2}\rangle$ in the Adams-Novikov
$E_{2}$-page. In the classical Adams $E_{2}$-page, the corresponding Massey
product is zero. Consequently, the Toda bracket
$\epsilon=\langle\nu,\eta,\nu\rangle$ has an entirely algebraic proof, but
only in the Adams-Novikov spectral sequence. See 2.18 for more detail on this
example.

In order to obtain one key Adams-Novikov differential, we use Bruner’s theorem
on the interaction between algebraic Steenrod operations [May70] and Adams
differentials in the context of the Adams spectral sequence. We refer to
[Bru84, Theorem 2.2] for a precise readable statement; see also [BMMS86] and
[Mäk73]. The practical implementation of Bruner’s theorem requires only
algebraic information in the form of algebraic Steenrod operations on
$\operatorname{Ext}$ groups. These operations can be computed by machine,
although not as effectively as the additive and multiplicative structure of
the $\operatorname{Ext}$ groups. The algebraic Steenrod operations are
additional structure on top of what topologists usually think of as “standard
homological algebra".

In the context of the Adams-Novikov spectral sequence, we also rely on the
Leibniz rule in the form $d_{r}(x^{k})=kx^{k-1}d_{r}(x)$. Philosophically,
this formula is connected to Bruner’s theorem, although we do not know how to
make a precise connection. As in the case of Bruner’s theorem, it feels like
slightly more information than is usually considered in standard homological
algebra.

We also draw attention to 4.5, in which we establish a hidden $2$ extension in
the $110$-stem. Here we use some information about the homotopy groups of
$\mathit{mmf}/\tau^{2}$. One might argue that this information is not entirely
of an algebraic nature. By comparison, the corresponding $2$ extension in the
Adams spectral sequence is hidden, but not particularly difficult [BR21,
Theorem 9.8(110)].

###  1.2. Techniques

Section 2.10 describes a particularly powerful method for studying the
$\mathbb{C}$-motivic Adams-Novikov spectral sequence in a way that has no
classical analogue. There is a map
$q:\mathit{mmf}/\tau\rightarrow\Sigma^{1,-1}\mathit{mmf}$ that can be viewed
as projection to the top cell of the $2$-cell $\mathit{mmf}$-module
$\mathit{mmf}/\tau$. The homotopy of $\mathit{mmf}/\tau$ is entirely
understood in an algebraic sense since it is isomorphic to the classical
Adams-Novikov $E_{2}$-page for $\mathit{tmf}$. Moreover, the map $q$ maps onto
the homotopy of $\mathit{mmf}$ that is annihilated by $\tau$. Thus $q$ can be
used to detect structure in $\mathit{mmf}$ that is related to classes that are
annihilated by $\tau$.

In practice, many specific questions about hidden extensions do not directly
involve elements that are annihilated by $\tau$. Frequently, if we multiply
these elements by a power of $\tau$ and a power of $g$, then we end up with
elements that are annihilated by $\tau$. We can use $q$ to understand these
latter elements, and finally deduce information about the original elements.
LABEL:hidtaumethod lists numerous specific examples of this process. The
majority of hidden extensions can be handled very easily in this way, although
a few extensions require more complicated arguments.

We avoid the use of Toda brackets whenever possible, but occasionally they are
inevitable. In those cases where we must compute a Toda bracket, we once again
rely exclusively on algebraic techniques. Namely, our Toda brackets arise from
corresponding Massey products in either the Adams or Adams-Novikov
$E_{2}$-page. The Moss Convergence Theorem [Mos70] says that such algebraic
Massey products detect Toda brackets in “well-behaved" situations. In
practice, all of the situations that we study are well-behaved.

###  1.3. The differentials on $\Delta^{k}$

Having carried out the entire analysis of the motivic Adams-Novikov spectral
sequence for $\mathit{mmf}$, we can see in hindsight that there are a few key
steps from which all of the other miscellaneous computations follow. Our
experience shows that the key steps involve the differentials on elements of
the form $2^{j}\Delta^{k}$. This is not particularly surprising; we expect the
element $\Delta$ to play a dominant role since it represents
$v_{2}$-periodicity.

First, we establish $d_{5}(\Delta)=\tau^{2}h_{2}g$ in 3.7. This follows
immediately by comparison to the Adams spectral sequence, in which
$\tau^{2}h_{2}g$ is already zero in the $E_{2}$-page. Thus, we have an
algebraic proof for $d_{5}(\Delta)$. Then the Leibniz rule implies that
$d_{5}(\Delta^{2})=2\tau^{2}\Delta h_{2}g$.

The Leibniz rule also implies that
$d_{5}(\Delta^{4})=4\tau^{2}\Delta^{3}h_{2}g$. However,
$4\tau^{2}\Delta^{3}h_{2}g$ is zero in the Adams-Novikov $E_{2}$-page. Because
of the hidden $2$ extension from $2\tau^{2}h_{2}$ to $\tau^{3}h_{1}^{3}$, the
element $\tau^{3}\Delta^{3}h_{1}^{3}g$ ought to play the role of
$4\tau^{2}\Delta^{3}h_{2}g$. This strongly suggests that there should be a
differential $d_{7}(\Delta^{4})=\tau^{3}\Delta^{3}h_{1}^{3}g$. In fact, this
formula is correct (see 3.13), but it requires some work to give a precise
proof.

Our solution, once again, is to play the Adams and Adams-Novikov spectral
sequences against each other. We used the Adams $E_{2}$-page to obtain the
Adams-Novikov differential $d_{5}(\Delta)$. Then we used the Leibniz rule in
the Adams-Novikov spectral sequence to obtain $d_{5}(\Delta^{2})$. In turn,
this last Adams-Novikov differential implies an Adams differential
$d_{2}(\Delta^{2})$, or $d_{2}(w_{2})$ in the notation of [BR21]. Next, we
obtain an Adams differential $d_{3}(\Delta^{4})$, or $d_{3}(w_{2}^{2})$ in the
notation of [BR21], by applying Bruner’s theorem on the interaction between
squaring operations and Adams differentials [BMMS86] [Bru84]. Finally, the
Adams differential $d_{3}(\Delta^{4})$ implies that there is an Adams-Novikov
differential $d_{7}(\Delta^{4})$. For more details, see Sections 3.3 and 3.4.
Curiously, precise statements about the Adams-Novikov differential
$d_{7}(\Delta^{4})$ are missing from [Bau08] [HM14] [Rez02].

###  1.4. Main results

Our main results are expressed in the charts in Section 7. For completeness,
we express this in the form of a main theorem.

######  Theorem 1.1.

The charts in Section 7 represent the $\mathbb{C}$-motivic Adams-Novikov
spectral sequence for the motivic modular forms spectrum $\mathit{mmf}$,
including complete descriptions of

  * •

the $E_{2}$-page.

  * •

all differentials.

  * •

the $E_{\infty}$-page.

  * •

all hidden extensions by $2$, $\eta$, and $\nu$.

The proof of 1.1 consists of the sum of a long list of miscellaneous
computations, which are carried out throughout the manuscript. See especially
the tables in Section 6. These tables summarize the main computational facts,
and they give cross-references to more detailed proofs of each fact.

Our work is not as complete as [BR21] because we have not completely analyzed
the multiplicative structure. In principle, this could be done using the same
techniques. We do study one family of multiplicative relations in more detail.
Bruner and Rognes identify a family $\nu_{k}$ of elements in the homotopy of
$\mathit{tmf}$. They mostly determine the products among these elements, but
they leave one case unresolved. Our techniques settle this last detail about
the $2$-primary multiplicative structure of the homotopy of $\mathit{tmf}$.

######  Theorem 1.2.

In the context of [BR21], $\nu_{4}\nu_{6}=\nu\nu_{2}M$.

1.2 is proved later as 5.12. In fact, it is a consequence of the more general
5.10, which offers a graceful simultaneous analysis of products
$\nu_{j}\nu_{k}$. Bruner and Rognes empirically observed the formula

$\nu_{i}\nu_{j}=(i+1)\nu\nu_{i+j}.$

Our proof shows that the coefficients $(i+1)$ arise naturally from the Leibniz
rule

$d_{5}(\Delta^{i+1})=(i+1)\Delta^{i}d_{5}(\Delta).$

###  1.5. Future directions

Our work raises some questions that deserve further study.

######  Problem 1.1.

Compute the $\overline{\kappa}$-periodic $\mathbb{C}$-motivic spectrum
$\mathit{mmf}[\overline{\kappa}^{-1}]$.

Frequently, we detect elements and relations by first computing their products
with various powers of $g$ or $\overline{\kappa}$. In other words, much of the
structure of $\mathit{mmf}$ is reflected in the $\overline{\kappa}$-periodic
spectrum $\mathit{mmf}[\overline{\kappa}^{-1}]$. This motivic spectrum is non-
trivial, but its homotopy is entirely annihilated by $\tau^{11}$.
Consequently, its Betti realization is trivial, and it represents purely
“exotic" motivic phenomena. We mention that [BBC23] also studies $g$-periodic
phenomena in $\mathit{tmf}$, although not in a way that is particularly close
to our perspective.

######  Problem 1.2.

Develop better technology to deduce the differential
$d_{7}(\Delta^{4})=\tau^{3}\Delta^{3}h_{1}^{3}g$ directly from the
differential $d_{5}(\Delta)=\tau^{2}h_{2}g$.

It is conceivable that $d_{7}(\Delta^{4})$ could be deduced directly from
$d_{5}(\Delta)$ using a variant of Bruner’s theorem that would apply in the
Adams-Novikov spectral sequence, but we have not even formulated a precise
statement of such a variant. There is a connection between Bruner’s theorem
and the Leibniz rule $d_{r}(x^{2})=2xd_{r}(x)$, but the precise relationship
is not clear to us.

Another possible approach to 1.2 might involve an enriched $E_{2}$-page in
which the $2$ extension from $2\tau^{2}h_{2}$ to $\tau^{3}h_{1}^{3}$ is not
hidden.

######  Problem 1.3.

Construct a spectral sequence whose $E_{2}$-page reflects the algebraic
structure of both the Adams and Adams-Novikov $E_{2}$-pages.

We frequently pass back and forth between the Adams and Adams-Novikov spectral
sequences. In order to facilitate these transitions, Section 2.5 introduces a
notion of correspondence between elements of the Adams spectral sequence and
elements of the Adams-Novikov spectral sequence.

This setup feels like a preliminary attempt to describe a richer connection
between the two spectral sequences. It would be much more convenient and
effective to compute in just a single spectral sequence that reflects the
algebraic structure of both the Adams and Adams-Novikov spectral sequences.
There are some preliminary indications that “bimotivic homotopy theory" (also
known as $H\mathbb{F}_{2}$-synthetic $BP$-synthetic homotopy theory) provides
a context for this.

###  1.6. Outline

We begin in Section 2 with a discussion of tools that we will use to carry out
our explicit computations. We describe both the motivic Adams and motivic
Adams-Novikov spectral sequences for $\mathit{mmf}$, and we establish notation
for elements in these spectral sequences. We also establish notation for
certain homotopy elements that we will use later. We draw particular attention
to Sections 2.9 and 2.10, which establish a powerful tool for detecting hidden
extensions. The basic idea is to use the motivic spectrum $\mathit{mmf}/\tau$,
whose homotopy is entirely algebraic.

Our explicit computations begin in Section 3, where we establish all of the
Adams-Novikov differentials. The propositions in this section are mostly in
order of increasing length of differentials. However, we make some exceptions
to this general rule to preserve the logical order, so each result only
depends on previously proved results.

Once the Adams-Novikov differentials are computed, we proceed to compute all
hidden extensions by $2$, $\eta$, and $\nu$ in Section 4. Most of these
extensions follow immediately by comparison to the homotopy of
$\mathit{mmf}/\tau$, but there are several cases with more difficult proofs.

Finally, in Section 5, we consider an explicit family of products that are
particularly interesting. Our results on these products fill a gap in the
product structure of $\pi_{*}\mathit{tmf}$, as described in [BR21].

###  1.7. Conventions

We work exclusively at the prime $2$. There are interesting aspects to the
computation of $\mathit{tmf}$ at the prime $3$ ([Bau08, Chapter 5], [DFHH14],
[BR21, Chapter 13]), but we do not address that topic. We use the motivic
Adams-Novikov spectral sequence to compute the homotopy groups of the
$2$-localization of $\mathit{mmf}$. We also use the $E_{2}$-page of the
motivic Adams spectral sequence, which actually converges to the homotopy
groups of the $2$-completion of $\mathit{mmf}$. The distinction between
localization and completion is not essential since only finitely generated
abelian groups appear in our work. For expository simplicity, these
localizations or completions do not appear in our notation. For example, the
symbol $\mathbb{Z}$ refers to the integers localized at $2$, or to the
$2$-adic integers. Similarly, $\pi_{*,*}\mathit{mmf}$ refers to the motivic
stable homotopy groups of the $2$-localization (or $2$-completion) of
$\mathit{mmf}$.

The adjective “motivic" always refers exclusively to the $\mathbb{C}$-motivic
context. We consider no base fields other than $\mathbb{C}$.

Many of our explicit results are labelled with the degrees in which they
occur. These degrees may help the reader navigate the overall computation,
especially in finding the relevant elements on Adams-Novikov charts.

###  1.8. Acknowledgements

We thank Tilman Bauer, Robert Bruner, and John Rognes for various discussions
related to the production of this manuscript. We also appreciate stimulating
discussions with the participants of the Winter 2023 eCHT reading seminar on
the Adams spectral sequence for tmf.

##  2\. Background

In this section, we discuss the techniques that we will use later to carry out
our computations.

###  2.1. The $\mathbb{C}$-motivic modular forms spectrum $\mathit{mmf}$

There is a $\mathbb{C}$-motivic $E_{\infty}$-ring spectrum $\mathit{mmf}$ that
can be viewed as the analogue of the classical topological modular forms
spectrum $\mathit{tmf}$ [GIKR22]. The Betti realization of $\mathit{mmf}$ is
the classical spectrum $\mathit{tmf}$. Moreover, the cohomology of
$\mathit{mmf}$ is $A\sslash A(2)$, where $A$ denotes the $\mathbb{C}$-motivic
Steenrod algebra and $A(2)$ is the subalgebra generated by
$\operatorname{Sq}^{1}$, $\operatorname{Sq}^{2}$, and $\operatorname{Sq}^{4}$.

###  2.2. The $\mathbb{C}$-motivic Adams spectral sequence for $\mathit{mmf}$

We abbreviate the motivic Adams spectral sequence for $\mathit{mmf}$ by mAss.
The cohomology of $\mathbb{C}$-motivic $A(2)$ is the $E_{2}$-page of the mAss.
The manuscript [Isa09] computes the cohomology of $\mathbb{C}$-motivic $A(2)$
using the motivic May spectral sequence, and it gives a complete description
of its ring structure. The mAss $E_{2}$-page consists entirely of algebraic
information, which we take as given. We grade the mAss $E_{2}$-page in the
form $(s,f,w)$, where $s$ is the topological stem, $f$ is the Adams
filtration, and $w$ is the motivic weight.

The motivic Adams differentials are recorded in [Isa18]. However, this
manuscript does not depend on previous knowledge of any Adams differentials,
neither classical nor motivic. For completeness, we provide self-contained
proofs for two Adams differentials in 3.18.

We adopt the notation of [Isa09] and [Isa18] for the mAss. For the reader’s
convenience, LABEL:tab:mAss-notation provides a concordance between our
notation and the notation of [BR21]. Beware that the motivic generators $u$
and $\Delta u$ have no classical counterparts because they are annhilated by
$\tau$.

Table 1. Generators of the motivic Adams $E_{2}$-page for $\mathit{mmf}$ $(s,f,w)$ | [Isa09] | [BR21]
---|---|---
$(0,1,0)$ | $h_{0}$ | $h_{0}$
$(1,1,1)$ | $h_{1}$ | $h_{1}$
$(3,1,2)$ | $h_{2}$ | $h_{2}$
$(8,3,5)$ | $c$ | $c_{0}$
$(8,4,4)$ | $P$ | $w_{1}$
$(11,3,7)$ | $u$ |
$(12,3,6)$ |  $a$ or $\alpha$ | $\alpha$
$(14,4,8)$ | $d$ | $d_{0}$
$(15,3,8)$ |  $n$ or $\nu$ | $\beta$
$(17,4,10)$ | $e$ | $e_{0}$
$(20,4,12)$ | $g$ | $g$
$(25,5,13)$ | $\Delta h_{1}$ | $\gamma$
$(32,7,17)$ | $\Delta c$ | $\delta$
$(35,7,19)$ | $\Delta u$ |
$(48,8,24)$ | $\Delta^{2}$ | $w_{2}$

###  2.3. The $\mathbb{C}$-motivic Adams-Novikov spectral sequence for
$\mathit{mmf}$

The $E_{2}$-page of the classical Adams-Novikov spectral sequence for
$\mathit{tmf}$ is given by
$\operatorname{Ext}^{**}_{{BP}_{*}{BP}}({BP}_{*},{BP}_{*}\mathit{tmf})$, where
$BP$ denotes the Brown-Peterson spectrum. Analogously to the classical Adams-
Novikov spectral sequence, one can construct a motivic Adams-Novikov spectral
sequence by resolving with respect to the motivic Brown-Peterson spectrum. We
abbreviate the motivic Adams-Novikov spectral sequence by mANss. We grade the
mANss $E_{2}$-page in the form $(s,f,w)$, where $s$ is the topological stem,
$f$ is the Adams-Novikov filtration, and $w$ is the motivic weight.

The mANss is easy to describe in classical terms. The motivic $E_{2}$-page can
be obtained from its classical analogue by first assigning a third degree,
called the weight, to be half of the total degree for each class, then
adjoining a polynomial generator $\tau$ of degree $(0,0,-1)$ (see, e.g.
[HKO11][Isa19]). More explicitly, a classical element $x$ in degree $(s,f)$
corresponds to a family of elements $\\{\tau^{n}x|n\geq 0\\}$ in the mANss,
where the motivic element $x$ has degree $\left(s,f,\frac{s+f}{2}\right)$.

The $E_{2}$-page of the mANss consists entirely of algebraic information,
which we take as given. For our purposes, the best way to compute this
$E_{2}$-page is by the algebraic Novikov spectral sequence, which is worked
out in detail in [Bae].

