# The topological modular forms of $\mathbb{R}P^{2}$ and
$\mathbb{R}P^{2}\wedge\mathbb{C}P^{2}$

Agnès Beaudry  Department of Mathematics, University of Colorado, Boulder,
Boulder, CO, Campus Box 395, Boulder, CO, 80309, USA
<EMAIL_ADDRESS>,  Irina Bobkova  Department of Mathematics, Texas
A&M University, College Station, TX, 77843, USA<EMAIL_ADDRESS>,
Viet-Cuong Pham  Max-Planck Institute for Mathematics, Vivatsgasse 7, 53111
Bonn, Germany<EMAIL_ADDRESS>and  Zhouli Xu  Department of
Mathematics, UC San Diego, La Jolla, CA 92093, USA<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the elliptic spectral sequence computing
$tmf_{*}(\mathbb{R}P^{2})$ and
$tmf_{*}(\mathbb{R}P^{2}\wedge\mathbb{C}P^{2})$. Specifically, we compute all
differentials and resolve exotic extensions by $2$, $\eta$, and $\nu$. For
$tmf_{*}(\mathbb{R}P^{2}\wedge\mathbb{C}P^{2})$, we also compute the effect of
the $v_{1}$-self maps of $\mathbb{R}P^{2}\wedge\mathbb{C}P^{2}$ on
$tmf$-homology.

This material is based upon work supported by the National Science Foundation
under grants No. DMS–1906227, DMS-2005627 and DMS-2043485.

###### Contents

  1. 1 Introduction
  2. 2 Background
  3. 3 $tmf_{*}V(0)$: The $E_{2}$-page
  4. 4 $tmf_{*}V(0)$: The differentials and extensions
  5. 5 $tmf_{*}Y$: The $E_{2}$-page
  6. 6 $tmf_{*}Y$: The differentials and extensions

##  1\. Introduction

###  1.1. Motivation

Topological modular forms ($tmf$) are ubiquitous in algebraic topology and
homotopy theory. The goal of this paper is to compute the $tmf$-homology of
two spaces, namely $\mathbb{R}P^{2}$ and
$\mathbb{R}P^{2}\wedge\mathbb{C}P^{2}$, and to determine the differentials and
extensions in their elliptic spectral sequences.

We approach this problem from the point of view of stable homotopy theory. As
is common, we let $V(0)$ denote the cofiber of multiplication by 2 on the
sphere spectrum. Then

$V(0)\simeq\Sigma^{-1}\Sigma^{\infty}\mathbb{R}P^{2}$

and, via the suspension isomorphism, computing
$tmf_{*}V(0)\cong\pi_{*}tmf\wedge V(0)$ is equivalent to computing the
$tmf$-homology of $\mathbb{R}P^{2}$. Similarly, let $Y$ be the smash product
of $V(0)$ with $C_{\eta}$, the cofiber of the stable Hopf map $\eta$. Then

$Y\simeq\Sigma^{-3}\mathbb{R}P^{2}\wedge\mathbb{C}P^{2}$

and computing $tmf_{*}Y$ is equivalent to computing the $tmf$-homology of
$\mathbb{R}P^{2}\wedge\mathbb{C}P^{2}$. In this paper, we compute the elliptic
spectral sequence for both $tmf\wedge V(0)$ and $tmf\wedge Y$. From this
computation, we deduce $tmf_{*}V(0)$ and $tmf_{*}Y$ and provide information
about their module structure over $tmf_{*}$. In particular, we resolve all
exotic $2,\eta,\nu$ extensions as well as compute the effect of $v_{1}$-self
maps of $Y$ on $tmf_{*}Y$. Note that determining the $tmf_{*}$-module
structure is much less straightforward than a simple degree-wise computation
of $tmf_{*}V(0)$ or $tmf_{*}Y$.

Knowing the homology of basic spaces is part of a full understanding of any
generalized homology theory. So we see these computations as having
independent and fundamental interest. They are, at the very least, an addition
to the slim bank of examples of computations in $tmf$-homology theory of
spaces and finite spectra.

However, our motivation for doing this runs deeper and this computation is
part of a more ambitious program, coming from chromatic homotopy theory.
Specifically, our main goal in doing this computation is not just to
understand the structure of $tmf_{*}V(0)$ and $tmf_{*}Y$ as $tmf_{*}$-modules,
but more-so _to fully compute their elliptic spectral sequences_. To explain
this, we let $K(2)$ denote the Morava $K$-theory spectrum and $E_{2}$ the
Lubin-Tate spectrum (also often called Morava $E$-theory).

In the sequence of papers [GHM04, GHMR05, HKM13, GHMR15, GH16, GHM14, Hen07],
Goerss, Henn, Karamanov, Mahowald and Rezk carry out a program for studying
$K(2)$–local homotopy theory at $p=3$ using the theory of _finite
resolutions_. These are sequences of spectra built from the
$K(2)$-localization of $tmf$ (and $tmf$ with level structures) that resolve
the $K(2)$-local sphere. Finite resolutions give rise to Bousfield-Kan
spectral sequences. Let us call these _finite resolution spectral sequences_.
The input is $K(2)$-local $tmf$-homology, possibly with level structures, and
the output is $K(2)$-local homotopy groups. The ultimate goal is to use finite
resolutions to compute $\pi_{*}L_{K(2)}S^{0}$, but an intermediate step is the
computations of the homotopy groups of $L_{K(2)}F$ for some key finite spectra
$F$, such as the prime 3 Moore spectrum $V(0)$ [HKM13] and the cofiber of its
$v_{1}$-self map, commonly denoted $V(1)$ [GHM04]. So, to use the finite
resolution approach to $K(2)$-local homotopy, a key input is
$\pi_{*}L_{K(2)}(tmf\wedge F)$. This can be computed via the $K(2)$-local
$E_{2}$-based Adams-Novikov spectral sequence (which can also be cast as a
homotopy fixed point spectral sequence). This spectral sequence receives a map
from the elliptic spectral sequence of $tmf\wedge F$. Understanding the
elliptic spectral sequence of $tmf\wedge F$ thus provides key input for
$K(2)$-local computations.

Recently, there have been significant advancements towards carrying out an
analogous program at the prime $p=2$. See [Bea15, Bea17, BG18, BGH17]. But the
program is still in progress. For example, the only complete computation of
the $K(2)$-local homotopy groups of a finite spectrum at $p=2$ is the
computation of $\pi_{*}L_{K(2)}Z$ for $Z\in\mathcal{Z}$, where $\mathcal{Z}$
is the class of Bhattacharya-Egger spectra admitting a $v_{2}$-self map. See
[BE20a, BE20b] and also [BBB+19]. The motivation for this project is to add to
this bank of computations, namely, to study $L_{K(2)}V(0)$, $L_{K(2)}Y$, but
also $L_{K(2)}A_{1}$ where $A_{1}$ is the cofiber of a $v_{1}$-self map of
$Y$. For this, we found the need to understand the elliptic spectral sequence
of $tmf\wedge V(0)$, $tmf\wedge Y$ and $tmf\wedge A_{1}$. In [Pha18], the
third author computes a $K(2)$-local $E_{2}$-based Adams-Novikov spectral
sequence converging to $\pi_{*}L_{K(2)}(tmf\wedge A_{1})$. From this
computation, one can deduce that of the elliptic spectral sequence of
$tmf\wedge A_{1}$.

Here, we study instead the elliptic spectral sequences of $tmf\wedge V(0)$ and
$tmf\wedge Y$. For $F$ either $V(0)$ or $Y$, $tmf_{*}F=0$ for $*<0$ and
$tmf_{*}F$ is determined by its values in the range $0\leq*<192$. In this
paper, we obtain the following result, where the definition of what we mean by
_exotic extensions_ is given in 2.18.

######  Theorem 1.1.

The elliptic spectral sequence for $tmf\wedge V(0)$ is depicted in Figures 4,
5, 6 and 7. The $tmf$-homology of
$V(0)\simeq\Sigma^{-1}\Sigma^{\infty}\mathbb{R}P^{2}$, namely

$tmf_{*}V(0)\cong\widetilde{tmf}_{*+1}\mathbb{R}P^{2},$

together with all exotic $2,\eta$ and $\nu$ extensions in the corresponding
elliptic spectral sequence is as displayed in Figures 8 and 9 in degrees
$0\leq*<192$.

Similarly, the elliptic spectral sequence for $tmf\wedge Y$ is depicted in
Figures 14, 15, 16, 17, 19 and 20. The $tmf$-homology of
$Y\simeq\Sigma^{-3}\Sigma^{\infty}\mathbb{R}P^{2}\wedge\mathbb{C}P^{2}$,
namely

$tmf_{*}Y\cong\widetilde{tmf}_{*+3}\mathbb{R}P^{2}\wedge\mathbb{C}P^{2},$

together with all exotic $2,\eta$ and $\nu$ extensions and almost all exotic
$v_{1}$-extensions in the corresponding elliptic spectral sequence is as
displayed in Figures 21 and 22 in degrees $0\leq*<192$. In particular,

$2(\widetilde{tmf}_{*}(\mathbb{R}P^{2}\wedge\mathbb{C}P^{2}))=0.$

######  Remark 1.2.

Computing exotic extensions in this sense of 2.18 can (and does in some places
here) leave ambiguity about the module structure. However, this definition of
exotic extensions, which we borrowed from [IWX20], is very standard in these
kinds of large spectral sequence computations.

###  1.2. Methods and comparison with existing work

To say a few words about our techniques, the major input in our computation is
the elliptic spectral sequence of $tmf$, which was first computed by Hopkins
and Mahowald [DFHH14, Ch. 15], and later by Bauer [Bau08]. The computation of
the spectral sequence for $tmf_{*}V(0)$ is straightforward given that data,
while that of $tmf_{*}Y$ is more intricate. The technique we use for the
latter relies on an observation of the third author from [Pha18]. For both
$V(0)$ and $Y$, computation of the exotic extensions requires work and new
input. Several techniques are used to achieve this, and the most interesting
among these is probably the Brown-Comenetz “self-duality” of $tmf_{*}V(0)$ and
$tmf_{*}Y$. See 2.5.

In [BR] (soon to be published), Bruner and Rognes do a thorough investigation
of $tmf$. A main tool used in [BR] to answer computational questions about
$tmf$ and its modules is the _classical Adams spectral sequence_. (Note that
the study of the classical Adams spectral sequence of $tmf$ probably goes back
to Hopkins and Mahowald, and later to Henriques in [DFHH14, Chapter 13].)
Among many other topics, including duality for topological modular forms which
is relevant for our approaches, they study the classical Adams spectral
sequence of $tmf$ smashed with many finite spectra, including a study of $tmf$
smashed with $V(0)$. In particular, they also compute $tmf_{*}V(0)$,
determining all but a few $2,\eta,\nu$-multiplications as well as
$v_{1}^{4}$-multiplications. Here, we deliberately use the word
_multiplication_ in contrast to the word _extension_ discussed above to
emphasize that Bruner–Rognes name all classes, which leads them to a more
precise determination of multiplicative relations. Recently, Bruner and Rognes
shared their charts and an advanced copy of some of the chapters of their
forthcoming book with us. However, our results were obtained independently
from theirs and via different methods. So the two approaches complement one
another. We also use a few results on the classical Adams spectral sequence of
$tmf_{*}$ which we verified against both [DFHH14, Chapter 13] and [BR,
Chapters 5,9]. Furthermore, [BR, Chapter 10] is a direct reference of 2.5,
which is used extensively in this paper.

Finally, we reiterate that for our applications, namely, as an input in the
finite resolution approach to $K(2)$-local homotopy theory, it is important to
understand specifically the elliptic spectral sequence instead of the
classical Adams spectral sequence because of its close relationship to the
homotopy fixed point spectral sequence, a key tool in chromatic homotopy
theory (see the discussion above).

###  1.3. Organization of the paper

In Section 2, we discuss the elliptic spectral sequences and other key tools
used later in the paper. In Section 3, we review the computation of the
$E_{2}$-term of the elliptic spectral sequence for $tmf\wedge V(0)$. In
Section 4 we compute the differentials and some exotic extensions. In Section
5 we turn to the computation of the $E_{2}$-term of the elliptic spectral
sequence for $tmf\wedge Y$ and in Section 6 we compute the differentials and
exotic extensions.

###  1.4. Acknowledgements

We thank Robert Bruner and John Rognes for useful discussions and their
generosity in sharing some charts and chapters as well as the front matter of
their book [BR]. We are extremely grateful to Hans-Werner Henn and Vesna
Stojanoska for useful conversations along the way. In particular, Henn could
very well have been a co-author given the extent of interactions we had with
him on this project.