######  Remark 2.1.

The $E_{2}$-page of the classical Adams-Novikov spectral sequence for
$\mathit{tmf}$ is the cohomology of a version of the elliptic curve Hopf
algebroid ([Rez02][Bau08]). By the change-of-rings theorem [Rez02, Theorem
15.3], this is the same as the cohomology of the Hopf algebroid
$(BP_{*}tmf,BP_{*}BP\otimes_{BP_{*}}BP_{*}tmf)$. See [Rez02, Proposition 15.7
and Section 20] for more details. We do not rely on this perspective.

###  2.4. Notation for the motivic Adams-Novikov spectral sequence

LABEL:generators lists the multiplicative generators for the mANss
$E_{2}$-page for $\mathit{mmf}$. These generators are the starting point of
our computation.

Table 2. Generators of the motivic Adams-Novikov $E_{2}$-page for $\mathit{mmf}$ $(s,f,w)$ | generator
---|---
$(0,0,-1)$ | $\tau$
$(1,1,1)$ | $h_{1}$
$(3,1,2)$ | $h_{2}$
$(5,1,3)$ | $h_{1}v_{1}^{2}$
$(8,0,4)$ | $P$
$(8,2,5)$ | $c$
$(12,0,6)$ | $4a$
$(14,2,8)$ | $d$
$(20,4,12)$ | $g$
$(24,0,12)$ | $\Delta$

One must be slightly careful with the definitions of some of these elements
because they belong to cyclic groups of order greater than $2$. In these
cases, there is more than one possible generator. Specifically, this issue
arises for the elements $h_{2}$, $P$, $4a$, $g$, and $\Delta$. For $P$, $4a$,
and $g$, we simply choose arbitrary generators.

######  Remark 2.2.

$(3,1,2)$ The choice of $h_{2}$ makes little practical difference to us, as
long as it is a generator of the mANss $E_{2}$-page in degree $(3,1,2)$. For
definiteness, we take $h_{2}$ to represent the homotopy element $\nu$,
assuming an a priori definition of $\nu$ (for example, by appealing to the
homotopy of the sphere spectrum or by appealing to a geometric construction of
$\nu$ involving quaternionic multiplication).

The choice of $\Delta$ also makes little practical difference. We choose
$\Delta$ in such a way to make our formulas easier to write. See 3.8 and 5.8
for more details. Note that the choice of $\Delta$ depends on a previous
choice of $h_{2}$.

######  Remark 2.3.

$(12,0,6)$ The notation $4a$ does not appear to be natural and deserves some
explanation. There are two closely related reasons why we find this notation
to be convenient. First, the element $4a$ is detected in the algebraic Novikov
spectral sequence [Bae] by an element $h_{0}^{2}a$. Second, the element
$2\cdot 4a$ turns out to be a permanent cycle that detects an element in
$\pi_{12,6}\mathit{mmf}$. This same homotopy element is detected by
$h_{0}^{3}a$ in the Adams spectral sequence for $\mathit{mmf}$.

The element $g$ is a permanent cycle and therefore represents a homotopy class
$\overline{\kappa}$. Multiplication by $g$ provides regular structure to the
mANss for $\mathit{mmf}$. We typically sort elements into families that are
related by $g$ multiplication. In other words, when we consider a particular
element $x$, we also typically consider the elements $xg^{k}$ for all $k\geq
0$ at the same time.

Taken together, Figures 1 and 3 depict the $E_{2}$-page of the mANss for
$\mathit{mmf}$ graphically. The careful reader should superimpose these
figures in order to obtain a full picture of the mANss. Figure 1 depicts a
regular $v_{1}$-periodic pattern in the $E_{2}$-page, to be discussed in
detail in Section 2.7. Figure 3 depicts the remaining classes.

###  2.5. Comparison between the mANss and the mAss

######  Definition 2.4.

Let $a$ be a permanent cycle in the mANss for $\mathit{mmf}$, and let $b$ be a
permanent cycle in the mAss for $\mathit{mmf}$. The elements $a$ and $b$
correspond if there exists a non-zero element in $\pi_{*,*}\mathit{mmf}$ that
is detected by $a$ in the mANss for $\mathit{mmf}$ and is detected by $b$ in
the mAss for $\mathit{mmf}$.

######  Remark 2.5.

Beware that a permanent cycle may detect more than one element in
$\pi_{*,*}\mathit{mmf}$, depending on the presence of permanent cycles in
higher filtration. We ask only that the cosets detected by $a$ and $b$
intersect; they need not coincide. We give an explicit example.

The element $P$ of the mANss $E_{\infty}$-page detects two elements of
$\pi_{8,4}\mathit{mmf}$ because of the presence of $\tau c$ in higher
filtration. On the other hand, the element $P$ of the mAss $E_{\infty}$-page
detects infinitely many elements (which differ only by a $2$-adic unit factor)
because of the presence of $Ph_{0}^{k}$ in higher filtration for $k\geq 1$.
This is an example of a corresponding pair of elements that do not detect
precisely the same coset of homotopy elements.

######  Remark 2.6.

It is possible that a single element of the mANss corresponds to two different
elements of the mAss. For example, the element $P$ of the mANss detects two
elements of $\pi_{8,4}\mathit{mmf}$ because of the presence of $\tau c$ in
higher filtration. These two homotopy elements are detected by $\tau c$ and by
$P$ in the mAss. Consequently, the mANss element $P$ corresponds to the mAss
element $P$, and it also corresponds to the mAss element $\tau c$.
Fortunately, this kind of complication never arises for us in practice. For
example, none of the correspondences listed in LABEL:table:ASSANSS exhibit
this type of behavior.

######  Remark 2.7.

The element $2$ of the mANss $E_{\infty}$-page detects a single element in
homotopy since there are no elements in higher filtration. On the other hand,
the element $h_{0}$ of the mAss $E_{\infty}$-page detects infinitely many
elements in homotopy, all of which differ by a $2$-adic unit factor, because
of the presence of $h_{0}^{k}$ in higher filtration. Consequently, while $2$
and $h_{0}$ are a corresponding pair, they do not detect the same sets of
homotopy elements. Rather, the homotopy elements detected by $2$ form a subset
of the homotopy elements detected by $h_{0}$.

Among the corresponding pairs listed in LABEL:table:ASSANSS, the same
phenomenon occurs for $h_{2}$, $g$, $\Delta h_{1}$, and $4\Delta^{2}$. In all
of these cases, the homotopy elements detected by the mANss $E_{\infty}$-page
element form a subset of the homotopy elements detected by the mAss
$E_{\infty}$-page element.

Multiplicative structure respects corresponding pairs. The following
proposition establishes this principle precisely.

######  Proposition 2.8.

Let $a$ and $a^{\prime}$ be elements of the mANss $E_{\infty}$-page, and let
$b$ and $b^{\prime}$ be elements of the mAss $E_{\infty}$-page. If $a$
corresponds to $a^{\prime}$, $b$ corresponds to $b^{\prime}$, and $ab$ and
$a^{\prime}b^{\prime}$ are non-zero; then $ab$ corresponds to
$a^{\prime}b^{\prime}$.

###### Proof.

Let $a$ and $a^{\prime}$ detect a homotopy element $\alpha$, and let $b$ and
$b^{\prime}$ detect a homotopy element $\beta$. Then $ab$ and
$a^{\prime}b^{\prime}$ detect the product $\alpha\beta$. ∎

######  Remark 2.9.

The motivic Thom reduction map $BP\rightarrow H\mathbb{F}_{2}$ induces a map
from the mANss for $\mathit{mmf}$ to the mAss for $\mathit{mmf}$. This map
detects some corresponding pairs but not all of them. Namely, it detects the
pairs involving $h_{1}$, $h_{2}$, and $g$. These are the elements for which
there is no filtration shift between the mANss and the mAss.

###  2.6. Homotopy elements

LABEL:htpy-name lists some notation that we use for elements in the homotopy
of $\mathit{mmf}$. We use the same symbols as in [BR21] for our motivic
versions. Beware that some of our homotopy elements may not be exactly
compatible under Betti realization with the ones in [BR21]. We discuss the
details of these ambiguities in the following paragraphs.

We define elements in homotopy by specifying the elements in the mANss
$E_{\infty}$-page that detect them. In some cases, it is already easy to see
that these detecting elements survive to the $E_{\infty}$-page. For example,
there are no possible targets for differentials on $h_{1}$ and $h_{2}$; nor
can they be hit by differentials. Beware that we do not yet know that some of
these detecting elements actually survive to the $E_{\infty}$-page. This will
only become apparent after our analysis of Adams-Novikov differentials.

In some cases, there are $E_{\infty}$-page elements in higher filtration. When
this occurs, the specified element in the $E_{\infty}$-page detects more than
one element in homotopy. For example, the element $\tau h_{1}^{3}$ lies in
filtration higher than the filtration of $h_{2}$. Therefore, $h_{2}$ detects
two distinct elements in homotopy. In LABEL:htpy-name, this ambiguity occurs
only for $\nu$, $\kappa_{4}$, and the elements of the form $\nu_{k}$.

The choice of $\nu$ is of little practical signficance to us. For
definiteness, we may use an a priori definition of $\nu$, as discussed in 2.2.
The choices of $\nu_{k}$ will be discussed later in 5.4. The choice of
$\kappa_{4}$ is immaterial for our purposes, so it can be an arbitrary
generator of $\pi_{110,56}$.

Table 3. Some elememts of $\pi_{*,*}\mathit{mmf}$ $(s,w)$ | name | detected by
---|---|---
$(1,1)$ | $\eta$ | $h_{1}$
$(3,2)$ | $\nu$ | $h_{2}$
$(8,5)$ | $\epsilon$ | $c$
$(14,8)$ | $\kappa$ | $d$
$(20,12)$ | $\bar{\kappa}$ | $g$
$(25,13)$ | $\eta_{1}$ | $\Delta h_{1}$
$(27,14)$ | $\nu_{1}$ | $2\Delta h_{2}$
$(51,26)$ | $\nu_{2}$ | $\Delta^{2}h_{2}$
$(96,48)$ | $D_{4}$ | $2\Delta^{4}$
$(99,50)$ | $\nu_{4}$ | $\Delta^{4}h_{2}$
$(110,56)$ | $\kappa_{4}$ | $\Delta^{4}d$
$(123,62)$ | $\nu_{5}$ | $2\Delta^{5}h_{2}$
$(147,74)$ | $\nu_{6}$ | $\Delta^{6}h_{2}$
$(192,96)$ | $M$ | $\Delta^{8}$

######  Remark 2.10.

$(20,4,12)$ Bruner and Rognes choose $\overline{\kappa}$ by reference to the
unit map $S\rightarrow\mathit{tmf}$, together with a prior choice of
$\overline{\kappa}$ in $\pi_{20}S$. For our purposes, we only need that
$\overline{\kappa}$ is detected by $g$ in the mANss $E_{\infty}$-page, so we
may choose $\overline{\kappa}$ to be compatible with the one in [BR21].

There is a slight complication with $\overline{\kappa}$. In [Isa19] and
[IWX20], the symbol $\overline{\kappa}$ is used for an element of
$\pi_{20,11}S^{0,0}$ that is detected by $\tau g$ in the motivic Adams
spectral sequence. The point is that $g$ does not survive the May spectral
sequence, so it does not exist in the motivic Adams spectral sequence.

Here, we use $\overline{\kappa}$ for an element of $\pi_{20,12}\mathit{mmf}$.
This element is detected by $g$ in the Adams spectral sequence for
$\mathit{mmf}$. The unit map $S^{0,0}\to\mathit{mmf}$ takes
$\overline{\kappa}$ to $\tau\overline{\kappa}$.

######  Remark 2.11.

Bruner and Rognes refer to the “edge homomorphism" in order to specify certain
elements in $\pi_{*}\mathit{tmf}$. From the perspective of the Adams-Novikov
spectral sequence, this edge homomorphism takes a particularly convenient form
that can be easily described as a surjection followed by an injection. The
surjection takes $\pi_{*}\mathit{tmf}$ onto its quotient by elements that are
detected in strictly positive Adams-Novikov filtration. In other words, the
surjection maps $\pi_{*}\mathit{tmf}$ onto the Adams-Novikov $E_{\infty}$-page
in filtration $0$. Then the injection is the inclusion of the Adams-Novikov
$E_{\infty}$-page into the Adams-Novikov $E_{2}$-page in filtration $0$. In
other words, the edge homomorphism detects the homotopy elements that are
detected in Adams-Novikov filtration $0$. This description of the edge
homomorphism applies equally well in the setting of $\pi_{*,*}\mathit{mmf}$
and the motivic Adams-Novikov spectral sequence.

The edge homomorphism depends on the choice of $\Delta$ (see 3.8). Beware that
our choice of $\Delta$ does not guarantee that our edge homomorphism is
identical to the one discussed in [BR21]. Consequently, our definitions of the
homotopy elements $D_{4}$ and $M$ in LABEL:htpy-name may not be the same as
[BR21, Definition 9.22]. All possible choices of $\Delta$ differ by multiples
of $2$, so $\Delta^{k}$ is well-defined up to multiples of $2^{k}$. Therefore,
our choices of $D_{4}$ and $M$ agree with the Bruner-Rognes definitions up to
multiples of $16$ and $256$ respectively.

###  2.7. $v_{1}$-periodicity

Part of the mANss for $\mathit{mmf}$ reflects $v_{1}$-periodic homotopy. The
pattern of differentials in this part is similar to the Adams-Novikov
differentials for $\mathit{ko}$ (see [Bau08, page 31]). We consider this part
separately and omit them from computations of higher differentials. Beware
that we are not employing an intrinsic definition of $v_{1}$-periodic
homotopy. Rather, we are simply observing some specific structure in the mANss
for $\mathit{mmf}$.

In the mANss $E_{2}$-page, consider elements of the form
$\tau^{a}h_{1}^{b}P^{m}(4a)^{\epsilon}\Delta^{n}$, where $\epsilon$ equals $0$
or $1$ and $m+\epsilon>0$. We refer to these elements as the $v_{1}$-periodic
classes.

Note that $1$ and $\Delta^{n}$ (as well as their $\tau$ multiples and $h_{1}$
multiples) are excluded from this family of elements. The knowledgeable reader
may observe that these powers of $\Delta$ satisfy an intrinsic definition of
$v_{1}$-periodicity. Our family is constructed for its practical convenience,
not for its intrinsic properties. The $v_{1}$-periodic elements, as we have
defined them, only interact with each other through the Adams-Novikov
differentials. However, the powers of $\Delta$ support Adams-Novikov
differentials that take values outside of the $v_{1}$-periodic family.
Consequently, we consider them in conjunction with the non-$v_{1}$-periodic
elements.

Figures 1 and 2 display the $v_{1}$-periodic portions of the mANss
$E_{2}$-pages and $E_{\infty}$-pages respectively. Our other charts exclude
the $v_{1}$-periodic family.

###  2.8. The spectrum $\mathit{mmf}/\tau$

Consider the cofiber sequence

(2.1)
$\Sigma^{0,-1}\mathit{mmf}\xrightarrow{\tau}\mathit{mmf}\xrightarrow{i}\mathit{mmf}/\tau\xrightarrow{q}\Sigma^{1,-1}\mathit{mmf}$

of $\mathit{mmf}$-modules. The spectrum $\mathit{mmf}/\tau$ is a 2-cell
$\mathit{mmf}$-module, in the sense that it is built from two copies of
$\mathit{mmf}$. We refer to $i$ as inclusion of the bottom cell, and we refer
to $q$ as projection to the top cell.

The mANss for $\mathit{mmf}/\tau$ has a particularly simple algebraic form.
The $E_{2}$-page is isomorphic to the $E_{2}$-page of the classical Adams-
Novikov spectral sequence for $\mathit{tmf}$, except that it has a third
degree. However, this additional degree carries no extra information since it
equals half of the total degree, i.e., the sum of the stem and the Adams-
Novikov filtration.

Moreover, the mANss for $\mathit{mmf}/\tau$ collapses. There are no
differentials, so the $E_{\infty}$-page equals the $E_{2}$-page. Even better,
there are no possible hidden extensions for degree reasons. Consequently, the
homotopy of $\mathit{mmf}/\tau$ is isomorphic to the classical Adams-Novikov
$E_{2}$-page for $\mathit{tmf}$. Therefore, we take the homotopy of
$\mathit{mmf}/\tau$ as given since it is entirely algebraic information. The
results discussed in this paragraph are $\mathit{tmf}$ versions of the results
in [Isa19, Section 6.2], which are stated for the sphere spectrum.

We use the notation of LABEL:generators in order to describe homotopy elements
in $\pi_{*,*}\mathit{mmf}/\tau$. On the other hand, we need to be more careful
about notation for elements in $\pi_{*,*}\mathit{mmf}$. We can specify
elements in $\pi_{*,*}\mathit{mmf}$ by giving detecting elements in the mANss
$E_{\infty}$-page, but this only specifies homotopy elements up to higher
filtration. See Section 2.6 for more discussion of choices of elements in
$\pi_{*,*}\mathit{mmf}$.

The mAss for $\mathit{mmf}/\tau$ is isomorphic to the algebraic Novikov
spectral sequence, for which we have complete information [Bae]. This is a
$\mathit{tmf}$ version of the results in [GWX21], which are stated for the
sphere spectrum.

###  2.9. Inclusion and projection

We discuss the inclusion $i$ and the projection $q$ from Equation 2.1 in more
detail. Many of these ideas first appeared in [Isa19, Chapter 5] in more
primitive forms.

We already observed that both $i$ and $q$ are $\mathit{mmf}$-module maps. Note
that the inclusion $i$ is a ring map, but the projection $q$ is not. They
induce maps of motivic Adams-Novikov spectral sequences. These spectral
sequence maps are in fact module maps over the mANss for $\mathit{mmf}$.
Similarly, the induced maps of homotopy groups are
$\pi_{*,*}\mathit{mmf}$-module maps.

We describe the inclusion $i:\mathit{mmf}\rightarrow\mathit{mmf}/\tau$ of the
bottom cell in computational terms. If $\alpha$ is a homotopy element that is
not a multiple of $\tau$, then $i(\alpha)$ is an element of the mANss
$E_{2}$-page that detects $\alpha$. On the other hand, if $\alpha$ is a
multiple of $\tau$, then $i(\alpha)$ is zero. This fact is closely related to
the observation that the motivic Adams-Novikov spectral sequence is the same
as the $\tau$-Bockstein spectral sequence.

LABEL:htpy-name gives a number of values of $i$. For example, we have
$i(\eta)=h_{1}$. In fact, we have defined the elements in the middle column of
the table to have the appropriate values under $i$.

For later use, we describe the computational implication that
$q:\mathit{mmf}/\tau\rightarrow\Sigma^{1,-1}\mathit{mmf}$ is an
$\mathit{mmf}$-module map. Let $\alpha$ be an element of
$\pi_{*,*}\mathit{mmf}$, and let $x$ be an element of
$\pi_{*,*}\mathit{mmf}/\tau$. The object $\mathit{mmf}/\tau$ is a right
$\mathit{mmf}$-module, and

$x\cdot\alpha=x\cdot i(\alpha),$

where the dot on the left side represents the module action and the dot on the
right side represents the multiplication of the ring spectrum
$\mathit{mmf}/\tau$. Then we have that

(2.2) $q(x)\cdot\alpha=q(x\cdot\alpha)=q(x\cdot i(\alpha)),$

where the dot on the left represents multiplication in $\mathit{mmf}$; the dot
in the center represents the $\mathit{mmf}$-module action on
$\mathit{mmf}/\tau$; and the dot on the right represents multiplication in
$\mathit{mmf}/\tau$.

We need a precise statement about the values of $q$. Our desired statement has
essentially the same content as [BHS19, Theorem 9.19(1c)], which we
reformulate into a form that is more convenient for us.

######  Proposition 2.12.

Let $x$ be an element of the mANss $E_{2}$-page that is not divisible by
$\tau$, and suppose that there is a non-zero motivic Adams-Novikov
differential $d_{2r+1}(x)=\tau^{r}y$. If we consider $x$ as an element of
$\pi_{*,*}\mathit{mmf}/\tau$, then the element $q(x)$ of
$\pi_{*,*}\mathit{mmf}$ is detected by $-\tau^{r-1}y$ in the mANss
$E_{\infty}$-page.

###### Proof.

The proof is a chase of the right side of the diagram

$\textstyle{\mathit{mmf}/\tau\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau^{r}}$$\textstyle{\mathit{mmf}/\tau^{r+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathit{mmf}/\tau^{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{\mathit{mmf}/\tau}$$\textstyle{\mathit{mmf}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{\tau^{r-1}}$$\scriptstyle{\tau^{r}}$$\textstyle{\mathit{mmf}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{\mathit{mmf}/\tau^{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{\mathit{mmf}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{\tau^{r-1}}$$\textstyle{\mathit{mmf}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\mathit{mmf}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mathit{mmf}/\tau\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{\mathit{mmf},}$

in which the rows are cofiber sequences. We start with the element $x$ in
$\pi_{*,*}\mathit{mmf}/\tau$ in the bottom row. This element lifts to
$\mathit{mmf}/\tau^{r}$ in the middle row by [BHS19, Theorem 9.19] because $x$
survives to the $E_{2r+1}$-page. The map $\beta$ is the “Bockstein" mentioned
in [BHS19, Theorem 9.19], so we have that $\beta(x)$ equals $-y$ in the upper
right corner of the diagram. Then $-y$ lifts to an element of
$\pi_{*,*}\mathit{mmf}$ in the middle row that is detected by $-y$. Finally,
multiply by $\tau^{r-1}$ to obtain $q(x)$. ∎

######  Remark 2.13.

2.12 requires that $x$ supports a non-zero Adams-Novikov differential. On the
other hand, suppose that $x$ is a permanent cycle. Then $x$ is in the image of
$i$, and $q(x)=0$ since the composition $qi$ is zero.

###  2.10. Hidden extensions

We briefly review the notion of hidden extensions in spectral sequences. We
adopt the following definition of hidden extensions.

######  Definition 2.14.

[Isa19, Definition 4.1.2] Let $\alpha$ be an element in the target of a
multiplicative spectral sequence, and suppose that $\alpha$ is detected by an
element $a$ in the $E_{\infty}$-page of the spectral sequence. A hidden
extension by $\alpha$ is a pair of elements $b$ and $c$ of the
$E_{\infty}$-page such that:

  1. (1)

the product $a\cdot b$ equals zero in the $E_{\infty}$-page.

  2. (2)

the element $b$ detects an element $\beta$ in the target such that $c$ detects
the product $\alpha\cdot\beta$.

  3. (3)

if there exists an element $\beta^{\prime}$ of the target that is detected by
$b^{\prime}$ such that $\alpha\cdot\beta^{\prime}$ is detected by $c$, then
the filtration of $b^{\prime}$ is less than or equal to the filtration of $b$.

We will use projection $q$ to simplify our analysis of hidden extensions. We
shall show that two different products in $\pi_{*,*}\mathit{mmf}$ are the
image of the same element in $\pi_{*,*}\mathit{mmf}/\tau$. Therefore, they are
equal.

######  Method 2.15.

Suppose that $\alpha$ is not divisible by $\tau$, so $i(\alpha)=a$, where $a$
is an element of the mANss that detects $\alpha$. Consider a possible hidden
$\alpha$ extension from $b$ to $c$ in the mANss for $\mathit{mmf}$. If $b$ and
$c$ detect classes $\beta$ and $\gamma$ that are annihilated by $\tau$, then
$\beta$ and $\gamma$ are in the image of projection $q$ to the top cell. Let
$\overline{b}$ and $\overline{c}$ be their pre-images in
$\pi_{*,*}(\mathit{mmf}/\tau)$. Since this latter object is algebraic and
completely known, we can determine whether $\overline{b}$ and $\overline{c}$
are related by an extension by mere inspection.

Equation 2.2 shows that

$q(\overline{b}\cdot a)=q(\overline{b}\cdot
i(\alpha))=q(\overline{b})\cdot\alpha=\beta\cdot\alpha,$

where the first two dots represent multiplication in $\mathit{mmf}/\tau$,
while the last two dots represent multiplication in $\mathit{mmf}$. If
$\overline{b}\cdot a$ equals $\overline{c}$, then $\beta\cdot\alpha$ equals
$q(\overline{c})=\gamma$, and there is a hidden $\alpha$ extension from $b$ to
$c$.

On the other hand, if $\overline{b}\cdot a$ equals zero, then
$\beta\cdot\alpha$ equals zero, and there is not a hidden $\alpha$ extension
from $b$ to $c$.

In practice, 2.15 is very effective for determining hidden extensions. The
main restriction is that it only applies to extensions between classes that
are annihilated by $\tau$.

######  Example 2.16.

$(54,2,28)$ We illustrate 2.15 with a concrete example of the hidden $2$
extension from $\Delta^{2}h_{2}^{2}$ to $\tau^{4}dg^{2}$ in the 54-stem. In
this example, we assume some knowledge of the relevant Adams-Novikov
differentials (see Section 3). Consequently, one should view this example as a
deduction of a hidden extension from previously determined differentials.

First, multiply by $\tau g$. If we establish a hidden $2$ extension from
$\tau\Delta^{2}h_{2}^{2}g$ to $\tau^{5}dg^{3}$ in the 74-stem, then we can
immediately conclude the desired extension in the 54-stem. This step already
requires motivic technology, since both $\Delta^{2}h_{2}^{2}g$ and $dg^{3}$
are hit by classical Adams-Novikov differentials.

The key point is that the two elements under consideration in the 74-stem are
non-zero but annihilated by $\tau$. They are annihilated by $\tau$ because of
the differentials $d_{5}(\Delta^{3}h_{2})=\tau^{2}\Delta^{2}h_{2}^{2}g$ and
$d_{13}(2\Delta^{3}h_{2})=\tau^{6}dg^{3}$, to be proved later in Propositions
3.7 and 3.15.

The elements $\tau\Delta^{2}h_{2}^{2}g$ and $\tau^{5}dg^{3}$ represent classes
in $\pi_{74,39}\mathit{mmf}$ that are annihilated by $\tau$. Therefore, these
elements lie in the image of
$q:\pi_{75,38}\mathit{mmf}/\tau\rightarrow\pi_{74,39}\mathit{mmf}$.

By 2.12, the preimages in $\pi_{75,38}\mathit{mmf}/\tau$ are $\Delta^{3}h_{2}$
and $2\Delta^{3}h_{2}$ respectively. These two elements are connected by a $2$
extension. Therefore, their images under $q$ are also connected by a $2$
extension.

###  2.11. Toda brackets

For background on Massey products and Toda brackets, including statements of
the May convergence theorem and the Moss convergence theorem, we refer readers
to [Tod62], [May69], [Mos70] and also [Isa19], [BK21].

Massey products in the $E_{2}$-page of an Adams or Adams-Novikov spectral
sequence are algebraic information since they are part of the structure of Ext
groups. Some Toda brackets in homotopy can be deduced directly from these
Massey products using the Moss convergence theorem. In order to apply this
theorem, one must establish the absence of crossing differentials. Whenever we
apply the Moss convergence theorem, there will be no possible crossing
differentials. In other words, the crossing differentials condition is
satisfied for algebraic reasons. Thus, the Toda brackets that we use are
algebraic in the sense that they can be deduced directly from the algebraic
structure of Ext.

######  Remark 2.17.

In general, Massey products and Toda brackets are defined as sets, not
elements. An equality of the form $\langle\alpha,\beta,\gamma\rangle=\delta$
means that

  1. (1)

$\delta$ is contained in the bracket;

  2. (2)

the bracket has zero indeterminacy.

The following lemma gives an explicit example of an algebraic deduction of a
Toda bracket. See LABEL:htpy-name for an explanation of the notation.

######  Lemma 2.18.

$(8,3,5)$ The Toda bracket $\langle\nu,\eta,\nu\rangle$ in
$\pi_{8,5}\mathit{mmf}$ is detected by $c$ and has no indeterminacy.

###### Proof.

The proof follows several steps:

  1. (1)

Establish the Massey product $c=\langle h_{2},h_{1},h_{2}\rangle$ in the
$E_{2}$-page of the mANss.

  2. (2)

Check that there are no crossing differentials.

  3. (3)

Check that the Toda bracket $\langle\nu,\eta,\nu\rangle$ is well-defined and
that it has no indeterminacy.

  4. (4)

Apply the Moss convergence theorem to the Massey product and deduce the
desired Toda bracket.

For step (1), we check the following statements:

  1. (a)

The Massey product is well-defined because of the relation $h_{1}h_{2}=0$ in
the $E_{2}$-page of the mANss for $\mathit{mmf}$ (see Figure 3).

  2. (b)

The element $c$ is contained in the Massey product [Bau08, Equation (7.3)]
[Bae].

  3. (c)

The indeterminacy is trivial by inspection. In more detail, the indeterminacy
equals $h_{2}\cdot E_{2}^{5,1,3}$. The only non-zero element of
$E_{2}^{5,1,3}$ is $h_{1}v_{1}^{2}$, and $h_{2}\cdot h_{1}v_{1}^{2}=0$. This
last relation holds already in the $E_{2}$-page of the motivic algebraic
Novikov spectral sequence [Bae].

For step (2), we need to check for crossing differentials for the relation
$h_{1}h_{2}$ in degree $(4,2,3)$. We are looking for non-zero Adams-Novikov
differentials in degrees $(5,f,3)$, where $f<1$. There are no possible sources
for such differentials (see Figure 3).

For step (3), we check that the Toda bracket is well-defined because $\eta\nu$
is zero in $\pi_{4,3}\mathit{mmf}$ for degree reasons. The indeterminacy
equals $\nu\cdot\pi_{5,3}\mathit{mmf}$, which is zero for degree reasons.

For step (4), we apply the Moss convergence theorem. The theorem implies that
there exists an element in $\langle h_{2},h_{1},h_{2}\rangle$ that is a
permanent cycle and that detects an element in $\langle\nu,\eta,\nu\rangle$.
Since there are no indeterminacies for both the Massey product and the Toda
bracket, the permanent cycle must be $c$. ∎

##  3\. Differentials

In this section, we compute all differentials in the mANss for $\mathit{mmf}$,
proving hidden extensions and Toda brackets only as needed along the way. Our
results are presented in logical order, so each proof only depends on earlier
results. We return to a more exhaustive study of hidden extensions later in
Section 4.

######  Theorem 3.1.

LABEL:diff lists all of the non-zero differentials on all of the
indecomposable elements of each mANss $E_{r}$-page.

###### Proof.

The differentials are proved in the various propositions later in this
section. The last column of LABEL:diff indicates the specific proposition that
proves each differential.

Some indecomposables do not support differentials. In most cases, this follows
for degree reasons, i.e., because there are no possible targets. 3.29 handles
two slightly more difficult cases. ∎

All differentials follow from straightforward applications of the Leibniz rule
to the ones listed in LABEL:diff.

###  3.1. $d_{3}$ differentials

######  Proposition 3.2.

$(5,1,3)$ $d_{3}(h_{1}v_{1}^{2})=\tau h_{1}^{4}$.

###### Proof.

In the mAss $E_{2}$-page, $h_{1}^{4}$ is a non-zero element that is
annihilated by $\tau$. By inspection, $h_{1}^{4}$ corresponds to the element
of the same name in the mANss. Therefore, $\tau h_{1}^{4}$ must be hit by an
Adams-Novikov differential, and there is only one possibility. ∎

######  Proposition 3.3.

$(12,0,6)$ $d_{3}(4a)=\tau Ph_{1}^{3}$.

###### Proof.

For degree reasons, $d_{3}(P)=0$. Thus 3.2 implies that $d_{3}(P\cdot
h_{1}v_{1}^{2})=\tau Ph_{1}^{4}.$ We have the relation $P\cdot
h_{1}v_{1}^{2}=h_{1}\cdot 4a$ in the Adams-Novikov $E_{2}$-page. Note that
this relation arises from a hidden $h_{1}$ extension from $h_{0}^{2}a$ to
$P\overline{h_{1}^{4}}$ in the algebraic Novikov spectral sequence [Bae].
Therefore, $4a$ must also support a $d_{3}$ differential, and there is only
one possibility. ∎

The Leibniz rule, combined with 3.2 and 3.3, implies some additional $d_{3}$
differentials. By inspection, the other multiplicative generators do not
support $d_{3}$ differentials.

######  Remark 3.4.

All of the $d_{3}$ differentials are $h_{1}$-periodic, in the sense that they
can be computed in the localization of the mANss $E_{2}$-page in which $h_{1}$
is inverted. This localized spectral sequence computes the homotopy of the
$\eta$-periodic spectrum $\mathit{mmf}[\eta^{-1}]$. See [GI15, Section 6.1]
for a related discussion.

###  3.2. Corresponding pairs

Earlier in Section 2.5, we discussed the notion of elements from the mANss and
from the mAss that correspond. Having computed the $d_{3}$ differentials, we
are now in a position to establish a number of corresponding pairs that will
be used in later arguments.

######  Theorem 3.5.

LABEL:table:ASSANSS lists some pairs of elements that correspond.

Table 4. Some corresponding elements in the motivic Adams and motivic Adams-Novikov spectral sequences  mANss degree | mANss element | mAss element | mAss degree
---|---|---|---
$(0,0,0)$ | $2$ | $h_{0}$ | $(0,1,0)$
$(1,1,1)$ | $h_{1}$ | $h_{1}$ | $(1,1,1)$
$(3,1,2)$ | $h_{2}$ | $h_{2}$ | $(3,1,2)$
$(14,2,8)$ | $d$ | $d$ | $(14,4,8)$
$(20,4,12)$ | $g$ | $g$ | $(20,4,12)$
$(25,1,13)$ | $\Delta h_{1}$ | $\Delta h_{1}$ | $(25,5,13)$
$(27,1,14)$ | $2\Delta h_{2}$ | $an$ | $(27,6,14)$
$(48,0,24)$ | $4\Delta^{2}$ | $\Delta^{2}h_{0}^{2}$ | $(48,10,24)$
$(110,2,56)$ | $\Delta^{4}d$ | $\Delta^{4}d$ | $(110,20,56)$

###### Proof.

We discuss the correspondence between $2\Delta h_{2}$ and $an$ in detail. Most
of the other corresponding pairs are established with essentially the same
argument. Some slightly more difficult cases are established later in Lemmas
3.9 and 3.33.

For degree reasons, the element $2\Delta h_{2}$ of the mANss for
$\mathit{mmf}$ cannot support an Adams-Novikov differential, nor can it be hit
by an Adams-Novikov differential. (Beware that $\Delta h_{2}$ does support a
differential.) Therefore, $2\Delta h_{2}$ detects some element $\alpha$ in
$\pi_{27,14}\mathit{mmf}$.

The inclusion $i:\mathit{mmf}\rightarrow\mathit{mmf}/\tau$ induces a map

(3.1)
$\textstyle{E_{2}(\mathit{mmf})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{2}(\mathit{mmf}/\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{*,*}\mathit{mmf}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{*,*}\mathit{mmf}/\tau}$

of motivic Adams spectral sequences. The spectral sequence on the right is
identified with the algebraic Novikov spectral sequence that converges to the
classical Adams-Novikov $E_{2}$-page for $\mathit{tmf}$ [GWX21].

The element $\alpha$ in the lower left corner maps to $2\Delta h_{2}$ in the
lower right corner. This latter element is detected by $an$ in filtration $6$
in the upper right corner [Bae]. Therefore, $\alpha$ is detected in the upper
left corner in filtration at most $6$. The only possible value is $an$. ∎

######  Remark 3.6.

Previous knowledge of the $d_{3}$ differentials is required in order to
conclude that $2\Delta h_{2}$ (and other elements as well) does not support an
Adams-Novikov differential. For example, it is conceivable that
$d_{25}(2\Delta h_{2})=\tau^{12}h_{1}^{26}$. However, we already know that
$\tau^{12}h_{1}^{26}$ is hit by the differential
$d_{3}(\tau^{11}h_{1}^{22}\cdot h_{1}v_{1}^{2})$.

###  3.3. $d_{5}$ differentials

Having determined all $d_{3}$ differentials, one can mechanically compute the
$E_{4}$-page. Through the 22-stem, no additional differentials are possible
for degree reasons, so the $E_{4}$-page equals the $E_{\infty}$-page in that
range.