Computations like these are much harder without effective drawing tools and
spectral sequence programs. We are thankful to Tilman Bauer (luasseq) and Hood
Chatham (spectralsequences) for their LaTeX spectral sequence programs. While
the charts in this paper have mostly been re-drawn with Hood’s program, early
versions of our computations (before spectralsequences was written) were
facilitated by Bauer’s program and his kindness in helping us make it work in
such large scales. Classic but not least, we thank Bruner for his Ext-program
which is an ever-useful tool.

Finally, the second and third authors also thank l’Université de Strasbourg
for its support during part of the project.

##  2\. Background

In this section, we review some of the key tools that will be used in the
paper.

###  2.1. The elliptic spectral sequence

We begin with the elliptic spectral sequence. Let

$(A,\Lambda)=(\mathbb{Z}[a_{1},a_{2},a_{3},a_{4},a_{6}],\mathbb{Z}[a_{1},a_{2},a_{3},a_{4},a_{6},s,r,t])$

with

$|a_{i}|=2i,|r|=4,|s|=2,|t|=6$

be the Hopf algebroid of Weierstrass elliptic curves. Then the elliptic
spectral sequence has the form [Bau08]

$E_{2}^{s,t-s}=\mathop{\mathrm{Ext}}\nolimits^{s,t}_{\Lambda}(A,A)\Longrightarrow\pi_{t-s}tmf.$

Consider the map

$\Omega SU(4)\rightarrow\Omega SU\simeq BU$

induced by the usual inclusion $SU(4)\rightarrow SU$. Let $X(4)$ be the Thom
spectrum of the associated virtual vector bundle over $\Omega SU(4)$. These
spectra play a crucial role in the study of nilpotence and periodicity in
chromatic homotopy theory, in particular, in the work of Ravenel [Rav87]. As
outlined in [DFHH14, Ch. 9], the elliptic spectral sequence is the
$X(4)$-based Adams spectral sequence for $tmf$. See also [Rez].

Let us spell this out. We let $R=tmf$ and $E=tmf\wedge X(4)$. Then

$E\wedge_{R}E\simeq tmf\wedge X(4)\wedge X(4).$

Let $\overline{E}$ be the fiber of the unit map $R\rightarrow E$. For any
$tmf$-module $M$, one can construct the Adams tower

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\overline{E}\wedge_{R}M}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\overline{E}\wedge_{R}\overline{E}\wedge_{R}M}$$\textstyle{...\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\wedge_{R}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\wedge_{R}\overline{E}\wedge_{R}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\wedge_{R}\overline{E}\wedge_{R}\overline{E}\wedge_{R}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

by splicing together the cofiber sequences

$\overline{E}^{\wedge_{R}(n+1)}\wedge_{R}M\rightarrow\overline{E}^{\wedge_{R}n}\wedge_{R}M\rightarrow
E\wedge_{R}\overline{E}^{\wedge_{R}n}\wedge_{R}M.$

We abbreviate

$\displaystyle X_{k}$
$\displaystyle:=\overline{E}^{\wedge_{R}k}\wedge_{R}M\simeq\overline{X(4)}^{\wedge
k}\wedge M,$ $\displaystyle I_{k}$
$\displaystyle:=E\wedge_{R}\overline{E}^{\wedge_{R}k}\wedge_{R}M\simeq
X(4)\wedge\overline{X(4)}^{\wedge k}\wedge M$

where $\overline{X(4)}$ is the fiber of the unit map $S^{0}\rightarrow X(4)$.
As a consequence, the associated spectral sequence is identified with the
$X(4)$-based Adams spectral sequence for $M$.

However, we have that the Hopf algebroid

$(\pi_{*}(E),\pi_{*}(E\wedge_{R}E))=(\pi_{*}(tmf\wedge X(4)),\pi_{*}(tmf\wedge
X(4)\wedge X(4)))$

is isomorphic to $(A,\Lambda)$. In particular, it is flat. Therefore, the
$E_{2}$-term of the associated spectral sequence is identified with

$E_{2}^{s,t}(M)\cong\mathrm{Ext}^{s,t}_{\Lambda}(A,\pi_{*}(E\wedge_{R}M)).$

See [BL01]. When $M=tmf$, this is precisely the elliptic spectral sequence,
and more generally, this is the elliptic spectral sequence for the
$tmf$-module $M$.

According to Bousfield [Bou79, Theorem 6.5], since $X(4)$ is connected and
$\pi_{0}(X(4))\cong\mathbb{Z}$, if $M$ is connective, then $L_{X(4)}M\simeq M$
and the spectral sequence converges to $\pi_{*}(M)$. In particular, if $F$ is
a finite spectrum, then the elliptic spectral sequence for $tmf\wedge F$ reads
as

$E_{2}^{s,t-s}=\mathrm{Ext}^{s,t}_{\Lambda}(A,\pi_{*}(tmf\wedge X(4)\wedge
F))\Longrightarrow\pi_{t-s}(tmf\wedge F).$

To simplify the notation, we put

$\mathcal{F}_{*}(F):=\pi_{*}(tmf\wedge X(4)\wedge F)$

noting that this is a $\Lambda$-comodule.

###  2.2. (co)Truncated spectral sequences

We will use the (co-)truncation of the spectral sequence associated to a tower
of cofibrations. We will recall the constructions and their basic properties.
Let

---
$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
X_{1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
X_{2}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
X_{3}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ldots}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
X_{n-1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
X_{n}}$$\textstyle{...\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

be a tower of cofibrations of spectra. Let $(E_{r}^{*,*},d_{r})_{r\geq 1}$ be
the associated spectral sequence.

Let $X_{i}/X_{n}$ be the cofiber of the evident map $X_{n}\to X_{i}$. For any
$n\in\mathbb{N}$, there is a tower of fibrations, which we call the _$n$
-truncated tower_:

$\textstyle{X_{0}/X_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
X_{1}/X_{n}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
X_{2}/X_{n}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ldots}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
X_{n-1}/X_{n}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{pt}}$$\textstyle{I_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

We denote the terms of the resulting spectral sequence by $E_{r,<n}^{s,t}$.
This spectral sequence computes the homotopy groups of

$\mathrm{sk}_{n-1}X_{0}:=X_{0}/X_{n}.$

There is a natural map from the original tower to the $n$-truncated tower. Let

$T_{r}^{s,t}\colon{E}_{r}^{s,t}\to{E}_{r,<n}^{s,t}$

be the induced map between the respective ${E}_{r}$-terms. Then
${E}_{2,<n}^{s,t}=0$ for $s\geq n$, while $T_{2}^{s,t}$ is an isomorphism if
$s<n-1$ and an injection if $s=n-1$. More generally, we have:

######  Lemma 2.1.

For every $r\geq 2$, the map $T_{r}^{s,t}$ has the following properties:

  * (i)

$T^{s,t}_{r}$ is injective for $s\leq n-1$, and

  * (ii)

$T^{s,t}_{r}$ is bijective for $s\leq n-1-(r-1)$.

###### Proof.

We prove this by induction on the $r$. From the above discussion, (i) and (ii)
hold for $r=2$. Suppose both hold for some $r\geq 2$.

We prove that (i) holds at $E_{r+1}$. Let $[x]\in E_{r+1}^{s,t}$ be
represented by an element $x\in E_{r}^{s,t}$ such that $s\leq n-1$ and
$T_{r+1}^{s,t}([x])=0$. So $T_{r}^{s,t}(x)$ is the target of a
$d_{r}$-differential. That is, there exists $y\in E^{s-r,t-r-1}_{r,<n}$ such
that $d_{r}(y)=T_{r}^{s,t}(x)$. Since $s-r\leq n-r$, $T_{r}^{s-r,*}$ is
bijective by the induction hypothesis. It follows that there exists
$\overline{y}\in E^{s-r,t-r-1}_{r}$ such that
$T_{r}^{s-r,t-r-1}(\overline{y})=y$. So, by naturality and the hypothesis that
$T_{r}^{s,t}$ is injective, $d_{r}(\overline{y})=x$. This means that $[x]=0$,
and hence $T_{r+1}^{s,t}$ is injective when $s\geq n-1$.

Now, we prove that (ii) holds at $E_{r+1}$. Let $[x]\in E_{r+1,<n}^{s,t}$ with
$s\leq n-r-1$. We need to show that $[x]$ is in the image of $T_{r+1}^{s,t}$.
By the induction hypothesis, there is a class $\overline{x}\in E_{r}^{s,t}$
such that $T_{r}^{s,t}(\overline{x})=x$. It suffices to prove that
$\overline{x}$ is a $d_{r}$-cycle. By naturality,

$T_{r}^{s+r,t+r-1}(d_{r}(\overline{x}))=d_{r}(T_{r}^{s,t}(\overline{x}))=d_{r}(x)=0.$

Since $d_{r}(x)\in E_{r,<n}^{s+r,t+r-1}$ and $s+r\leq n-1$, the induction
hypothesis implies that $d_{r}(\overline{x})=0$. ∎

Next, we look at the co-truncated spectral sequence. Consider the following
tower of fibrations, which we call the _$n$ -co-truncated tower_,

---
$\textstyle{Y_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
Y_{1}}$$\scriptstyle{\mathop{\mathrm{id}}\nolimits}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ldots}$$\scriptstyle{\mathop{\mathrm{id}}\nolimits}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
Y_{n}=X_{n}}$$\scriptstyle{\mathop{\mathrm{id}}\nolimits}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
X_{n+1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
X_{n+2}}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J_{n}=I_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{n+2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $Y_{0}=\ldots=Y_{n}=X_{n}$ and $J_{0}=\ldots=J_{n-1}=pt$. We denote by
$E_{r,\geq n}^{s,t}$ the $r$-term of the spectral sequence associated to this
tower. There is an obvious map from the $n$-co-truncated tower to the original
one. This map induces a map of spectral sequences:

$cT_{r}^{s,t}\colon E_{r,\geq n}^{s,t}\rightarrow E_{r}^{s,t}.$

We observe that $E_{r,\geq n}^{s,t}=0$ for $s<n$, and that $cT_{2}^{s,*}$ is a
bijection for $s\geq n+1$ and a surjection for $s=n$. The following lemma is
proved as in 2.1.

######  Lemma 2.2.

For every $r\geq 2$, the map $cT_{r}^{s,t}$ has the following properties:

  * (i)

$cT^{s,t}_{r}$ is surjective for $s\geq n$, and

  * (ii)

$cT^{s,t}_{r}$ is bijective for $s\geq n+r-1$.

We will be applying this technology to $2$-local spectra. As described in
[Bau08, Section 7], one can simplify the computation of the cohomology of the
Weierstrass Hopf algebroid

$(A_{(2)},\Lambda_{(2)})\cong(A\otimes\mathbb{Z}_{(2)},\Lambda\otimes\mathbb{Z}_{(2)})$

as follows. Let $A^{\prime}$ denote $\mathbb{Z}_{(2)}[a_{1},a_{3}]$ and
$f\colon A\rightarrow A^{\prime}$ the evident projection. Let
$\Lambda^{\prime}$ denote $A^{\prime}\otimes_{A}\Lambda\otimes_{A}A^{\prime},$
which is isomorphic to $A^{\prime}[s,t]/{\sim}$, where the relations ${\sim}$
are generated by

$s^{4}-6st+a_{1}s^{3}-3a_{1}t-3a_{3}s=0$
$s^{6}-27t^{2}+3a_{1}s^{5}-9a_{1}s^{2}t+3a_{1}^{2}s^{4}-9a_{1}^{2}st+a_{1}^{3}s^{3}-27a_{3}t=0.$

The map between Hopf algebroids

$f\colon(A_{(2)},\Lambda_{(2)})\to(A^{\prime},\Lambda^{\prime})$

induces an equivalence of the associated categories of comodules [Bau08,
Sections 2 & 7], where

$N\mapsto A^{\prime}\otimes_{A_{(2)}}N$

for an $(A_{(2)},\Lambda_{(2)})$-comodule $N$. When $F$ is the
$2$-localization of a finite spectrum, the ${E}_{2}$-term of the elliptic
spectral sequence for

$tmf\wedge F\simeq tmf_{(2)}\wedge F$

is isomorphic to

$E_{2}^{s,t}(tmf\wedge
F)\cong\mathop{\mathrm{Ext}}\nolimits^{s,t}_{\Lambda^{\prime}}(A^{\prime},A^{\prime}\otimes_{A}\mathcal{F}_{*}(F)).$

######  Remark 2.3.

The spectrum $tmf\wedge X(4)$ is a complex oriented ring spectrum (e.g.,
$A=\pi_{*}(tmf\wedge X(4))$ is concentrated in even degrees). Let us denote by

$H\colon MU\rightarrow tmf\wedge X(4)$

the map of ring spectra inducing the complex orientation of $tmf\wedge X(4)$
given by the completion of the universal Weierstrass curve at the origin. Then
$H$ induces a homomorphism of Hopf algebroids

$H_{*}\colon(MU_{*},MU_{*}MU)\rightarrow((tmf\wedge X(4))_{*},(tmf\wedge
X(4)\wedge X(4))_{*})=(A,\Lambda).$

Recall that $MU_{*}\cong\mathbb{Z}[x_{1},x_{2},\ldots]$ with $|x_{i}|=2i$ and
$MU_{*}MU\cong MU_{*}[m_{1},m_{2},\ldots]$ with $|m_{i}|=2i$. We note that
$H_{*}(x_{1})=\pm a_{1}$. This is discussed in [Bau08, (3.2)].