######  Proposition 3.7.

$(24,0,12)$ There exists a generator $\Delta$ of the mANss $E_{2}$-page in
degree $(24,0,12)$ such that $d_{5}(\Delta)=\tau^{2}h_{2}g.$

###### Proof.

The mAss element $h_{2}g$ is annihilated by $\tau^{2}$ in the $E_{2}$-page.
Moreover, $\tau h_{2}g$ does not support a hidden $\tau$ extension in the mAss
because of the presence of $\overline{\tau h_{2}g}$ in the homotopy of
$\mathit{mmf}/\tau$. More precisely, projection to the top cell takes
$\overline{\tau h_{2}g}$ to $\tau h_{2}g$, so $\tau h_{2}g$ must detect
homotopy elements that are annihilated by $\tau$.

The mANss element $h_{2}g$ corresponds to the mAss element $h_{2}g$ because of
LABEL:table:ASSANSS and 2.8. Therefore, $\tau^{2}h_{2}g$ must be hit by an
Adams-Novikov differential. The only possibility is a $d_{5}$ differential
whose source is in degree $(24,0,12)$. Since $\tau^{2}h_{2}g$ is not a
multiple of $2$, the source of the differential must be a generator. ∎

######  Remark 3.8.

$(24,0,12)$ 3.7 does not uniquely specify $\Delta$. Since $4\tau^{2}h_{2}g$ is
zero in the mANss $E_{2}$-page, $\Delta$ is only well-defined up to multiples
of $4$. Later in 5.8 we will make a further refinement in the definition of
$\Delta$. Also note that the choice of $\Delta$ depends on a previous choice
of $h_{2}$, as in 2.2.

The Leibniz rule, together with 3.7, implies additional $d_{5}$ differentials.
The other multiplicative generators of the $E_{5}$-page do not support
differentials.

Of particular note is the differential

$d_{5}(\Delta^{2})=2\Delta d_{5}(\Delta)=2\tau^{2}\Delta h_{2}g.$

This easy computation is an Adams-Novikov version of Bruner’s theorem on the
interaction between Adams differentials and algebraic squaring operations
[BMMS86] [Bru84]. However, its corresponding Adams differential
$d_{2}(\Delta^{2})=\tau^{2}ang$ is not as easy to obtain by direct analysis of
the Adams spectral sequence [BR21]. The difficulty is that $\Delta^{2}$ is not
the value of an algebraic squaring operation since $\Delta$ is not present in
the Adams $E_{2}$-page. By “postponing" the differential that hits
$\tau^{2}h_{2}g$ from algebra to topology, we obtain an easier argument for
the differential on $\Delta^{2}$.

######  Lemma 3.9.

$(48,0,24)$ The element $4\Delta^{2}$ of the mANss for $\mathit{mmf}$
corresponds to $\Delta^{2}h_{0}^{2}$ in the mAss for $\mathit{mmf}$.

###### Proof.

Having established that $d_{5}(\Delta^{2})=2\tau^{2}\Delta h_{2}g$ as a
consequence of the Leibniz rule and 3.7, we conclude that $4\Delta^{2}$ does
not support an Adams-Novikov differential for degree reasons. (Beware that
$2\Delta^{2}$ does support a differential, but we do not need to know that
already.) Note that $4\Delta^{2}$ is detected in the algebraic Novikov
spectral sequence by $\Delta^{2}h_{0}^{2}$, which has filtration $10$. Using
the argument in the proof of 3.5, we conclude that $4\Delta^{2}$ corresponds
to an element in the mAss with filtration at most $10$. However, there are
three possibilities: $\Delta^{2}$, $\Delta^{2}h_{0}$, and
$\Delta^{2}h_{0}^{2}$.

The top horizontal map of Diagram (3.1) takes $\Delta^{2}$ and
$\Delta^{2}h_{0}$ to elements of the same name. These elements detect
$\Delta^{2}$ and $2\Delta^{2}$ in the Adams-Novikov $E_{2}$-page. This means
that $4\Delta^{2}$ cannot correspond to $\Delta^{2}$ or $\Delta^{2}h_{0}$. ∎

###  3.4. $d_{7}$ differentials

The main goal of this section is to establish some $d_{7}$ differentials in
3.13 and 3.20. In order to obtain these differentials, we will need some
hidden extensions and some later differentials. We establish these other
results first, in order to preserve strict logical order.

######  Lemma 3.10.

$(3,1,2)$ There is a hidden $2$ extension from $2h_{2}$ to $\tau h_{1}^{3}.$

###### Proof.

According to LABEL:table:ASSANSS and 2.8, the mANss element $2h_{2}$
corresponds to the mAss element $h_{0}h_{2}$. The element $h_{0}h_{2}$
supports an $h_{0}$ extension in the mAss $E_{2}$-page, so $2h_{2}$ must
support a $2$ extension in the mANss. There is only one possible target for
this extension. ∎

######  Remark 3.11.

The hidden extension of 3.10 is the first in an infinite family of similar
hidden extensions from the elements $2h_{2}g^{k}$ to the elements $\tau
h_{1}^{3}g^{k}$. For $k\geq 1$, these extensions are “exotic” in the sense
that they do not occur classically, since both $2h_{2}g^{k}$ and
$h_{1}^{3}g^{k}$ are the targets of classical Adams-Novikov differentials.

######  Lemma 3.12.

$(27,1,14)$ There is a hidden $2$ extension from $2\Delta h_{2}$ to
$\tau\Delta h_{1}^{3}$.

###### Proof.

We already observed in LABEL:table:ASSANSS that $2\Delta h_{2}$ and $\Delta
h_{1}\cdot h_{1}^{2}$ correspond to $an$ and $\Delta h_{1}^{3}$ in the mAss.
In the mAss $E_{2}$-page, we have the relation $h_{0}\cdot an=\tau\Delta
h_{1}^{3}$. Therefore, there must be a hidden $2$ extension between the
corresponding Adams-Novikov elements. ∎

######  Proposition 3.13.

  1. (1)

$(24,0,12)$ $d_{7}(4\Delta)=\tau^{3}h_{1}^{3}g$.

  2. (2)

$(48,0,24)$ $d_{7}(2\Delta^{2})=\tau^{3}\Delta h_{1}^{3}g$.

###### Proof.

3.7 says that $\tau^{2}h_{2}g$ is hit by an Adams-Novikov differential, so
$2\tau^{2}h_{2}g$ is also hit by an Adams-Novikov differential. 3.11 says that
there is a hidden $2$ extension from $2h_{2}g$ to $\tau h_{1}^{3}g$.
Therefore, $\tau^{3}h_{1}^{3}g$ is hit by a differential, and there is just
one possible source for this differential.

The proof for the second differential is essentially the same. We need a
hidden $2$ extension from $2\Delta h_{2}g$ to $\tau\Delta h_{1}^{3}g$, which
follows from 3.12 and multiplication by $g$. ∎

######  Remark 3.14.

3.7 and 3.13 show that both $2\tau h_{2}g^{k}$ and $\tau^{2}h_{1}^{3}g^{k}$
are annihilated by $\tau$. In hindsight, we can see that the hidden $2$
extensions connecting them are examples of 2.15. Their pre-images in
$\mathit{mmf}/\tau$ are $2\Delta g^{k-1}$ and $4\Delta g^{k-1}$, which are
related by $2$ extensions.

However, beware that we needed the hidden $2$ extension from $2h_{2}$ to $\tau
h_{1}^{3}$ in order to establish the differential $d_{7}(4\Delta)$. An
independent proof of 3.10 is necessary in order to avoid a circular argument.

Before finishing the analysis of the $d_{7}$ differential in 3.20, we deduce
some higher differentials.

######  Proposition 3.15.

$(75,1,38)$ $d_{13}(2\Delta^{3}h_{2})=\tau^{6}dg^{3}$.

###### Proof.

We have the relation $ang\cdot an=\tau^{4}dg^{3}$ in the mAss $E_{2}$-page
because of the relations $a^{2}n=\tau d\cdot\Delta h_{1}$ and $\Delta
h_{1}\cdot n=\tau^{3}g^{2}$ [Isa09, Theorem 4.13]. According to
LABEL:table:ASSANSS and 2.8, the mANss elements $2\Delta h_{2}g$, $2\Delta
h_{2}$, $d$, and $g$ correspond to the mAss elements $ang$, $an$, $d$, and
$g$. This means that there is a hidden $2\Delta h_{2}$ extension from $2\Delta
h_{2}g$ to $\tau^{4}dg^{3}$ in the mANss.

Using the Leibniz rule and 3.7, we already know that $2\tau^{2}\Delta h_{2}g$
is hit by the differential $d_{5}(\Delta^{2})$. Therefore, $\tau^{6}dg^{3}$
must also be hit by a differential. There are two possibilities for this
differential: $d_{11}(\tau\Delta^{3}h_{1}^{3})$ and $d_{13}(\Delta^{3}h_{2})$.
However, $\tau\Delta^{3}h_{1}^{3}$ is a product $\tau(\Delta h_{1})^{3}$ of
permanent cycles, so it cannot support a differential. ∎

######  Remark 3.16.

The proof of 3.15 contains an example of 2.15. There is a hidden $2\Delta
h_{2}$ extension from $2\tau\Delta h_{2}g$ to $\tau^{5}dg^{3}$. Both of these
elements are annihilated by $\tau$. Their pre-images under projection to the
top cell of $\mathit{mmf}/\tau$ are $\Delta^{2}$ and $2\Delta^{3}h_{2}$
respectively, which are related by a $2\Delta h_{2}$ extension.

######  Proposition 3.17.

$(56,2,29)$ $d_{9}(\Delta^{2}c)=\tau^{4}h_{1}dg^{2}$.

###### Proof.

Recall from Example 2.16 that there is a hidden $2$ extension from
$\Delta^{2}h_{2}^{2}$ to $\tau^{4}dg^{2}$. The argument for this hidden
extension uses 3.7 and 3.15. Therefore, $\tau^{4}h_{1}dg^{2}$ must be hit by a
differential because $2h_{1}=0$. There is only one possible differential. ∎

######  Proposition 3.18.

In the mAss for $\mathit{mmf}$, we have the Adams differentials:

  1. (1)

$(48,8,24)$ $d_{2}(\Delta^{2})=\tau^{2}ang$.

  2. (2)

$(96,16,48)$ $d_{3}(\Delta^{4})=\tau^{8}ng^{4}$.

###### Proof.

We start with the Adams-Novikov differential
$d_{5}(\Delta^{2})=2\tau^{2}\Delta h_{2}g$. We know from LABEL:table:ASSANSS
and 2.8 that $2\Delta h_{2}g$ corresponds to the element $ang$ in the mAss.
Therefore, $\tau^{2}ang$ must be hit by some Adams differential, and the only
possibility is that $d_{2}(\Delta^{2})$ equals $\tau^{2}ang$.

Next, we apply Bruner’s theorem on the interaction between Adams differentials
and algebraic squaring operations. We refer to [BR21, Theorem 5.6] for a
precise readable statement, although [Bru84], [BMMS86] and [Mäk73] are
preceding references. We apply Bruner’s theorem with $x=\Delta^{2}$, $r=2$,
and $i=8$; so $s=8$, $t=56$, $v=v(48)=1$, and $\overline{a}=h_{0}$. We obtain
that

$d_{*}\operatorname{Sq}^{8}(\Delta^{2})=\operatorname{Sq}^{9}d_{2}(\Delta^{2})\dotplus
h_{0}\cdot\Delta^{2}\cdot
d_{2}(\Delta^{2})=\operatorname{Sq}^{9}(\tau^{2}ang)+h_{0}\cdot\Delta^{2}\cdot\tau^{2}ang=\operatorname{Sq}^{9}(\tau^{2}ang).$

Next, we compute that
$\operatorname{Sq}^{9}(\tau^{2}ang)=\tau^{4}\cdot\tau\Delta h_{1}\cdot
n^{2}\cdot g^{2}$, using the Cartan formula for algebraic squaring operations,
as well as the formulas $\operatorname{Sq}^{2}(a)=\tau\Delta h_{1}$,
$\operatorname{Sq}^{3}(n)=n^{2}$, and $\operatorname{Sq}^{4}(g)=g^{2}$ [BR21,
Theorem 1.20]. Finally, apply the relation $\Delta h_{1}\cdot n=\tau^{3}g^{2}$
to obtain the Adams differential $d_{3}(\Delta^{4})=\tau^{8}ng^{4}$. ∎

######  Remark 3.19.

The careful reader may object to our use of a motivic version of Bruner’s
theorem in the proof of 3.18, while only the classical version of the theorem
has a published proof. In fact, this concern is irrelevant here. One can use
the classical Bruner’s theorem to establish the classical Adams $d_{3}$
differential and then deduce the motivic version of the differential.

######  Proposition 3.20.

$(96,0,48)$ $d_{7}(\Delta^{4})=\tau^{3}\Delta^{3}h_{1}^{3}g$.

###### Proof.

LABEL:table:ASSANSS shows that the mANss element $4\Delta^{2}$ corresponds to
the mAss element $\Delta^{2}h_{0}^{2}$. Therefore, 2.8 shows that the mANss
element $16\Delta^{4}$ corresponds to the mAss element $\Delta^{4}h_{0}^{4}$.

3.18 shows that $\Delta^{4}$ does not survive the mAss. Therefore,
$\Delta^{4}h_{0}^{4}$ does not detect homotopy elements that are divisible by
$16$. Consequently, the corresponding element $16\Delta^{4}$ in the mANss does
not detect homotopy elements that are divisible by $16$. This means that
$\Delta^{4}$ must support an Adams-Novikov differential.

There are two possible values for this differential:
$\tau^{3}\Delta^{3}h_{1}^{3}g$ and $\tau^{9}h_{1}dg^{4}$. However, 3.17 shows
that the latter element is already hit by the differential
$d_{9}(\tau^{5}\Delta^{2}cg^{2})=\tau^{9}h_{1}dg^{4}$. ∎

###  3.5. $d_{9}$ differentials

At this point, we have determined all differentials $d_{r}$ for $r\leq 7$. It
remains to study higher differentials, although some higher differentials have
already been determined in earlier propositions. We continue to proceed
roughly in order of increasing values of $r$, although we occasionally need
some Toda brackets, hidden extensions, and later differentials to preserve
strict logical order.

######  Proposition 3.21.

$(171,1,86)$ $d_{13}(2\Delta^{7}h_{2})=\tau^{6}\Delta^{4}dg^{3}$.

###### Proof.

The argument is nearly identical to the proof of 3.15. The mAss $E_{2}$-page
relation $\Delta^{4}ang\cdot an=\tau^{4}\Delta^{4}dg^{3}$ implies that there
is a hidden $2\Delta h_{2}$ extension from $2\Delta^{5}h_{2}g$ to
$\tau^{4}\Delta^{4}dg^{3}$ in the mANss. We already know that
$2\tau^{2}\Delta^{5}h_{2}g$ is hit by the differential $d_{5}(\Delta^{6})$.
Therefore, $\tau^{6}\Delta^{4}dg^{3}$ must also be hit by a differential.

There are two possibilities for this differential:
$d_{11}(\tau\Delta^{7}h_{1}^{3})$ and $d_{13}(2\Delta^{7}h_{2})$. The former
possibility is ruled out by the decomposition
$\tau\Delta^{6}h_{1}^{2}\cdot\Delta h_{1}$ and the observation that both
$\Delta^{6}h_{1}^{2}$ and $\Delta h_{1}$ survive past the $E_{11}$-page for
degree reasons. ∎

######  Lemma 3.22.

$(150,2,76)$ There is a hidden $2$ extension from $\Delta^{6}h_{2}^{2}$ to
$\tau^{4}\Delta^{4}dg^{2}$.

###### Proof.

The proof is similar to the argument in Example 2.16. We already know the
differentials $d_{5}(\Delta^{7}h_{2})=\tau^{2}\Delta^{6}h_{2}^{2}g$ and
$d_{13}(2\Delta^{7}h_{2})=\tau^{6}\Delta^{4}dg^{3}$ from Propositions 3.7 and
3.21. Therefore, projection to the top cell detects a hidden $2$ extension
from $\tau\Delta^{6}h_{2}^{2}g$ to $\tau^{5}\Delta^{4}dg^{3}$. Finally, use
$\tau g$ multiplication to deduce the hidden $2$ extension on
$\Delta^{6}h_{2}^{2}$. ∎

######  Proposition 3.23.

  1. (1)

$(80,2,41)$ $d_{9}(\Delta^{3}c)=\tau^{4}\Delta h_{1}dg^{2}$.

  2. (2)

$(176,2,89)$ $d_{9}(\Delta^{7}c)=\tau^{4}\Delta^{5}h_{1}dg^{2}$.

###### Proof.

We saw in Example 2.16 that $\tau^{4}dg^{2}$ detects a multiple of $2$.
Therefore, $\Delta h_{1}\cdot\tau^{4}dg^{2}$ must detect zero since $\Delta
h_{1}$ does not support a $2$ extension for degree reasons. Therefore,
$\tau^{4}\Delta h_{1}dg^{2}$ must be hit by a differential, and there is only
one possibility.

The argument for the second differential is nearly identical. 3.22 shows that
the element $\tau^{4}\Delta^{4}dg^{2}$ detects a multiple of $2$. Therefore,
$\Delta h_{1}\cdot\tau^{4}\Delta^{4}dg^{2}$ must detect zero, and there is
only one differential that can hit it. ∎

######  Proposition 3.24.

$(152,2,77)$ $d_{9}(\Delta^{6}c)=\tau^{4}\Delta^{4}h_{1}dg^{2}$.

###### Proof.

The argument is similar to the proof of 3.17. 3.22 shows that
$\tau^{4}\Delta^{4}dg^{2}$ detects a multiple of $2$. Therefore,
$\tau^{4}\Delta^{4}h_{1}dg^{2}$ must be hit by a differential because
$2h_{1}=0$. There is only one possible differential. ∎

######  Lemma 3.25.

$(25,1,13)$ The Toda bracket $\langle\eta,\nu,\tau^{2}\bar{\kappa}\rangle$ is
detected by $\Delta h_{1}$ and has indeterminacy detected by $P^{3}h_{1}$.

###### Proof.

By inspection, the Toda bracket is well-defined and has indeterminacy detected
by $P^{3}h_{1}$ (which is a $v_{1}$-periodic element).

We use the Moss convergence theorem in the mAss for $\mathit{mmf}$. By [Isa09,
Definition 4.4(1)], we have the Massey product $\Delta h_{1}=\langle
h_{1},h_{2},\tau^{2}g\rangle$ in the $E_{2}$-page of the mAss for
$\mathit{mmf}$. There are no possible crossing differentials in the mAss for
$\mathit{mmf}$.