For any finite spectrum $F$, $H$ also induces a map from the Adams–Novikov
spectral sequence for $\pi_{*}(F)$ to the elliptic spectral sequence for
$\pi_{*}(tmf\wedge F)$, which converges to the Hurewitz map
$h\colon\pi_{*}(F)\to\pi_{*}(tmf\wedge F)$. Moreover, the induced map at the
$E_{2}$ is induced by $H_{*}$.

###  2.3. Duality

In this section, we discuss Brown-Comenetz duality for $tmf$. This will be
used for determining some of the exotic extensions in our spectral sequences.
First, we introduce the following notation.

######  Notation 2.4.

Let $A$ be a graded module over a graded commutative ring $S$ and $x\in S$. We
let $\Sigma^{r}A$ be the module determined by $(\Sigma^{r}A)_{t}=A_{t-r}$. We
denote by $\Gamma_{x}A$ the $x$-power torsion of $A$, i.e.,

$\Gamma_{x}A=\\{m\in A\mid x^{i}m=0,i\gg 0\\},$

and by $A/(x^{\infty})$ the module that fits into the exact sequence of
$S$-modules

$A\rightarrow A\left[\frac{1}{x}\right]\rightarrow A/(x^{\infty})\rightarrow
0.$

We will also denote by $A^{\vee}$ the Pontryagin dual $A$, i.e.,

$(A^{\vee})_{*}=\mathop{\mathrm{Hom}}\nolimits((A)_{-*},\mathbb{Q}/\mathbb{Z})$

with the $S$-module structure given by $(r.f)(m)=(-1)^{|r||f|}f(rm)$ for every
$r\in S_{|r|}$, $f\in(A^{\vee})_{|f|}$ and $m\in A_{|m|}$.

Now suppose that $R$ is a commutative ring spectrum (e.g., $R=tmf$) and $M$ is
a $R$-module. For any $x\in\pi_{*}(R)$, we define $M\left[\frac{1}{x}\right]$
to be

$M\left[\frac{1}{x}\right]=\mathop{\mathrm{hocolim}}\nolimits(M\xrightarrow{x}\Sigma^{-|x|}M\xrightarrow{x}\Sigma^{-2|x|}M\xrightarrow{x}...).$

We define $M/(x^{\infty})$ to be the cofiber of the natural map $M\rightarrow
M\left[\frac{1}{x}\right]$. Inductively, if $(x_{1},x_{2},...,x_{n})$ is a
sequence of element of $\pi_{*}R$, then we define

$M/(x_{1}^{\infty},x_{2}^{\infty},...,x_{n}^{\infty})=(M/(x_{1}^{\infty},x_{2}^{\infty},...,x_{n-1}^{\infty}))/(x_{n}^{\infty}).$

With this notation, using the long exact sequence on homotopy groups, we see
that the cofiber sequence

$M\rightarrow M\left[\frac{1}{x}\right]\rightarrow M/(x^{\infty})$

gives rise to the short exact sequence of $\pi_{*}(R)$-modules

$0\rightarrow\pi_{*}(M)/(x^{\infty})\rightarrow\pi_{*}(M/(x^{\infty}))\rightarrow\Gamma_{x}(\pi_{*-1}(M))\rightarrow
0.$

Let $I_{\mathbb{Q}/\mathbb{Z}}$ be the spectrum representing the Pontryagin
dual of stable homotopy groups, so that for a spectrum $X$,

$I_{\mathbb{Q}/\mathbb{Z}}^{q}(X):=\mathop{\mathrm{Hom}}\nolimits(\pi_{q}X,\mathbb{Q}/\mathbb{Z}).$

Then the the Brown-Comenetz dual of a spectrum $X$ is defined to be

$I_{\mathbb{Q}/\mathbb{Z}}(X)=F(X,I_{\mathbb{Q}/\mathbb{Z}}).$

The literature contains a variety of references and methods for studying
dualities of $tmf$ and related spectra. To name a few, we note work of
Mahowald–Rezk [MR99], of Stojanoska [Sto12, Sto14] and of Greenlees [Gre16].
While the identification of $I_{\mathbb{Q}/\mathbb{Z}}(tmf)$ is known to
experts, there is no direct reference in the literature. (The work of
Greenlees and Stojanoska [GS18] describes the relationship between various
forms of duality, but this work does not directly apply to $tmf$.) Upcoming
work of Bruner–Rognes [BR, Chapter 10] and Bobkova–Stojanoska will soon fill
this gap and provide a reference for the following result.

######  Theorem 2.5.

There is an equivalence of $tmf$-modules

$I_{\mathbb{Q}/{\mathbb{Z}}}(tmf/(2^{\infty},c_{4}^{\infty},\Delta^{\infty}))\simeq\Sigma^{20}tmf.$

######  Remark 2.6.

Here and below, “$-/\Delta^{\infty}$”, we really mean
$-/(\Delta^{8})^{\infty}$ as $\Delta$ is an element of the $E_{2}$-term of the
elliptic spectral sequence but it does not survive to the $E_{\infty}$-term.
However, $\Delta^{8}$ survives and detects a class in $\pi_{192}tmf$. Note
also that the class $c_{4}\in\pi_{8}tmf$ reduces to $v_{1}^{4}\in tmf\wedge
V(0)$ and so $c_{4}$-power torsion is the same as $v_{1}$-power torsion when
the latter makes sense.

######  Corollary 2.7.

There are equivalences of $tmf$-modules

  1. (1)

$I_{\mathbb{Q}/{\mathbb{Z}}}(tmf\wedge
V(0)/(2^{\infty},c_{4}^{\infty},\Delta^{\infty}))\simeq\Sigma^{19}tmf\wedge
V(0)$, and

  2. (2)

$I_{\mathbb{Q}/{\mathbb{Z}}}(tmf\wedge
Y/(2^{\infty},c_{4}^{\infty},\Delta^{\infty}))\simeq\Sigma^{17}tmf\wedge Y$.

In the proof of the result below, we use the following lemma.

######  Lemma 2.8.

For $\mathcal{X}=tmf\wedge V(0)$ or $tmf\wedge Y$ and
$a\in\pi_{*}\mathcal{X}$, $c_{4}a$ is divisible by $\Delta^{8}$ if and only if
$a$ is divisible by $\Delta^{8}$.

######  Remark 2.9.

The proof makes use of the structure of the $E_{\infty}$-terms of the elliptic
spectral sequences as a module over $\mathbb{F}_{2}[c_{4},\Delta^{8}]$. So
this is a bit premature but we want to have this result here to gather all our
techniques in one place. We note that the logic of the argument is not
circular as the determination of the $E_{\infty}$-terms do not require this
lemma.

###### Proof.

Let $\mathcal{X}$ be $tmf\wedge V(0)$. The homotopy groups of $\mathcal{X}$
decompose as

$0\to T_{c_{4}}\to\pi_{*}\mathcal{X}\to F_{c_{4}}\to 0$

Here, $T_{c_{4}}$ is the subgroup of $c_{4}$-torsion elements, and
$F_{c_{4}}=\pi_{*}(\mathcal{X})/T_{c_{4}}$. By the calculation of the
$E_{\infty}$-term of the elliptic spectral sequence for $\mathcal{X}$,
multiplication by $\Delta^{8}$ induces a bijective endormorphism of
$T_{c_{4}}$ in every stem and an injective endormorphism of $F_{c_{4}}$.
Furthermore, there are no non-trivial $c_{4}$-torsion elements in the stems
between $176$ and $191$, and hence $T_{c_{4}}$ satisfies the conclusion of the
lemma. Any element that maps non-trivially to $F_{c_{4}}$ is detected in
filtrations less than or equal to 2 of the $E_{\infty}$-term of the elliptic
spectral sequence. This part of the $E_{\infty}$-term is free as a module over
$\mathbb{F}_{2}[v_{1}^{4},\Delta^{8}]$, and hence satisfies the conclusion of
the lemma. (Note that in the elliptic spectral sequence $c_{4}$ is detected by
$v_{1}^{4}$.) Now, suppose we have $a\in\pi_{*}\mathcal{X}$ such that
$a\not\in T_{c_{4}}$ and $c_{4}a$ is divisible by $\Delta^{8}$. Then by our
remarks on $F_{c_{4}}$, $a=\Delta^{8}b+c$ for some element $c\in T_{c_{4}}$.
Since $\Delta^{8}$ is surjective on $T_{c_{4}}$, we see that $c$ is in the
image of $\Delta^{8}$ and so the claim holds.

For $\mathcal{X}=tmf\wedge Y$, a similar argument applies. ∎

######  Corollary 2.10.

We have the following isomorphisms of $\pi_{*}tmf$-modules

  1. (1)

$\Gamma_{c_{4}}(\pi_{*}(tmf\wedge
V(0))/(\Delta^{\infty}))^{\vee}\cong\Gamma_{c_{4}}(\pi_{*-21}(tmf\wedge
V(0)))$, and

  2. (2)

$\Gamma_{c_{4}}(\pi_{*}(tmf\wedge
Y)/(\Delta^{\infty}))^{\vee}\cong\Gamma_{c_{4}}(\pi_{*-19}(tmf\wedge Y))$.

###### Proof.

In this proof, we let $\mathcal{X}=tmf\wedge V(0)$. Since $\pi_{*}\mathcal{X}$
is $2$-power torsion, we have $\mathcal{X}[1/2]\simeq*$. Thus,

(2.11) $\mathcal{X}/(2^{\infty})\simeq\Sigma\mathcal{X}.$

The long exact sequence in homotopy associated to the cofiber sequence

$\mathcal{X}/(2^{\infty})\rightarrow\mathcal{X}/(2^{\infty})\left[\frac{1}{c_{4}}\right]\rightarrow\mathcal{X}/(2^{\infty},c_{4}^{\infty}),$

gives an exact sequence

(2.12)
$0\rightarrow(\pi_{*}\mathcal{X}/(2^{\infty}))/(c_{4}^{\infty})\rightarrow\pi_{*}(\mathcal{X}/(2^{\infty},c_{4}^{\infty}))\rightarrow\Gamma_{c_{4}}\pi_{*-1}(\mathcal{X}/(2^{\infty}))\rightarrow
0.$

By (2.11), we have that

$(\pi_{*}(\mathcal{X}/(2^{\infty})))/(c_{4}^{\infty})\cong(\pi_{*-1}\mathcal{X})/(c_{4}^{\infty})$

and that

$\Gamma_{c_{4}}(\pi_{*-1}(\mathcal{X}/(2^{\infty})))\cong\Gamma_{c_{4}}(\pi_{*-2}\mathcal{X}).$

Since $\Delta^{8}$ acts injectively on $\pi_{*}\mathcal{X}$, it also acts
injectively on $\Gamma_{c_{4}}(\pi_{*-2}\mathcal{X})$. Moreover, $\Delta^{8}$
acts injectively on $(\pi_{*}\mathcal{X})/(c_{4}^{\infty})$ by 2.8. The short
exact sequence (2.12) then shows that $\Delta^{8}$ acts injectively on
$\pi_{*}(\mathcal{X}/(2^{\infty},c_{4}^{\infty}))$. Therefore, we have that

$\pi_{*}(\mathcal{X}/(2^{\infty},c_{4}^{\infty},\Delta^{\infty}))\cong(\pi_{*}\mathcal{X}/(2^{\infty},c_{4}^{\infty}))/(\Delta^{\infty}).$

The 9-lemma then implies that the following is a short exact sequence of
$\pi_{*}tmf$-modules:

(2.13)
$0\rightarrow(\pi_{*-1}\mathcal{X})/(c_{4}^{\infty},\Delta^{\infty})\rightarrow\pi_{*}(\mathcal{X}/(2^{\infty},c_{4}^{\infty},\Delta^{\infty}))\rightarrow\Gamma_{c_{4}}(\pi_{*-2}\mathcal{X})/(\Delta^{\infty})\rightarrow
0.$

By applying $\mathop{\mathrm{Hom}}\nolimits(-,\mathbb{Q}/\mathbb{Z})$ to this
exact sequence, we obtain that

$0\to(\Gamma_{c_{4}}(\pi_{*-2}\mathcal{X})/(\Delta^{\infty}))^{\vee}\rightarrow\pi_{*}(\mathcal{X}/(2^{\infty},c_{4}^{\infty},\Delta^{\infty}))^{\vee}\rightarrow((\pi_{*-1}\mathcal{X})/(c_{4}^{\infty},\Delta^{\infty}))^{\vee}\rightarrow
0,$

is an exact sequence of $\pi_{*}tmf$-modules.