Finally, LABEL:table:ASSANSS implies that the mAss elements $h_{1}$, $h_{2}$,
and $\tau^{2}g$ detect $\eta$, $\nu$, and $\tau^{2}\overline{\kappa}$
respectively (see also LABEL:htpy-name). ∎

######  Lemma 3.26.

$(25,1,13)$ There is a hidden $\nu$ extension from $\Delta h_{1}$ to
$\tau^{2}cg$.

###### Proof.

Lemmas 2.18 and 3.25 show that the Toda brackets $\langle\nu,\eta,\nu\rangle$
and $\langle\eta,\nu,\tau^{2}\bar{\kappa}\rangle$ are detected by $c$ and
$\Delta h_{1}$ respectively.

The hidden $\nu$ extension follows from the shuffling relation

$\nu\langle\eta,\nu,\tau^{2}\bar{\kappa}\rangle=\langle\nu,\eta,\nu\rangle\tau^{2}\bar{\kappa}.$

∎

######  Lemma 3.27.

$(25,1,13)$ There is a hidden $\eta$ extension from $2\Delta h_{2}$ to
$\tau^{2}cg$.

###### Proof.

As in the proof of 3.26, the element $\tau^{2}cg$ detects
$\langle\eta,\nu,\tau^{2}\overline{\kappa}\rangle\nu$, which equals
$\eta\langle\nu,\tau^{2}\overline{\kappa},\nu\rangle$. Therefore, $\tau^{2}cg$
is the target of a hidden $\eta$ extension. There are two possible sources for
such an extension: $\tau\Delta h_{1}^{3}$ and $2\Delta h_{2}$. The former
possibility is ruled out by 3.12, which shows that $\tau\Delta h_{1}^{3}$ is
the target of a hidden $2$ extension. ∎

######  Proposition 3.28.

  1. (1)

$(49,1,25)$ $d_{9}(\Delta^{2}h_{1})=\tau^{4}cg^{2}$.

  2. (2)

$(73,1,37)$ $d_{9}(\Delta^{3}h_{1})=\tau^{4}\Delta cg^{2}$.

  3. (3)

$(145,1,73)$ $d_{9}(\Delta^{6}h_{1})=\tau^{4}\Delta^{4}cg^{2}$.

  4. (4)

$(169,1,85)$ $d_{9}(\Delta^{7}h_{1})=\tau^{4}\Delta^{5}cg^{2}$.

###### Proof.

It follows from 3.27 that there is a hidden $\eta$ extension from $2\Delta
h_{2}g$ to $\tau^{2}cg^{2}$. 3.7 and the Leibniz rule imply that
$d_{5}(\Delta^{2})=2\tau^{2}\Delta h_{2}g$. Therefore, $\tau^{4}cg^{2}$ must
be hit by some differential, and there is only one possibility.

Having established the first differential, we can compute that

$d_{9}(\Delta^{3}h_{1}^{2})=\Delta h_{1}\cdot
d_{9}(\Delta^{2}h_{1})=\tau^{4}\Delta h_{1}cg^{2}.$

Since $\Delta^{3}h_{1}^{2}=\Delta^{3}h_{1}\cdot h_{1}$, it follows that
$d_{9}(\Delta^{3}h_{1})$ equals $\tau^{4}\Delta cg^{2}$.

Similarly,

$d_{9}(\Delta^{7}h_{1}^{2})=\Delta^{5}h_{1}\cdot
d_{9}(\Delta^{2}h_{1})=\tau^{4}\Delta^{5}h_{1}cg^{2},$

from which it follows that $d_{9}(\Delta^{7}h_{1})$ equals
$\tau^{4}\Delta^{5}cg^{2}$. However, we need to observe that
$d_{9}(\Delta^{5}h_{1})$ is zero. The only possible non-zero value for
$d_{9}(\Delta^{5}h_{1})$ is $\tau^{4}\Delta^{3}cg^{2}$, but this is ruled out
by the observation that $\tau^{4}\Delta^{3}cg^{2}$ supports a $d_{9}$
differential by 3.23.

Finally, note that $d_{9}(\Delta^{7}h_{1}^{2})=\Delta h_{1}\cdot
d_{9}(\Delta^{6}h_{1})$. The value of $d_{9}(\Delta^{7}h_{1}^{2})$ was
computed in the previous paragraph. It follows that $d_{9}(\Delta^{6}h_{1})$
equals $\tau^{4}\Delta^{4}cg^{2}$. ∎

######  Proposition 3.29.

  1. (1)

$d_{9}(\Delta^{4}c)=0$.

  2. (2)

$d_{9}(\Delta^{5}c)=0$.

###### Proof.

It follows from 3.28 that $\tau^{4}\Delta^{4}cg^{2}$ and
$\tau^{4}\Delta^{5}cg^{2}$ are targets of $d_{9}$ differentials, so they
cannot support $d_{9}$ differentials. This implies that $\Delta^{4}c$ and
$\Delta^{5}c$ cannot support $d_{9}$ differentials. ∎

The Leibniz rule, together with the differentials given in Propositions 3.23,
3.24, 3.28, and 3.29, determines all $d_{9}$ differentials.

###  3.6. $d_{11}$ differentials

######  Lemma 3.30.

$(14,2,8)$ There is a hidden $\epsilon$ extension from $d$ to $\tau
h_{1}^{2}g$.

###### Proof.

We will show that there is a hidden $\epsilon$ extension from $h_{1}d$ to
$\tau h_{1}^{3}g$. The desired extension follows immediately.

The relation $h_{1}c=h_{2}^{3}$ in the mANss $E_{2}$-page implies that
$\eta\epsilon$ equals $\nu^{3}$. Also, the relation $h_{2}^{2}d=4g$ implies
that $\nu^{2}\kappa=4\overline{\kappa}$. Then

$\eta\epsilon\kappa=\nu^{3}\kappa=4\nu\overline{\kappa}=\tau\eta^{3}\overline{\kappa}.$

The last equality uses the hidden $2$ extension from $2h_{2}$ to $\tau
h_{1}^{3}$, as shown in 3.10. ∎

######  Lemma 3.31.

$(39,3,21)$ There is a hidden $\nu$ extension from $\Delta h_{1}d$ to
$\tau^{3}h_{1}^{2}g^{2}.$

###### Proof.

The element $\Delta h_{1}d$ detects the product $\eta_{1}\cdot\kappa$. 3.26
implies that $\nu\cdot\eta_{1}\cdot\kappa$ equals
$\tau^{2}\epsilon\kappa\overline{\kappa}$. 3.30 implies that this last product
equals $\tau^{3}\eta^{2}\overline{\kappa}^{2}$, which is detected by
$\tau^{3}h_{1}^{2}g^{2}$. ∎

######  Proposition 3.32.

  1. (1)

$(62,2,32)$ $d_{11}(\Delta^{2}d)=\tau^{5}h_{1}g^{3}$.

  2. (2)

$(158,2,80)$ $d_{11}(\Delta^{6}d)=\tau^{5}\Delta^{4}h_{1}g^{3}$.

###### Proof.

The element $\tau^{5}h_{1}^{2}g^{3}$ detects
$\tau^{5}\eta^{2}\overline{\kappa}^{3}$. 3.31 implies that
$\tau^{5}\eta^{2}\overline{\kappa}^{3}$ equals
$\tau^{2}\nu\overline{\kappa}\cdot\eta_{1}\cdot\kappa$. Because of 3.7, we
know that $\tau^{2}\nu\overline{\kappa}$ is zero. Therefore,
$\tau^{5}h_{1}^{2}g^{3}$ is hit by some differential. The only possibility is
that $d_{11}(\Delta^{2}h_{1}d)=\tau^{5}h_{1}^{2}g^{3}$. It follows that
$d_{11}(\Delta^{2}d)=\tau^{5}h_{1}g^{3}$.

For the second formula, multiply by the permanent cycle $\Delta^{4}h_{1}$ to
see that $d_{11}(\Delta^{6}h_{1}d)$ equals $\tau^{5}\Delta^{4}h_{1}^{2}g^{3}$.
It follows that $d_{11}(\Delta^{6}d)$ equals $\tau^{5}\Delta^{4}h_{1}g^{3}$. ∎

###  3.7. $d_{13}$ differentials

We have already established some $d_{13}$ differentials in Propositions 3.15
and 3.21 because we needed those results in order to compute shorter
differentials. We now finish the computation of the $d_{13}$ differentials.

######  Lemma 3.33.

$(110,2,56)$ The element $\Delta^{4}d$ of the mANss for $\mathit{mmf}$
corresponds to the element of the same name in the mAss for $\mathit{mmf}$.

###### Proof.

We have already analyzed all possible Adams-Novikov differentials of length
$11$ or less, and there are no other possible values for a differential on
$\Delta^{4}d$. Therefore, $\Delta^{4}d$ is a permanent cycle in the mANss for
$\mathit{mmf}$.

Now the argument given in the proof of 3.5 applies. The mANss element
$\Delta^{4}d$ is detected in filtration $20$ in the Adams $E_{2}$-page for
$\mathit{mmf}/\tau$. Therefore, $\Delta^{4}d$ corresponds to an element of the
mAss with Adams filtration at most $20$. There is only one possible element in
the mAss with sufficiently low filtration. ∎

######  Lemma 3.34.

  1. (1)

$(39,3,21)$ There is a hidden $\eta$ extension from $\Delta h_{1}d$ to
$2\tau^{2}g^{2}$.

  2. (2)

$(135,3,69)$ There is a hidden $\eta$ extension from $\Delta^{5}h_{1}d$ to
$2\tau^{2}\Delta^{4}g^{2}$.

###### Proof.

LABEL:table:ASSANSS shows that the elements $\Delta h_{1}$ and $d$ in the
mANss for $\mathit{mmf}$ correspond to elements of the same name in the mAss
for $\mathit{mmf}$. The product $\Delta h_{1}\cdot h_{1}d$ is non-zero in the
mAss $E_{2}$-page and also in the mAss $E_{\infty}$-page because there are no
possible differentials that could hit it. (Note that this product is non-zero
in the motivic context, but the corresponding classical product is zero in the
$E_{2}$-page of the Adams spectral sequence for $\mathit{tmf}$.)

Therefore, $\Delta h_{1}d$ must support a hidden $\eta$ extension in the mANss
for $\mathit{mmf}$. There are three possible targets for this extension:
$\tau^{2}g^{2}$, $2\tau^{2}g^{2}$, and $3\tau^{2}g^{2}$. The first and last
possibilities are ruled out by the relation $2\eta=0$.

The argument for the second extension is nearly identical. LABEL:table:ASSANSS
and 2.8 imply that the mANss element $\Delta^{5}h_{1}d$ corresponds to the
mAss element $\Delta^{4}\cdot\Delta h_{1}\cdot d$. The product
$\Delta^{4}\cdot\Delta h_{1}\cdot h_{1}d$ is non-zero in the mAss
$E_{\infty}$-page, so $\Delta^{5}h_{1}d$ must support a hidden $\eta$
extension in the mANss. The only possible target for this extension is
$2\tau^{2}\Delta^{4}g^{2}$. ∎

######  Proposition 3.35.

  1. (1)

$(81,3,42)$ $d_{13}(\Delta^{3}h_{1}c)=2\tau^{6}g^{4}$.

  2. (2)

$(177,3,90)$ $d_{13}(\Delta^{7}h_{1}c)=2\tau^{6}\Delta^{4}g^{4}$.

###### Proof.

3.34 implies that there is a hidden $\eta$ extension from $\Delta h_{1}dg^{2}$
to $2\tau^{2}g^{4}$. 3.23 shows that $\tau^{4}\Delta h_{1}dg^{2}$ is hit by a
differential. Therefore, $2\tau^{6}g^{4}$ must also be hit by a differential.
There is only one possible source for this differential.

The proof for the second formula is similar. There is a hidden $\eta$
extension from $\Delta^{5}h_{1}dg^{2}$ to $2\tau^{2}\Delta^{4}g^{4}$. Since
$\tau^{4}\Delta^{5}h_{1}dg^{2}$ is hit by a differential,
$2\tau^{6}\Delta^{4}g^{4}$ must also be hit by a differential. ∎

###  3.8. $d_{23}$ differentials

######  Lemma 3.36.

$(75,3,38)$ There is a hidden $\eta_{1}$ extension from
$\tau\Delta^{3}h_{1}^{3}$ to $\tau^{9}g^{5}$.

###### Proof.

According to LABEL:table:ASSANSS, the mANss elements $\Delta h_{1}$ and $g$
correspond to elements of the same name in the mAss. In the mAss $E_{2}$-page,
the relations given in [Isa09, Theorem 4.13] imply that $\tau(\Delta
h_{1})^{4}=\tau^{9}g^{5}$. Therefore, in the mANss, $\tau^{9}g^{5}$ detects
the product $\tau\eta_{1}^{4}$. On the other hand, $\tau\Delta^{3}h_{1}^{3}$
detects the product $\tau\eta_{1}^{3}$ in the mANss. ∎

######  Remark 3.37.

$(75,3,39)$ Beware that $\Delta^{3}h_{1}^{3}$ does not support a hidden
$\eta_{1}$ extension. Rather, it supports a non-hidden extension since
$\Delta^{4}h_{1}^{4}$ is non-zero. However, $\Delta^{4}h_{1}^{4}$ is
annihilated by $\tau$, which allows for the hidden extension on
$\tau\Delta^{3}h_{1}^{3}$.

######  Proposition 3.38.

$(121,1,61)$ $d_{23}(\Delta^{5}h_{1})=\tau^{11}g^{6}$.

###### Proof.

The hidden extension of 3.36 implies that there is a hidden $\eta_{1}$
extension from $\tau\Delta^{3}h_{1}^{3}g$ to $\tau^{9}g^{6}$. We already know
that $\tau^{3}\Delta^{3}h_{1}^{3}g$ is zero because of the differential
$d_{7}(\Delta^{4})$ from 3.20. Therefore, $\tau^{11}g^{6}$ must be the value
of some differential, and there is only one possibility. ∎

##  4\. Hidden extensions

In Section 3, we established several hidden extensions in the mANss for
$\mathit{mmf}$ as steps towards computing differentials. In this section, we
finish the analysis of all hidden extensions by $2$, $\eta$, and $\nu$. Our
work does not completely determine the ring structure of
$\pi_{*,*}\mathit{mmf}$ because there exist hidden extensions by other
elements. Up to one minor uncertainty, the entire ring structure of
$\pi_{*}\mathit{tmf}$ is determined in [BR21].

######  Theorem 4.1.

Up to multiples of $g$ and $\Delta^{8}$, Tables LABEL:hidh0, LABEL:hidh1 and
LABEL:hidh2 list all hidden extensions by $2$, $\eta$, and $\nu$ in the mANss
for $\mathit{mmf}$.

###### Proof.

Some of the non-zero hidden extensions are established in the previous results
because we needed them to compute Adams-Novikov differentials. The remaining
non-zero hidden extensions are proved in the following results. The last
columns of the tables indicate the specific proofs for each extension.

There are some possible hidden extensions that turn out not to occur. Most of
these possibilities can be ruled out using 2.15. For example, consider the
possible hidden $\eta$ extension from $\tau\Delta h_{1}^{3}$ to $\tau^{2}cg$.
Because of multiplication by $\tau g$, we may instead consider the possible
hidden $\eta$ extension from $\tau^{2}\Delta h_{1}^{3}g$ to $\tau^{3}cg^{2}$.
These last two elements are annihilated by $\tau$, so they are in the image of
projection to the top cell. By inspection, there is no $\eta$ extension in the
homotopy of $\mathit{mmf}/\tau$ in the appropriate degree.

A few miscellaneous cases remain, but their proofs are straightforward. For
example,

  * •

$(65,3,34)$ there is no hidden $2$ extension from $\Delta^{2}h_{2}d$ to
$\tau^{3}\Delta h_{1}g^{2}$ because the latter element supports an $h_{1}$
extension.

  * •

$(24,0,12)$ there is no hidden $\nu$ extension from $8\Delta$ to $\tau\Delta
h_{1}^{3}$ because the first element is annihilated by $g$ while the second
element is not.

∎

######  Proposition 4.2.

LABEL:hidtaumethod lists some hidden extensions in the mANss for
$\mathit{mmf}$.