We see that the right most term is $c_{4}$-free and the left most term is
$c_{4}$-torsion. In particular, it follows that

$\displaystyle(\Gamma_{c_{4}}(\pi_{*-2}\mathcal{X})/(\Delta^{\infty}))^{\vee}$
$\displaystyle\cong\Gamma_{c_{4}}(\pi_{*}(\mathcal{X}/(2^{\infty},c_{4}^{\infty},\Delta^{\infty}))^{\vee})$
$\displaystyle\cong\Gamma_{c_{4}}(\pi_{*}I_{\mathbb{Q}/\mathbb{Z}}(\mathcal{X}/(2^{\infty},c_{4}^{\infty},\Delta^{\infty}))),$

where the second isomorphism comes from the definition of the Brown-Comenetz
dual
$I_{\mathbb{Q}/\mathbb{Z}}(\mathcal{X}/(2^{\infty},c_{4}^{\infty},\Delta^{\infty}))$.
Together with 2.7, we obtain an isomorphism of $\pi_{*}tmf$-modules

$\displaystyle(\Gamma_{c_{4}}(\pi_{*}\mathcal{X})/(\Delta^{\infty}))^{\vee}$
$\displaystyle\cong\Sigma^{2}(\Gamma_{c_{4}}(\pi_{*-2}\mathcal{X})/(\Delta^{\infty}))^{\vee}$
$\displaystyle\cong\Sigma^{2}\Gamma_{c_{4}}\pi_{*}(I_{\mathbb{Q}/\mathbb{Z}}(\mathcal{X}/(2^{\infty},c_{4}^{\infty},\Delta^{\infty})))$
$\displaystyle\cong\Sigma^{2}\Sigma^{19}\Gamma_{c_{4}}(\pi_{*}\mathcal{X})$
$\displaystyle\cong\Sigma^{21}\Gamma_{c_{4}}(\pi_{*}\mathcal{X}).$

Substituting $\mathcal{X}$ for $tmf\wedge Y$ and this last $19$ with $17$
gives the result for $Y$. ∎

######  Remark 2.14.

We will explain how to use 2.10 to compute extensions. Continue to let
$\mathcal{X}=tmf\wedge V(0)$. Let $K$ denote the kernel of the homomorphism
induced by multiplication by $\Delta^{8}$ on
$\Gamma_{c_{4}}(\pi_{*}\mathcal{X})/(\Delta^{\infty})$. Since multiplication
by $\Delta^{8}$ induces an isomorphism

(2.15)
$\Gamma_{c_{4}}(\pi_{*}\mathcal{X})\xrightarrow{\cong}\Gamma_{c_{4}}(\pi_{*+192}\mathcal{X})$

for $*\geq 0$, we see that, for $-192\leq t<0$,

$K_{t}\cong\Gamma_{c_{4}}(\pi_{*}\mathcal{X})/(\Delta^{\infty})_{t}.$

The Snake Lemma applied to the following diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma_{c_{4}}(\pi_{*}\mathcal{X})\left[\frac{1}{\Delta^{8}}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta^{8}}$$\textstyle{\Gamma_{c_{4}}(\pi_{*}\mathcal{X})\left[\frac{1}{\Delta^{8}}\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma_{c_{4}}(\pi_{*}\mathcal{X})/(\Delta^{\infty})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta^{8}}$$\textstyle{\Gamma_{c_{4}}(\pi_{*}\mathcal{X})/(\Delta^{\infty})}$

gives rise to the exact sequence

$0\rightarrow\Gamma_{c_{4}}(\pi_{*}\mathcal{X})\xrightarrow{\Delta^{8}}\Gamma_{c_{4}}(\pi_{*+192}\mathcal{X})\rightarrow
K\rightarrow 0.$

Using (2.15) again, the homomorphism
$\Gamma_{c_{4}}(\pi_{*+192}\mathcal{X})\rightarrow K$ in the above short exact
sequence induces an isomorphism

$\Gamma_{c_{4}}(\pi_{*}\mathcal{X})_{t}\rightarrow
K_{t-192}\cong\Gamma_{c_{4}}\pi_{*}(\mathcal{X}/(\Delta^{\infty}))_{t-192}$

for $0\leq t<192$.

Now let $r$ be an element of $\pi_{l}(tmf)$. If $0\leq k<192-l$,
multiplication by $r$ induces a commutative diagram

$\textstyle{\Gamma_{c_{4}}(\pi_{*}\mathcal{X})_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{r}$$\textstyle{K_{k-192}\cong\Gamma_{c_{4}}(\pi_{*}\mathcal{X})/(\Delta^{\infty})_{k-192}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\textstyle{\Gamma_{c_{4}}(\pi_{*}\mathcal{X})_{k+l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{K_{k+l-192}\cong\Gamma_{c_{4}}(\pi_{*}\mathcal{X})/(\Delta^{\infty})_{k+l-192}.}$

By applying the Pontryagin dual to this commutative diagram, together with
2.10, we obtain the commutative diagram

$\textstyle{\mathop{\mathrm{Hom}}\nolimits(\Gamma_{c_{4}}(\pi_{*}\mathcal{X})_{k},\mathbb{Q}/\mathbb{Z})}$$\textstyle{\Gamma_{c_{4}}(\pi_{*}\mathcal{X})_{171-k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathop{\mathrm{Hom}}\nolimits(\Gamma_{c_{4}}(\pi_{*}\mathcal{X})_{k+l},\mathbb{Q}/\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r^{\vee}}$$\textstyle{\Gamma_{c_{4}}(\pi_{*}\mathcal{X})_{171-k-l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{r}$$\scriptstyle{\cong}$

As a consequence, the cardinality of the image of

$r\colon\Gamma_{c_{4}}(\pi_{*}\mathcal{X})_{k}\rightarrow\Gamma_{c_{4}}(\pi_{*}\mathcal{X})_{k+l}$

is the same as that of

$r\colon\Gamma_{c_{4}}(\pi_{*}\mathcal{X})_{171-k-l}\rightarrow\Gamma_{c_{4}}(\pi_{*}\mathcal{X})_{171-k}.$

In particular, this means that a non-trivial multiplication by $r$ on stem $k$
forces a non-trivial multiplication by $r$ on stem $171-k-l$.

Similarly, for $tmf\wedge Y$ we obtain that a non-trivial multiplication by
$r$ on stem $k$ forces a non-trivial multiplication by $r$ on stem $173-k-l$.