Table 5. Some hidden extensions deduced from 2.15 $(s,f,w)$ | source | type | target | reason |
---|---|---|---|---|---
$(51,1,26)$ | $2\Delta^{2}h_{2}$ | $2$ | $\tau\Delta^{2}h_{1}^{3}$ | $d_{5}(2\Delta^{3})=2\tau^{2}\Delta^{2}h_{2}g$ | $d_{7}(4\Delta^{3})=\tau^{3}\Delta^{2}h_{1}^{3}g$
$(54,2,28)$ | $\Delta^{2}h_{2}^{2}$ | $2$ | $\tau^{4}dg^{2}$ | $d_{5}(\Delta^{3}h_{2})=\tau^{2}\Delta^{2}h_{2}^{2}g$ | $d_{13}(2\Delta^{3}h_{2})=\tau^{6}dg^{3}$
$(99,1,50)$ | $2\Delta^{4}h_{2}$ | $2$ | $\tau\Delta^{4}h_{1}^{3}$ | $d_{5}(2\Delta^{5})=2\tau^{2}\Delta^{4}h_{2}g$ | $d_{7}(4\Delta^{5})=\tau^{3}\Delta^{4}h_{1}^{3}g$
$(123,1,62)$ | $2\Delta^{5}h_{2}$ | $2$ | $\tau\Delta^{5}h_{1}^{3}$ | $d_{5}(\Delta^{6})=2\tau^{2}\Delta^{5}h_{2}g$ | $d_{7}(2\Delta^{6})=\tau^{3}\Delta^{5}h_{1}^{3}g$
$(147,1,74)$ | $2\Delta^{6}h_{2}$ | $2$ | $\tau\Delta^{6}h_{1}^{3}$ | $d_{5}(2\Delta^{7})=2\tau^{2}\Delta^{6}h_{2}g$ | $d_{7}(4\Delta^{7})=\tau^{3}\Delta^{6}h_{1}^{3}g$
$(51,1,26)$ | $\Delta^{2}h_{2}$ | $\eta$ | $\tau^{2}\Delta cg$ | $d_{5}(\Delta^{3})=\tau^{2}\Delta^{2}h_{2}g$ | $d_{9}(\Delta^{3}h_{1})=\tau^{4}\Delta cg^{2}$
$(99,1,50)$ | $\Delta^{4}h_{2}$ | $\eta$ | $\tau^{9}g^{5}$ | $d_{5}(\Delta^{5})=\tau^{2}\Delta^{4}h_{2}g$ | $d_{23}(\Delta^{5}h_{1})=\tau^{11}g^{6}$
$(123,1,62)$ | $2\Delta^{5}h_{2}$ | $\eta$ | $\tau^{2}\Delta^{4}cg$ | $d_{5}(\Delta^{6})=2\tau^{2}\Delta^{5}h_{2}g$ | $d_{9}(\Delta^{6}h_{1})=\tau^{4}\Delta^{4}cg^{2}$
$(124,6,63)$ | $\tau^{2}\Delta^{4}cg$ | $\eta$ | $\tau^{9}\Delta h_{1}g^{5}$ | $d_{9}(\Delta^{6}h_{1})=\tau^{4}\Delta^{4}cg^{2}$ | $d_{23}(\Delta^{6}h_{1}^{2})=\tau^{11}\Delta h_{1}g^{6}$
$(129,3,66)$ | $\Delta^{5}h_{1}c$ | $\eta$ | $\tau^{7}\Delta^{2}h_{1}^{2}g^{4}$ | $d_{9}(\Delta^{7}h_{1}^{2})\mathord{=}\tau^{4}\Delta^{5}h_{1}cg^{2}$ | $d_{23}(\Delta^{7}h_{1}^{3})=\tau^{11}\Delta^{2}h_{1}^{2}g^{6}$
$(147,1,74)$ | $\Delta^{6}h_{2}$ | $\eta$ | $\tau^{2}\Delta^{5}cg$ | $d_{5}(\Delta^{7})=\tau^{2}\Delta^{6}h_{2}g$ | $d_{9}(\Delta^{7}h_{1})=\tau^{4}\Delta^{5}cg^{2}$
$(161,3,82)$ | $\Delta^{6}h_{2}d$ | $\eta$ | $\tau^{3}\Delta^{5}h_{1}^{2}g^{2}$ | $d_{5}(\Delta^{7}d)=\tau^{2}\Delta^{6}h_{2}dg$ | $d_{11}(\Delta^{7}h_{1}d)\mathord{=}\tau^{5}\Delta^{5}h_{1}^{2}g^{3}$
$(0,0,0)$ | $4$ | $\nu$ | $\tau h_{1}^{3}$ | $d_{5}(\Delta h_{2}d)=4\tau^{2}g^{2}$ | $d_{7}(4\Delta g)=\tau^{3}h_{1}^{3}g^{2}$
$(48,0,24)$ | $4\Delta^{2}$ | $\nu$ | $\tau\Delta^{2}h_{1}^{3}$ | $d_{5}(\Delta^{3}h_{2}d)=4\tau^{2}\Delta^{2}g^{2}$ | $d_{7}(4\Delta^{3}g)=\tau^{3}\Delta^{2}h_{1}^{3}g^{2}$
$(51,1,26)$ | $2\Delta^{2}h_{2}$ | $\nu$ | $\tau^{4}dg^{2}$ | $d_{5}(2\Delta^{3})=2\tau^{2}\Delta^{2}h_{2}g$ | $d_{13}(2\Delta^{3}h_{2})=\tau^{6}dg^{3}$
$(57,3,30)$ | $\Delta^{2}h_{2}^{3}$ | $\nu$ | $2\tau^{4}g^{3}$ | $d_{5}(\Delta^{3}h_{2}^{2})=\tau^{2}\Delta^{2}h_{2}^{3}g$ | $d_{13}(\Delta^{3}h_{2}^{3})=2\tau^{6}g^{4}$
$(96,0,48)$ | $4\Delta^{4}$ | $\nu$ | $\tau\Delta^{4}h_{1}^{3}$ | $d_{5}(\Delta^{5}h_{2}d)=4\tau^{2}\Delta^{4}g^{2}$ | $d_{7}(4\Delta^{5}g)=\tau^{3}\Delta^{4}h_{1}^{3}g^{2}$
$(144,0,72)$ | $4\Delta^{6}$ | $\nu$ | $\tau\Delta^{6}h_{1}^{3}$ | $d_{5}(\Delta^{7}h_{2}d)=4\tau^{2}\Delta^{6}g^{2}$ | $d_{7}(4\Delta^{7}g)=\tau^{3}\Delta^{6}h_{1}^{3}g^{2}$
$(147,1,74)$ | $2\Delta^{6}h_{2}$ | $\nu$ | $\tau^{4}\Delta^{4}dg^{2}$ | $d_{5}(2\Delta^{7})=2\tau^{2}\Delta^{6}h_{2}g$ | $d_{13}(2\Delta^{7}h_{2})=\tau^{6}\Delta^{4}dg^{3}$
$(153,3,78)$ | $\Delta^{6}h_{2}^{3}$ | $\nu$ | $2\tau^{4}\Delta^{4}g^{3}$ | $d_{5}(\Delta^{7}h_{2}^{2})=\tau^{2}\Delta^{6}h_{2}^{3}g$ | $d_{13}(\Delta^{7}h_{2}^{3})=2\tau^{6}\Delta^{4}g^{4}$
Table 5. Some hidden extensions deduced from 2.15

###### Proof.

All of these extensions follow from 2.15, using the differentials in the last
two columns of LABEL:hidtaumethod. To illustrate, we discuss the first
extension in the table. In order to obtain the extension from
$2\Delta^{2}h_{2}$ to $\tau\Delta^{2}h_{1}^{3}$, we can establish a hidden $2$
extension from $2\tau\Delta^{2}h_{2}g$ to $\tau^{2}\Delta^{2}h_{1}^{3}g$. Then
the desired extension follows immediately.

The elements $2\tau\Delta^{2}h_{2}g$ and $\tau^{2}\Delta^{2}h_{1}^{3}g$ are
annihilated by $\tau$ in the $E_{\infty}$-page of the mANss for
$\mathit{mmf}$. Therefore, they detect elements in $\pi_{71,37}\mathit{mmf}$
that are in the image of $\pi_{72,36}\mathit{mmf}/\tau$ under projection to
the top cell. By inspection, these preimages are $2\Delta^{3}$ and
$4\Delta^{3}$. These latter elements are connected by a $2$ extension, so
their images are also connected by a $2$ extension.

The other extensions have essentially the same proof. First multiply by an
appropriate power of $g$. Then pull back to $\pi_{*,*}\mathit{mmf}/\tau$,
where the extension is visible by inspection. ∎

######  Remark 4.3.

$(124,6,63)$ The hidden $\eta$ extension from $\tau^{2}\Delta^{4}cg$ to
$\tau^{9}\Delta h_{1}g^{5}$ in LABEL:hidtaumethod deserves further discussion.
Note that $\Delta^{4}cg$ and $\tau\Delta^{4}cg$ support $\eta$ extensions that
are not hidden. However, $\tau^{2}\Delta^{4}h_{1}cg$ is zero, so
$\tau^{2}\Delta^{4}cg$ can support a hidden $\eta$ extension. This explains
why the $E_{\infty}$-page chart in Figure 5 shows both an $h_{1}$ extension
and a hidden $\eta$ extension on the element $\Delta^{4}cg$ in the 124-stem.

The subtleties of this situation are illuminated by consideration of homotopy
elements. Let $\alpha$ be an element of $\pi_{124,65}\mathit{mmf}$ that is
detected by $\Delta^{4}cg$. The element $\tau^{2}\alpha$ is detected by
$\tau^{2}\Delta^{4}cg$. The hidden $\eta$ extension implies that
$\tau^{2}\eta\alpha$ is detected by $\tau^{9}\Delta h_{1}g^{5}$.

Now let $\beta$ be an element in $\pi_{122,64}$ that is detected by
$\Delta^{4}h_{2}^{2}g$. Note that $\tau^{2}\beta$ must be zero because
$\tau^{2}\Delta^{2}h_{2}^{2}g$ is zero and because there are no
$E_{\infty}$-page elements in higher filtration. Then $\nu\beta$ is detected
by $h_{2}\cdot\Delta^{4}h_{2}^{2}g$, which equals $\Delta^{4}h_{1}cg$.

Both $\eta\alpha$ and $\nu\beta$ are detected by the same element of the
$E_{\infty}$-page, but they are not equal. The first product is not
annihilated by $\tau^{2}$, while the latter product is annihilated by
$\tau^{2}$. In fact, the difference between $\eta\alpha$ and $\nu\beta$ is
detected by $\tau^{7}\Delta h_{1}g^{5}$. This phenomenon corresponds to the
classical relation
$\nu^{2}\nu_{4}=\eta\epsilon_{4}+\eta_{1}\overline{\kappa}^{4}$ [BR21,
Proposition 9.17].

######  Remark 4.4.

$(65,3,34)$ The chart in [Bau08] shows a hidden $\eta$ extension from
$\Delta^{2}h_{2}d$ to $\Delta h_{1}^{2}g^{2}$ in the 66-stem. According to
2.14, this is not a hidden extension because of the presence of $\Delta
h_{1}g^{2}$ in higher filtration.

Nevertheless, there is a relevant point here about multiplicative structure.
Because of the presence of $\tau^{3}\Delta h_{1}g^{2}$ in higher filtration,
the element $\Delta^{2}h_{2}d$ detects two homotopy elements. One of these
elements is annihilated by $\eta$, and one is not. The product $\nu_{2}\kappa$
is one of the two homotopy elements that are detected by $\Delta^{2}h_{2}d$.
In fact, $\nu_{2}\kappa$ is the homotopy element that is not annihilated by
$\eta$. This follows from the hidden $\eta$ extension from $\Delta^{2}h_{2}$
to $\tau^{2}\Delta cg$ and the hidden $\kappa$ extension from $\Delta cg$ to
$\tau\Delta h_{1}^{2}g^{2}$.

######  Proposition 4.5.

$(110,2,56)$ There is a hidden $2$ extension from $\Delta^{4}d$ to
$\tau^{6}\Delta^{2}h_{1}^{2}g^{3}$.

###### Proof.

The proof is a variation on 2.15, in which we use the long exact sequence

${\pi_{*,*}\mathit{mmf}}$${\pi_{*,*}\mathit{mmf}/\tau^{2}}$${\pi_{*-1,*+2}\mathit{mmf}}$${\pi_{*-1,*}\mathit{mmf}}$$\scriptstyle{\tau^{2}}$

induced by the cofiber sequence

${\mathit{mmf}}$${\mathit{mmf}/\tau^{2}}$${\Sigma^{1,-2}\mathit{mmf}}$${\Sigma^{1,0}\mathit{mmf}.}$$\scriptstyle{\tau^{2}}$

We will show that there is a hidden $2$ extension from
$\tau^{4}\Delta^{4}dg^{3}$ to $\tau^{10}\Delta^{2}h_{1}^{2}g^{6}$. The desired
$2$ extension follows immediately by multiplication by $\tau^{4}g^{3}$.

Recall from 3.21 that there is a differential
$d_{13}(2\Delta^{7}h_{2})=\tau^{6}\Delta^{4}dg^{3}$. Also, it follows from
3.38 that there is a differential
$d_{23}(\Delta^{7}h_{1}^{3})=\tau^{11}\Delta^{2}h_{1}^{2}g^{6}$.

Therefore, $\tau^{4}\Delta^{4}dg^{3}$ and $\tau^{10}\Delta^{2}h_{1}^{2}g^{6}$
detect elements in $\pi_{170,88}\mathit{mmf}$ that are annihilated by
$\tau^{2}$. Hence they have preimages in $\pi_{171,86}\mathit{mmf}/\tau^{2}$
under projection to the top cell. By inspection, these preimages are
$2\Delta^{7}h_{2}$ and $\tau\Delta^{7}h_{1}^{3}$.

In the mANss for $\mathit{mmf}$, there is a differential
$d_{5}(\Delta^{7})=\tau^{2}\Delta^{6}h_{2}g$. However, in the mANss for
$\mathit{mmf}/\tau^{2}$, the element $\tau^{2}\Delta^{6}h_{2}g$ is already
zero in the $E_{2}$-page. Therefore, $\Delta^{7}$ is a permanent cycle in the
mANss for $\mathit{mmf}/\tau^{2}$.

Recall the hidden $2$ extension from $2h_{2}$ to $\tau h_{1}^{3}$ established
in 3.10. Multiplication by $\Delta^{7}$ gives a hidden $2$ extension in the
mANss $E_{\infty}$-page for $\mathit{mmf}/\tau^{2}$ from $2\Delta^{7}h_{2}$ to
$\tau\Delta^{7}h_{1}^{3}$.

Finally, apply projection to the top cell to obtain the hidden $2$ extension
from $\tau^{4}\Delta^{4}dg^{3}$ to $\tau^{10}\Delta^{2}h_{1}^{2}g^{6}$. ∎

######  Proposition 4.6.

$(50,2,26)$ There is a hidden $\nu$ extension from $\Delta^{2}h_{1}^{2}$ to
$\tau^{2}\Delta h_{1}cg$.

###### Proof.

This follows from $\Delta h_{1}$ multiplication on the hidden extension from
$\Delta h_{1}$ to $\tau^{2}cg$ established in 3.26. ∎

The next several lemmas establish some Toda brackets that we will use to
deduce further hidden extensions. All of these Toda brackets are deduced from
algebraic information, i.e., from Massey products in the mANss $E_{2}$-page.

######  Lemma 4.7.

$(32,2,17)$ The Toda bracket $\langle\nu^{2},2,\eta_{1}\rangle$ is detected by
$\Delta c$ and has no indeterminacy.

###### Proof.

We have the Massey product $c=\langle h_{2}^{2},h_{0},h_{1}\rangle$ in the
motivic algebraic Novikov $E_{2}$-page [Bae]. The May convergence theorem
[May69] [BK21, Theorem 4.16] implies that $c=\langle h_{2}^{2},2,h_{1}\rangle$
in the mANss $E_{2}$-page. Multiply by $\Delta$ to obtain

$\Delta c=\langle h_{2}^{2},2,h_{1}\rangle\Delta=\langle h_{2}^{2},2,\Delta
h_{1}\rangle.$

The second equality holds because there is no indeterminacy by inspection.

There are no crossing differentials, so the Moss convergence theorem [Mos70,
Theorem 1.2] [BK21, Theorem 4.16] implies that $\Delta c$ detects the Toda
bracket. By inspection, the bracket has no indeterminacy. ∎

######  Lemma 4.8.

$(128,2,65)$ The Toda bracket
$\left\langle\nu_{2}^{2},2,\eta_{1}\right\rangle$ is detected by $\Delta^{5}c$
and has no indeterminacy.

###### Proof.

As in the proof of 4.8, we have the Massey product $c=\langle
h_{2}^{2},2,h_{1}\rangle$ in the mANss $E_{2}$-page. Multiply by $\Delta^{5}$
to obtain

$\Delta^{5}c=\Delta^{4}\langle
h_{2}^{2},2,h_{1}\rangle\Delta=\langle\Delta^{4}h_{2}^{2},2,\Delta
h_{1}\rangle.$

The second equality holds because there is no indeterminacy by inspection.

There are no crossing differentials, so the Moss convergence theorem [Mos70,
Theorem 1.2] [BK21, Theorem 4.16] implies that $\Delta^{5}c$ detects the Toda
bracket. By inspection, the bracket has no indeterminacy. ∎

######  Lemma 4.9.

$(35,7,21)$ The Toda bracket $\langle\nu^{2},2,\epsilon\bar{\kappa}\rangle$ is
detected by $h_{1}dg$ and has no indeterminacy.

###### Proof.

We have the Massey product $h_{1}dg=\langle h_{2}^{2},h_{0},cg\rangle$ in the
motivic algebraic Novikov $E_{2}$-page [Bae]. The May convergence theorem
[May69] [BK21, Theorem 4.16] implies that $h_{1}dg=\langle
h_{2}^{2},2,cg\rangle$ in the mANss $E_{2}$-page.

There are no crossing differentials, so the Moss convergence theorem [Mos70,
Theorem 1.2] [BK21, Theorem 4.16] implies that $h_{1}dg$ detects the Toda
bracket. By inspection, the bracket has no indeterminacy. ∎

######  Lemma 4.10.

$(131,7,69)$ The Toda bracket
$\left\langle\nu_{2}^{2},2,\epsilon\bar{\kappa}\right\rangle$ is detected by
$\Delta^{4}h_{1}dg$ and has no indeterminacy.

###### Proof.

As in the proof of 4.9, we have the Massey product $h_{1}dg=\langle
h_{2}^{2},2,cg\rangle$ in the mANss $E_{2}$-page. Multiply by $\Delta^{4}$ to
obtain

$\Delta^{4}h_{1}dg=\Delta^{4}\langle
h_{2}^{2},h_{0},cg\rangle=\langle\Delta^{4}h_{2}^{2},h_{0},cg\rangle.$

The second equality holds because there is no indeterminacy by inspection.

There are no crossing differentials, so the Moss convergence theorem [Mos70,
Theorem 1.2] [BK21, Theorem 4.16] implies that $\Delta^{4}h_{1}dg$ detects the
Toda bracket. By inspection, the bracket has no indeterminacy. ∎

######  Proposition 4.11.

There are hidden $\nu$ extensions:

  1. (1)

$(32,2,17)$ from $\Delta c$ to $\tau^{2}h_{1}dg$.

  2. (2)

$(128,2,65)$ from $\Delta^{5}c$ to $\tau^{2}\Delta^{4}h_{1}dg$.

###### Proof.

Recall from 4.7 that the Toda bracket $\langle\nu^{2},2,\eta_{1}\rangle$ is
detected by $\Delta c$. We have

$\langle\nu^{2},2,\eta_{1}\rangle\nu=\langle\nu^{2},2,\nu\cdot\eta_{1}\rangle=\langle\nu^{2},2,\tau^{2}\epsilon\bar{\kappa}\rangle.$

The first equality holds because there is no indeterminacy by inspection. The
second equality follows from the hidden $\nu$ extension of 3.26. 4.9 implies
that $\tau^{2}h_{1}dg$ detects the last Toda bracket.

The proof for the second hidden extension is nearly identical. Consider the
equalities

$\langle\nu_{2}^{2},2,\eta_{1}\rangle\nu=\langle\nu_{2}^{2},2,\nu\cdot\eta_{1}\rangle=\langle\nu_{2}^{2},2,\tau^{2}\epsilon\bar{\kappa}\rangle,$

and use 4.8 and 4.10. ∎

######  Proposition 4.12.

There are hidden $\nu$ extensions:

  1. (1)

$(97,1,49)$ from $\Delta^{4}h_{1}$ to $\tau^{9}g^{5}$.

  2. (2)

$(122,2,62)$ from $\Delta^{5}h_{1}^{2}$ to $\tau^{9}\Delta h_{1}g^{5}$.

  3. (3)

$(147,3,75)$ from $\Delta^{6}h_{1}^{3}$ to $\tau^{9}\Delta^{2}h_{1}^{2}g^{5}$.

###### Proof.

We prove the third hidden extension. Then the first two hidden extensions
follow from multiplication by $\Delta h_{1}$.

4.5 and 3.22 imply that there is a hidden $4\nu$ extension from
$\Delta^{6}h_{2}$ to $\tau^{10}\Delta^{2}h_{1}^{2}g^{5}$. We also have a
hidden $2$ extension from $2\Delta^{6}h_{2}$ to $\tau\Delta^{6}h_{1}^{3}$, as
shown in 4.2. It follows that there must be a hidden $\nu$ extension from
$\Delta^{6}h_{1}^{3}$ to $\tau^{9}\Delta^{2}h_{1}^{2}g^{5}$. ∎

######  Proposition 4.13.

$(110,2,56)$ There is a hidden $\epsilon$ extension from $\Delta^{4}d$ to
$\tau\Delta^{4}h_{1}^{2}g$.

###### Proof.

We showed in 3.30 that there is a hidden $\epsilon$ extension from $d$ to
$\tau h_{1}^{2}g$. Multiply by $\Delta^{4}h_{1}$ to obtain a hidden $\epsilon$
extension from $\Delta^{4}h_{1}d$ to $\tau\Delta^{4}h_{1}^{2}g$. Finally, use
$h_{1}$ multiplication to obtain the hidden extension on $\Delta^{4}d$. ∎

######  Proposition 4.14.

$(135,3,69)$ There is a hidden $\nu$ extension from $\Delta^{5}h_{1}d$ to
$\tau^{3}\Delta^{4}h_{1}^{2}g^{2}.$

###### Proof.

By 3.25, the element $\Delta h_{1}$ detects the Toda bracket
$\langle\eta,\nu,\tau^{2}\bar{\kappa}\rangle$. Recall from LABEL:htpy-name
that $\kappa_{4}$ is an element of $\pi_{110,56}\mathit{mmf}$ that is detected
by the permanent cycle $\Delta^{4}d$. Then the element $\Delta^{5}h_{1}d$
detects $\langle\eta,\nu,\tau^{2}\bar{\kappa}\rangle\kappa_{4}$. Now shuffle
to obtain

$\nu\langle\eta,\nu,\tau^{2}\overline{\kappa}\rangle\kappa_{4}=\langle\nu,\eta,\nu\rangle\tau^{2}\overline{\kappa}\cdot\kappa_{4}.$

Recall from 2.18 that $\epsilon=\langle\nu,\eta,\nu\rangle$. Also recall from
4.13 that there is a hidden $\epsilon$ extension from $\Delta^{4}d$ to
$\tau\Delta^{4}h_{1}^{2}g$. We conclude that
$\epsilon\cdot\tau^{2}\overline{\kappa}\cdot\kappa_{4}$ is detected by
$\tau^{3}\Delta^{4}h_{1}^{2}g^{2}$. ∎

##  5\. The elements $\nu_{k}$

The multiplicative structure of classical $\pi_{*}\mathit{tmf}$ at the prime
$2$ has been completely computed, with one exception [BR21, p. 19]. We will
use the mANss for $\mathit{mmf}$ in order to resolve this last piece of
$2$-primary multiplicative structure.

As discussed in 2.11, our choices of homotopy elements are not necessarily
strictly compatible with the choices in [BR21]. However, our choices do agree
up to multiples of certain powers of $2$. Our computations below in 5.9, 5.10,
5.12, 5.13, and 5.15 lie in groups of order at most 8, so the possible
discrepancies are irrelevant.

We will frequently multiply by the element $\tau\overline{\kappa}$ in
$\pi_{20,11}\mathit{mmf}$ in order to detect elements and relations. Beware
that multiplication by $\tau\overline{\kappa}$ is not injective in general.
However, in all degrees that we study, multiplication by
$\tau\overline{\kappa}$ is in fact an isomorphism.

Recall the projection $q:\mathit{mmf}/\tau\rightarrow\mathit{mmf}$ to the top
cell that was discussed in detail in Section 2.9. We will rely heavily on this
map in order to transfer the algebraic information in
$\pi_{*,*}\mathit{mmf}/\tau$ into homotopical information about
$\pi_{*,*}\mathit{mmf}$.

######  Lemma 5.1.

The element $q(\Delta^{k+1})$ of $\pi_{*,*}\mathit{mmf}$ is detected by
$-(k+1)\tau\Delta^{k}h_{2}g$ in Adams-Novikov filtration $5$.

###### Proof.

If $k+1$ is not a multiple of $4$, then we have the non-zero differential
$d_{5}(\Delta^{k+1})=(k+1)\tau^{2}\Delta^{k}h_{2}g$. 2.12 implies that
$q(\Delta^{k+1})$ is detected by $-(k+1)\tau\Delta^{k}h_{2}g$.

If $k+1$ is congruent to $4$ modulo $8$, then we have the non-zero
differential $d_{7}(\Delta^{k+1})=\tau^{3}\Delta^{k}h_{1}^{3}g$. 2.12 implies
that $q(\Delta^{k+1})$ is detected by $\tau^{2}\Delta^{k}h_{1}^{3}g$ in
filtration $7$. This implies that $q(\Delta^{k+1})$ is detected by zero in
filtration $5$.

If $k+1$ is a multiple of $8$, then $\Delta^{k}$ is a permanent cycle, so
$q(\Delta^{k+1})$ equals zero. This implies that $q(\Delta^{k+1})$ is detected
by zero in filtration $5$. ∎

######  Remark 5.2.

For uniformity, we have stated 5.1 for all values of $k$. As shown in the
proof of the lemma, there are in fact three cases, depending on the value of
$k$. If $k+1$ is not a multiple of $4$, then $-(k+1)\tau\Delta^{k}h_{2}g$ is a
non-zero element in the mANss $E_{\infty}$-page.

On the other hand, if $k+1$ is a multiple of $4$, then
$-(k+1)\tau\Delta^{k}h_{2}g$ is zero in the $E_{\infty}$-page since
$\tau\Delta^{k}h_{2}g$ is an element of order $4$. In these cases, the lemma
says that $q(\Delta^{k+1})$ is detected by zero in filtration $5$. In other
words, $q(\Delta^{k+1})$ is detected in filtration strictly greater than 5, if
it is non-zero. In fact, $q(\Delta^{k+1})$ is detected by
$\tau^{2}\Delta^{k}h_{1}^{3}g$ in filtration $7$ when $k+1$ is congruent to
$4$ modulo $8$. Also, $q(\Delta^{k+1})$ is zero when $k+1$ is a multiple of
$8$ because $\Delta^{k+1}$ is a permanent cycle.

######  Lemma 5.3.

The element $q(\Delta^{k+1})$ is a multiple of $\tau\overline{\kappa}$.

###### Proof.

5.1 shows that $q(\Delta^{k+1})$ is detected by $-(k+1)\tau\Delta^{k}h_{2}g$.
By inspection, all possible values of $q(\Delta^{k+1})$ are multiples of
$\tau\overline{\kappa}$. ∎

######  Definition 5.4.

Let $\nu_{k}$ be the element of $\pi_{24k+3,12k+2}\mathit{mmf}$ such that
$q(\Delta^{k+1})$ equals $-\tau\overline{\kappa}\cdot\nu_{k}$.

Note that $\nu_{k}$ exists because of 5.3. Multiplication by
$\tau\overline{\kappa}$ is an isomorphism in the relevant degrees, so
$\nu_{k}$ is specified uniquely. We choose a minus sign in the defining
formula of 5.4 for later convenience.

######  Remark 5.5.

Bruner and Rognes consider $\nu_{3}$ and $\nu_{7}$ to be “honorary" members of
the family of elements $\nu_{k}$. They are not multiplicative generators;
$\nu_{3}$ is non-zero but decomposable, and $\nu_{7}$ equals zero. 5.4 also
implies that $\nu_{7}$ is zero. This follows from the observation that
$q(\Delta^{8})$ equals zero since $\Delta^{8}$ is a permanent cycle.

The careful reader will note that the elements $\nu_{k}$ were already
partially defined in LABEL:htpy-name in Section 2.6. The following lemma shows
that the two approaches to $\nu_{k}$ are compatible. LABEL:htpy-name leaves
some ambiguity in the definition of $\nu_{k}$, and 5.4 resolves that
ambiguity.

######  Lemma 5.6.

The element $\nu_{k}$ is detected by $(k+1)\Delta^{k}h_{2}$ in Adams-Novikov
filtration $1$.

###### Proof.

5.1 determines the mANss $E_{\infty}$-page elements that detect
$q(\Delta^{k+1})$. Then 5.4 means that $-\tau\overline{\kappa}\cdot\nu_{k}$ is
detected by those same elements. Multiplication by $\tau g$ is an isomorphism
in the relevant degrees, so the detecting elements for $\nu_{k}$ are then
determined. ∎

######  Remark 5.7.

Similarly to 5.2, 5.6 includes three cases. If $k+1$ is not a multiple of $4$,
then $(k+1)\Delta^{k}h_{2}$ is a non-zero element of the mANss
$E_{\infty}$-page. If $k+1$ is a multiple of $4$, then $(k+1)\Delta^{k}h_{2}$
is zero since $\Delta^{k}h_{2}$ is an element of order $4$. This means that
$\nu_{k}$ is detected in filtration strictly greater than $1$, if it is non-
zero. In fact, $\nu_{k}$ is detected by $\tau\Delta^{k}h_{1}^{3}$ in
filtration $3$ if $k+1$ is congruent to $4$ modulo $8$, and $\nu_{k}$ is zero
if $k+1$ is a multiple of $8$.

######  Remark 5.8.

Earlier in 2.2, we chose $h_{2}$ so that it detects the element $\nu$. 5.6
shows that $\nu_{0}$ is also detected by $h_{2}$, but that does not guarantee
that it equals $\nu$ because of the presence of $\tau h_{1}^{3}$ in higher
filtration. We can only conclude that $\nu$ and $\nu_{0}$ are equal up to
multiples of $4$.

If $\nu$ equals $5\nu_{0}$, then we compute that

$q(5\Delta)=-5\tau\overline{\kappa}\cdot\nu_{0}=-\tau\overline{\kappa}\cdot\nu.$

So we may replace $\Delta$ by $5\Delta$, if necessary, and assume without loss
of generality that $\nu_{0}$ equals $\nu$. This replacement is compatible with
our previous choice of $\Delta$ in 3.8, which specified $\Delta$ only up to
multiples of $4$.

######  Proposition 5.9.

$\nu_{k+8}=\nu_{k}\cdot M$.

###### Proof.

Using Equation 2.2, we have

$q(\Delta^{k+9})=q(\Delta^{k+1}\cdot\Delta^{8})=q(\Delta^{k+1}\cdot
i(M))=q(\Delta^{k+1})\cdot M=-\tau\overline{\kappa}\cdot\nu_{k}\cdot M.$

Here we are using that $i(M)=\Delta^{8}$, which is equivalent to the
definition that $M$ is detected by $\Delta^{8}$ (see LABEL:htpy-name).

On the other hand, $q(\Delta^{k+9})$ equals
$-\tau\overline{\kappa}\cdot\nu_{k+8}$ by 5.4. Finally, multiplication by
$-\tau\overline{\kappa}$ is an isomorphism in the relevant degrees. ∎

5.9 means that for practical purposes, we only need to consider the elements
$\nu_{k}$ for $0\leq k\leq 7$.

######  Theorem 5.10.

$\nu_{j}\nu_{k}=(k+1)\nu_{j+k}\nu_{0}.$

###### Proof.

The proof splits into two cases, depending on whether $k+1$ is a multiple of
$4$. First, we handle the (more interesting) situation when $k+1$ is not a
multiple of $4$. We address the case when $k+1$ is a multiple of $4$ below in
a separate 5.13. The proof techniques for the two cases are similar, but the
details are somewhat different.

Multiplication by $\tau\overline{\kappa}$ is an isomorphism in the relevant
degrees, so it suffices to establish our relation after multiplication by
$\tau\overline{\kappa}$.

Using Equation 2.2, we have

$\displaystyle
q((k+1)\Delta^{j+k+1}h_{2})=q(\Delta^{j+k+1}\cdot(k+1)h_{2})=q(\Delta^{j+k+1}\cdot
i((k+1)\nu_{0}))=$
$\displaystyle=q(\Delta^{j+k+1})\cdot(k+1)\nu_{0}=-\tau\overline{\kappa}\cdot\nu_{j+k}\cdot(k+1)\nu_{0}.$

Here we are using that $i((k+1)\nu_{0})=(k+1)h_{2}$; in other words,
$(k+1)\nu_{0}$ is detected by $(k+1)h_{2}$. This requires that $k+1$ is not a
multiple of $4$. Otherwise, $(k+1)\nu_{0}$ is a multiple of $\tau$, and
$i((k+1)\nu_{0})$ is zero.

We will now compute $q((k+1)\Delta^{j+k+1}h_{2})$ another way. We have
$i(\nu_{k})=(k+1)\Delta^{k}h_{2}$; in other words, $\nu_{k}$ is detected by
the non-zero element $(k+1)\Delta^{k}h_{2}$, as shown in 5.6. This requires
that $k+1$ is not a multiple of $4$. Otherwise, $\nu_{k}$ is a multiple of
$\tau$, and $i(\nu_{k})$ is zero.

Then we have

$q((k+1)\Delta^{j+k+1}h_{2})=q(\Delta^{j+1}\cdot(k+1)\Delta^{k}h_{2})=q(\Delta^{j+1}\cdot
i(\nu_{k}))=q(\Delta^{j+1})\cdot\nu_{k}=-\tau\overline{\kappa}\cdot\nu_{j}\cdot\nu_{k}.$

∎

######  Remark 5.11.

The exact form of the equation in 5.10 is guided by the structure of our
proof. One could also write

$\nu_{i}\nu_{j}=(i+1)\nu\nu_{i+j},$

which more closely aligns with the notation in [BR21]. All of the elements
$\nu_{k}$ are in odd stems, so they pairwise anti-commute.

######  Corollary 5.12.

$(246,2,124)$ $\nu_{4}\nu_{6}=\nu\nu_{2}M$.

###### Proof.

5.10 implies that $\nu_{4}\nu_{6}$ equals $7\nu_{10}\nu_{0}$, which equals
$-7\nu_{0}\nu_{10}$ by graded commutativity. By 5.8 and 5.9, the latter
expression equals $-7\nu\nu_{2}M$. Finally, $\nu\nu_{2}M$ belongs to a group
of order $4$, so $-7\nu\nu_{2}M$ equals $\nu\nu_{2}M$. ∎

We now return to the case of 5.10 in which $k+1$ is a multiple of $4$.

######  Proposition 5.13.

If $k+1$ is a multiple of $4$, then
$\nu_{j}\cdot\nu_{k}=(k+1)\nu_{j+k}\nu_{0}$.

###### Proof.

First, let $k+1$ be a multiple of $8$, so $\nu_{k}$ is zero. The element
$\nu_{j+k}\nu_{0}$ belongs to a group whose order divides $8$, so
$(k+1)\nu_{j+k}\nu_{0}$ is zero. In other words, the equality holds because
both sides are zero.

Next, let $k+1$ be congruent to $4$ modulo $8$. Let $\alpha$ be an element of
$\pi_{*,*}\mathit{mmf}$ that is detected by $\Delta^{k}h_{1}^{3}$. The element
$\nu_{k}$ is detected by $\tau\Delta^{k}h_{1}^{3}$, according to 5.7. Since
there are no elements in higher filtration, we can conclude that $\nu_{k}$
equals $\tau\alpha$. We have

$q(\Delta^{j+k+1}h_{1}^{3})=q(\Delta^{j+1}\cdot\Delta^{k}h_{1}^{3})=q(\Delta^{j+1}\cdot
i(\alpha))=q(\Delta^{j+1})\cdot\alpha=-\tau\overline{\kappa}\cdot\nu_{j}\cdot\alpha=-\overline{\kappa}\cdot\nu_{j}\cdot\nu_{k}.$

Now we add the assumption that $j+1$ is not congruent to $4$ modulo $8$. Given
the assumption that $k+1$ is congruent to $4$ modulo $8$, we get that $j+k+1$
is not congruent to $7$ modulo $8$. Then $\Delta^{j+k+1}h_{1}^{3}$ is a
permanent cycle, so $q(\Delta^{j+k+1}h_{1}^{3})$ is zero. Together with the
computation in the previous paragraph, this implies that $\nu_{j}\cdot\nu_{k}$
is zero since multiplication by $\overline{\kappa}$ is an isomorphism in the
relevant degrees. Note also that $(k+1)\nu_{j+k}\nu_{0}$ is zero because it
belongs to a group whose order divides $4$.

Finally, we must consider the case when $j+1$ is congruent to $4$ modulo $8$,
i.e., that $j+k+1$ is congruent to $7$ modulo $8$. Then
$q(\Delta^{j+k+1}h_{1}^{3})$ is detected by
$\tau^{10}\Delta^{j+k-4}h_{1}^{2}g^{6}$ because of 2.12 and the differential
$d_{23}(\Delta^{j+k+1}h_{1}^{3})=\tau^{11}\Delta^{j+k-4}h_{1}^{2}g^{6}$. This
means that $-\overline{\kappa}\cdot\nu_{j}\cdot\nu_{k}$ is detected by
$\tau^{10}\Delta^{j+k-4}h_{1}^{2}g^{6}$. It follows that $\nu_{j}\cdot\nu_{k}$
is detected by $\tau^{10}\Delta^{j+k-4}h_{1}^{2}g^{5}$. Finally, this latter
element also detects $(k+1)\nu_{j+k}\nu_{0}$ because of the hidden $2$
extensions in the $150$-stem and their multiples under $\Delta^{8}$
multiplication (see LABEL:hidh0). ∎

######  Remark 5.14.

As shown in the proof, most cases of 5.13 hold because both sides of the
equation are zero. Both sides of the equation are non-zero precisely when
$j+1$ and $k+1$ are congruent to $4$ modulo $8$.

Bruner and Rognes establish some relations that reduce the ambiguity in their
definitions of $\nu_{k}$. Finally, we will show that our elements defined in
5.4 satisfy those same relations. We have already discussed the choice of
$\nu_{0}$ in 5.8. The only additional requirements are the relations

$\displaystyle\nu_{0}D_{4}=2\nu_{4}$
$\displaystyle\nu_{1}\nu_{5}=2\nu_{0}\nu_{6}$
$\displaystyle\nu_{2}\nu_{4}=3\nu_{0}\nu_{6}.$

The first formula is proved in 5.15, while the last two are specific instances
of 5.10.

######  Proposition 5.15.

$(99,1,50)$ $\nu_{0}D_{4}=2\nu_{4}$.

###### Proof.

Because of 5.6, both products are detected by $2\Delta^{4}h_{2}$. However,
they are not necessarily equal because of the presence of
$\tau\Delta^{4}h_{1}^{3}$ in higher filtration. We will show that
$\tau\overline{\kappa}\cdot\nu D_{4}$ equals $\tau\overline{\kappa}\cdot
2\nu_{4}$. Our desired relation follows immediately because multiplication by
$\tau\overline{\kappa}$ is an isomorphism in the relevant degree.

Using Equation 2.2, we have

$q(2\Delta^{5})=q(\Delta\cdot 2\Delta^{4})=q(\Delta\cdot
i(D_{4}))=q(\Delta)\cdot D_{4}=-\tau\overline{\kappa}\cdot\nu\cdot D_{4}.$

Here we are using that $i(D_{4})=2\Delta^{4}$, which is equivalent to the
definition that $D_{4}$ is detected by $2\Delta^{4}$ (see LABEL:htpy-name). On
the other hand, we also have

$q(2\Delta^{5})=q(\Delta^{5}\cdot 2)=q(\Delta^{5}\cdot
i(2))=q(\Delta^{5})\cdot 2=-\tau\overline{\kappa}\cdot\nu_{4}\cdot 2.$

∎

##  6\. Tables

Table 6. Adams-Novikov differentials  $(s,f,w)$ | $x$ | $r$ | $d_{r}(x)$ | proof
---|---|---|---|---
$(5,1,3)$ | $h_{1}v_{1}^{2}$ | 3 | $\tau h_{1}^{4}$ | 3.2
$(12,0,6)$ | $4a$ | 3 | $\tau Ph_{1}^{3}$ | 3.3
$(24,0,12)$ | $\Delta$ | 5 | $\tau^{2}h_{2}g$ | 3.7
$(24,0,12)$ | $4\Delta$ | 7 | $\tau^{3}h_{1}^{3}g$ | 3.13
$(48,0,24)$ | $2\Delta^{2}$ | 7 | $\tau^{3}\Delta h_{1}^{3}g$ | 3.13
$(96,0,48)$ | $\Delta^{4}$ | 7 | $\tau^{3}\Delta^{3}h_{1}^{3}g$ | 3.20
$(49,1,25)$ | $\Delta^{2}h_{1}$ | 9 | $\tau^{4}cg^{2}$ | 3.28
$(56,2,29)$ | $\Delta^{2}c$ | 9 | $\tau^{4}h_{1}dg^{2}$ | 3.17
$(73,1,37)$ | $\Delta^{3}h_{1}$ | 9 | $\tau^{4}\Delta cg^{2}$ | 3.28
$(80,2,41)$ | $\Delta^{3}c$ | 9 | $\tau^{4}\Delta h_{1}dg^{2}$ | 3.23
$(145,1,73)$ | $\Delta^{6}h_{1}$ | 9 | $\tau^{4}\Delta^{4}cg^{2}$ | 3.28
$(169,1,85)$ | $\Delta^{7}h_{1}$ | 9 | $\tau^{4}\Delta^{5}cg^{2}$ | 3.28
$(152,2,77)$ | $\Delta^{6}c$ | 9 | $\tau^{4}\Delta^{4}h_{1}dg^{2}$ | 3.24
$(176,2,89)$ | $\Delta^{7}c$ | 9 | $\tau^{4}\Delta^{5}h_{1}dg^{2}$ | 3.23
$(62,2,32)$ | $\Delta^{2}d$ | 11 | $\tau^{5}h_{1}g^{3}$ | 3.32
$(158,2,80)$ | $\Delta^{6}d$ | 11 | $\tau^{5}\Delta^{4}h_{1}g^{3}$ | 3.32
$(75,1,38)$ | $2\Delta^{3}h_{2}$ | 13 | $\tau^{6}dg^{3}$ | 3.15
$(81,3,42)$ | $\Delta^{3}h_{1}c$ | 13 | $2\tau^{6}g^{4}$ | 3.35
$(171,1,86)$ | $2\Delta^{7}h_{2}$ | 13 | $\tau^{6}\Delta^{4}dg^{3}$ | 3.21
$(177,3,90)$ | $\Delta^{7}h_{1}c$ | 13 | $2\tau^{6}\Delta^{4}g^{4}$ | 3.35
$(121,1,61)$ | $\Delta^{5}h_{1}$ | 23 | $\tau^{11}g^{6}$ | 3.38
Table 6. Adams-Novikov differentials Table 7. Hidden $2$ extensions  $(s,f,w)$ | source | target | proof
---|---|---|---
$(3,1,2)$ | $2h_{2}$ | $\tau h_{1}^{3}$ | 3.10
$(27,1,14)$ | $2\Delta h_{2}$ | $\tau\Delta h_{1}^{3}$ | 3.12
$(51,1,26)$ | $2\Delta^{2}h_{2}$ | $\tau\Delta^{2}h_{1}^{3}$ | 4.2
$(54,2,28)$ | $\Delta^{2}h_{2}^{2}$ | $\tau^{4}dg^{2}$ | Example 2.16
$(99,1,50)$ | $2\Delta^{4}h_{2}$ | $\tau\Delta^{4}h_{1}^{3}$ | 4.2
$(110,2,56)$ | $\Delta^{4}d$ | $\tau^{6}\Delta^{2}h_{1}^{2}g^{3}$ | 4.5
$(123,1,62)$ | $2\Delta^{5}h_{2}$ | $\tau\Delta^{5}h_{1}^{3}$ | 4.2
$(147,1,74)$ | $2\Delta^{6}h_{2}$ | $\tau\Delta^{6}h_{1}^{3}$ | 4.2
$(150,2,76)$ | $\Delta^{6}h_{2}^{2}$ | $\tau^{4}\Delta^{4}dg^{2}$ | 4.2
Table 7. Hidden $2$ extensions

Table 8. Hidden $\eta$ extensions  $(s,f,w)$ | source | target | proof
---|---|---|---
$(27,1,14)$ | $2\Delta h_{2}$ | $\tau^{2}cg$ | 3.27
$(39,3,21)$ | $\Delta h_{1}d$ | $2\tau^{2}g^{2}$ | 3.34
$(51,1,26)$ | $\Delta^{2}h_{2}$ | $\tau^{2}\Delta cg$ | 4.2
$(99,1,50)$ | $\Delta^{4}h_{2}$ | $\tau^{9}g^{5}$ | 4.2
$(123,1,62)$ | $2\Delta^{5}h_{2}$ | $\tau^{2}\Delta^{4}cg$ | 4.2
$(124,6,63)$ | $\tau^{2}\Delta^{4}cg$ | $\tau^{9}\Delta h_{1}g^{5}$ | 4.2
$(129,3,66)$ | $\Delta^{5}h_{1}c$ | $\tau^{7}\Delta^{2}h_{1}^{2}g^{4}$ | 4.2
$(135,3,69)$ | $\Delta^{5}h_{1}d$ | $2\tau^{2}\Delta^{4}g^{2}$ | 4.2
$(147,1,74)$ | $\Delta^{6}h_{2}$ | $\tau^{2}\Delta^{5}cg$ | 4.2
$(161,3,82)$ | $\Delta^{6}h_{2}d$ | $\tau^{3}\Delta^{5}h_{1}^{2}g^{2}$ | 4.2
Table 8. Hidden $\eta$ extensions Table 9. Hidden $\nu$ extensions  $(s,f,w)$ | source | target | proof
---|---|---|---
$(0,0,0)$ | $4$ | $\tau h_{1}^{3}$ | 4.2
$(25,1,13)$ | $\Delta h_{1}$ | $\tau^{2}cg$ | 3.26
$(32,2,17)$ | $\Delta c$ | $\tau^{2}h_{1}dg$ | 4.11
$(39,3,21)$ | $\Delta h_{1}d$ | $\tau^{3}h_{1}^{2}g^{2}$ | 3.31
$(48,0,24)$ | $4\Delta^{2}$ | $\tau\Delta^{2}h_{1}^{3}$ | 4.2
$(50,2,26)$ | $\Delta^{2}h_{1}^{2}$ | $\tau^{2}\Delta h_{1}cg$ | 4.6
$(51,1,26)$ | $2\Delta^{2}h_{2}$ | $\tau^{4}dg^{2}$ | 4.2
$(57,3,30)$ | $\Delta^{2}h_{2}^{3}$ | $2\tau^{4}g^{3}$ | 4.2
$(96,0,48)$ | $4\Delta^{4}$ | $\tau\Delta^{4}h_{1}^{3}$ | 4.2
$(97,1,49)$ | $\Delta^{4}h_{1}$ | $\tau^{9}g^{5}$ | 4.12
$(122,2,62)$ | $\Delta^{5}h_{1}^{2}$ | $\tau^{9}\Delta h_{1}g^{5}$ | 4.12
$(128,2,65)$ | $\Delta^{5}c$ | $\tau^{2}\Delta^{4}h_{1}dg$ | 4.11
$(135,3,69)$ | $\Delta^{5}h_{1}d$ | $\tau^{3}\Delta^{4}h_{1}^{2}g^{2}$ | 4.14
$(144,0,72)$ | $4\Delta^{6}$ | $\tau\Delta^{6}h_{1}^{3}$ | 4.2
$(147,1,74)$ | $2\Delta^{6}h_{2}$ | $\tau^{4}\Delta^{4}dg^{2}$ | 4.2
$(147,3,75)$ | $\Delta^{6}h_{1}^{3}$ | $\tau^{9}\Delta^{2}h_{1}^{2}g^{5}$ | 4.12
$(153,3,78)$ | $\Delta^{6}h_{2}^{3}$ | $2\tau^{4}\Delta^{4}g^{3}$ | 4.2
Table 9. Hidden $\nu$ extensions Table 10. Some Toda brackets  $(s,f,w)$ | Toda bracket | detected by | indet | proof | used in
---|---|---|---|---|---
$(8,2,5)$ | $\langle\nu,\eta,\nu\rangle$ | $c$ | $0$ | 2.18 |  3.26, 4.14
$(25,1,13)$ | $\langle\eta,\nu,\tau^{2}\overline{\kappa}\rangle$ | $\Delta h_{1}$ | $P^{3}h_{1}$ | 3.25 |  3.26, 3.27, 4.14
$(32,2,17)$ | $\langle\nu^{2},2,\eta_{1}\rangle$ | $\Delta c$ | $0$ | 4.7 | 4.11
$(128,2,65)$ | $\langle\nu_{2}^{2},2,\eta_{1}\rangle$ | $\Delta^{5}c$ | $0$ | 4.8 | 4.11
$(35,7,21)$ | $\langle\nu^{2},2,\epsilon\overline{\kappa}\rangle$ | $h_{1}dg$ | $0$ | 4.9 | 4.11
$(131,7,69)$ | $\langle\nu_{2}^{2},2,\epsilon\overline{\kappa}\rangle$ | $\Delta^{4}h_{1}dg$ | $0$ | 4.10 | 4.11
Table 10. Some Toda brackets

##  7\. Charts

The following charts display the $E_{2}$-page, $E_{9}$-page, and
$E_{\infty}$-page of the mANss for $\mathit{mmf}$. Each of these pages is free
as a module over $\mathbb{Z}[\Delta^{8}]$, where $\Delta^{8}$ is a class in
the $192$-stem. For legibility, we display the $v_{1}$-periodic elements on
separate charts. See Section 2.7 for discussion of $v_{1}$-periodicity. To
obtain the full $E_{2}$-page, one must superimpose Figures 1 and 3. To obtain
the full $E_{\infty}$-page, one must superimpose Figures 2 and 5.

We describe each chart in slightly more detail.

  * •

Figure 1 shows the $v_{1}$-periodic portion of the mANss $E_{2}$-page,
together with all differentials that are supported by the displayed elements.

  * •

Figure 2 shows the $v_{1}$-periodic portion of the mANss $E_{\infty}$-page.

  * •

Figure 3 shows the non-$v_{1}$-periodic portion of the mANss $E_{2}$-page,
together with all $d_{3}$, $d_{5}$, and $d_{7}$ differentials that are
supported by the displayed elements.

  * •

Figure 4 shows the non-$v_{1}$-periodic portion of the mANss $E_{9}$-page,
together with all differentials that are supported by the displayed elements.

  * •

Figure 5 shows the non-$v_{1}$-periodic portion of the mANss
$E_{\infty}$-page, together with all hidden extensions by $2$, $\eta$, and
$\nu$.

###  7.1. Elements

For each fixed stem and filtration, the mANss consists of a
$\mathbb{Z}[\tau]$-module. We use a graphical notation to describe these
modules. Our notation represents the associated graded object of a filtration
that is related to the powers of $2$.

  * •

An open box  indicates a copy of $\mathbb{Z}[\tau]$ in the associated graded
object.

  * •

A solid gray dot  $\bullet$ indicates a copy of $\mathbb{F}_{2}[\tau]$ in the
associated graded object.

  * •

A solid colored dot indicates a copy of $\mathbb{F}_{2}[\tau]/\tau^{r}$ in the
associated graded object. The value of $r$ is encoded in the color of the dot,
as shown in LABEL:table:chartcolor.

  * •

Short vertical lines indicate extensions by $2$.

Our graphical notation has the advantages of flexibility, compactness, and
convenience. We illustrate with two examples.

######  Example 7.1.

In Figure 3 at degree $(48,0)$, one sees . This notation indicates a copy of
$\mathbb{Z}[\tau]$. More precisely, it represents the filtration
$4\mathbb{Z}[\tau]\subseteq 2\mathbb{Z}[\tau]\subseteq\mathbb{Z}[\tau]$ whose
filtration quotients are $\mathbb{Z}[\tau]$, $\mathbb{F}_{2}[\tau]$, and
$\mathbb{F}_{2}[\tau]$. This particular filtration is relevant for our mANss
computation because $2\mathbb{Z}[\tau]$ is the subgroup of $d_{5}$ cycles, and
$4\mathbb{Z}[\tau]$ is the subgroup of $d_{7}$ cycles.

######  Example 7.2.

In Figure 5 at degree $(120,24)$, one sees . This notation indicates the
$\mathbb{Z}[\tau]$-module

$\frac{\mathbb{Z}[\tau]}{8,4\tau^{2},2\tau^{6},\tau^{11}},$

which is somewhat cumbersome to describe in traditional notation. More
precisely, it represents the filtration

$\frac{4\mathbb{Z}[\tau]}{8,4\tau^{2}}\subseteq\frac{2\mathbb{Z}[\tau]}{8,4\tau^{2},2\tau^{6}}\subseteq\frac{\mathbb{Z}[\tau]}{8,4\tau^{2},2\tau^{6},\tau^{11}}.$

whose filtration quotients are $\mathbb{F}_{2}[\tau]/\tau^{2}$,
$\mathbb{F}_{2}[\tau]/\tau^{6}$, and $\mathbb{F}_{2}[\tau]/\tau^{11}$. The
blue, magenta, and orange dots correspond to these filtration quotients, as
shown in LABEL:table:chartcolor.

Table 11. Color interpretations for elements  $n$ | color
---|---
$\mathbb{F}_{2}[\tau]$ | $\bullet$ gray
$\mathbb{F}_{2}[\tau]/\tau$ | $\bullet$ red
$\mathbb{F}_{2}[\tau]/\tau^{2}$ | $\bullet$ blue
$\mathbb{F}_{2}[\tau]/\tau^{3}$ | $\bullet$ green
$\mathbb{F}_{2}[\tau]/\tau^{4}$ | $\bullet$ cyan
$\mathbb{F}_{2}[\tau]/\tau^{5}$ | $\bullet$ brown
$\mathbb{F}_{2}[\tau]/\tau^{6}$ | $\bullet$ magenta
$\mathbb{F}_{2}[\tau]/\tau^{11}$ | $\bullet$ orange
Table 11. Color interpretations for elements

###  7.2. Differentials

Lines of negative slope indicate Adams-Novikov differentials. The
differentials are colored according to their lengths, as described in
LABEL:table:diffcolor. These color choices are compatible with our choice of
colors for $\tau$ torsion in Section 7.1, in the following sense. An Adams-
Novikov $d_{2r+1}$ differential always takes the form $d_{2r+1}(x)=\tau^{r}y$,
and it creates $\tau^{r}$ torsion in the following page. We use matching
colors for $d_{2r+1}$ and for $\tau^{r}$ torsion.

Table 12. Color interpretations for Adams-Novikov differentials  color | slope | $d_{r}$
---|---|---
red | $-3$ | $d_{3}$
blue | $-5$ | $d_{5}$
green | $-7$ | $d_{7}$
cyan | $-9$ | $d_{9}$
brown | $-11$ | $d_{11}$
magenta | $-13$ | $d_{13}$
orange | $-23$ | $d_{23}$
Table 12. Color interpretations for Adams-Novikov differentials

###  7.3. Extensions

  * •

Solid lines of slope $1$ indicate $h_{1}$ multiplications. The colors of these
lines are determined by the $\tau$ torsion of the targets.

  * •

Arrows of slope $1$ indicate infinite families of elements that are connected
by $h_{1}$ multiplications. The colors of the arrows reflect the $\tau$
torsion of the elements.

  * •

Solid lines of slope $1/3$ indicate $h_{2}$ multiplications. The colors of
these lines are determined by the $\tau$ torsion of the targets.

  * •

Dashed lines indicate hidden extensions by $2$, $\eta$, and $\nu$. Some of
these lines are curved solely for the purpose of legibility.

  * •

The colors of dashed lines indicate the $\tau$ torsion of the targets of the