###  2.4. The Geometric Boundary Theorem

We also make use of the following result, due to Bruner [Bru78]. A standard
reference is Theorem 2.3.4 of [Rav86]. We apply this Theorem 2.3.4 to the
$X(4)$-based Adams-Novikov spectral sequence and the cofiber sequence

$\displaystyle tmf\wedge S^{0}\xrightarrow{2}tmf\wedge
S^{0}\xrightarrow{i}tmf\wedge V(0)\xrightarrow{p}tmf\wedge S^{1}.$

Using $X(4)_{*}tmf\cong A$ and $X(4)_{*}(tmf\wedge V(0))\cong A/2$, we have
$X(4)_{*}p=0$ and hence a short exact sequence

(2.16) $\displaystyle 0\to A\xrightarrow{2}A\to A/2\to 0.$

######  Theorem 2.17 (Geometric Boundary Theorem).

There are maps

$\delta_{r}\colon E_{r}^{s,t}(V(0))\to E_{r}^{s+1,t}(S^{0})$

such that

$\delta_{2}=\delta\colon E_{2}^{s,t}(V(0))\to E_{2}^{s+1,t}(S^{0})$

is the connecting homomorphism arising from (2.16). For all $r$,

$\delta_{r}d_{r}=d_{r}\delta_{r}$

and $\delta_{r+1}$ is induced by $\delta_{r}$. Furthermore, $\delta_{\infty}$
is a filtered form of

$p_{*}\colon\pi_{*}tmf\wedge V(0)\to\pi_{*+1}tmf.$

###  2.5. Further observations on extensions

Here, we collect a few classical but useful extension results. Note that, in
this paper, we use Definition 2.10 of [IWX20] as our definition of an _exotic
extension_. See Section 2.1 of that reference for a detailed discussion.
However, briefly, we have

######  Definition 2.18 (Definition 2.10 [IWX20]).

Let $\alpha\in\pi_{*}tmf$ be an element detected by $a$ on the
$E_{\infty}$-term of the elliptic spectral sequence for $tmf$. An _exotic
extension by $\alpha$_ is a pair of elements $b$ and $c$ on the
$E_{\infty}$-term of the elliptic spectral sequence for $M$ (where $M$ is a
$tmf$-module) such that

  1. (1)

$ab=0$ on the $E_{\infty}$-term,

  2. (2)

there is an element $\beta$ detected by $b$ such that $\alpha\beta$ is
detected by $c$,

  3. (3)

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

Note that this implies that if both $\alpha\beta$ and $\alpha\beta^{\prime}$
are detected by $c$ as in Figure 1, there is no exotic extension from
$b^{\prime}$ to $c$.

Figure 1.  Here there is no exotic extensions from $b^{\prime}$ to $c$, and so
the dashed line would not be drawn.

######  Lemma 2.19.

Let $X$ be a spectrum. Consider the long exact sequence in homotopy

$\ldots\to\pi_{n}X\xrightarrow{i}\pi_{n}(X\wedge
V(0))\xrightarrow{p}\pi_{n-1}X\xrightarrow{2}\ldots$

associated to the cofiber sequence $X\xrightarrow{2}X\to X\wedge V(0)$. Let
$a\in\pi_{n-1}X$ be an element of order $2$. If $a^{\prime}\in\pi_{n}(X\wedge
V(0))$ is such that $p_{*}(a^{\prime})=a$, then

$2a^{\prime}=i_{*}(\eta a)\in\pi_{n}X\wedge V(0).$

###### Proof.

This is a classical result. See, for example, [BGH17, Lemma 3.1.5.]. ∎

######  Remark 2.20.

Lemma 2.19 will be used with $X=tmf$ and $tmf\wedge C_{\eta}$ where $C_{\eta}$
is the cofiber of the Hopf map $\eta\colon S^{1}\to S^{0}$. This gives all
exotic 2-extensions in the elliptic spectral sequences for $tmf\wedge V(0)$
and $tmf\wedge Y$, since $Y\simeq C_{\eta}\wedge V(0)$.

Finally, we have the following classical result which is an analogue of Lemma
2.19.

######  Lemma 2.21.

Let $b\in\pi_{n}X$ be such that $\eta b=0$. If
$b^{\prime}\in\pi_{n+2}(C_{\eta}\wedge X)$ is such that $p_{*}b^{\prime}=b$ in
the long exact sequence on homotopy groups associated to

$\Sigma X\xrightarrow{\eta}X\xrightarrow{i}X\wedge
C_{\eta}\xrightarrow{p}\Sigma^{2}X,$

then $\eta b^{\prime}=i_{*}(\nu b)$.

###### Proof.

First, consider $b=\iota\in\pi_{0}C_{\eta}$ given by the inclusion $S^{0}\to
C_{\eta}$ of the bottom cell. We have a cofiber sequence

$C_{\eta}\xrightarrow{i}C_{\eta}\wedge
C_{\eta}\xrightarrow{p}\Sigma^{2}C_{\eta}$

which is not split because of the non-triviality of $Sq^{4}$ in
$H^{*}(C_{\eta}\wedge C_{\eta},\mathbb{Z}/2)$. We get a diagram

$\textstyle{\pi_{2}C_{\eta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{*}}$$\scriptstyle{\eta}$$\textstyle{\pi_{2}C_{\eta}\wedge
C_{\eta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{*}}$$\scriptstyle{\eta}$$\textstyle{\pi_{2}\Sigma^{2}C_{\eta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\pi_{3}C_{\eta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{*}}$$\textstyle{\pi_{3}(C_{\eta}\wedge
C_{\eta})}$

For any $b^{\prime}\in\pi_{2}(C_{\eta}\wedge C_{\eta})$ such that
$p_{*}b^{\prime}=\iota$, we must have $\eta b^{\prime}\neq 0$, else we could
split the above cofiber sequence. Since $\eta\iota=0$, $\eta b^{\prime}\in
i_{*}(\pi_{3}C_{\eta})$, where
$\pi_{3}C_{\eta}\cong\mathbb{Z}/4\\{\nu\iota\\}$. Now, in $\pi_{*}C_{\eta}$,
we can form the bracket

$\langle\iota,\eta,2\rangle\in\pi_{2}C_{\eta}$

with indeterminacy

$2\pi_{2}C_{\eta}+\iota\pi_{2}S^{0}=2\pi_{2}C_{\eta}\cong 2\mathbb{Z}.$

So, $\langle\iota,\eta,2\rangle\eta$ contains a unique element. In
$\pi_{*}S^{0}$, we also have $2\nu\in\langle\eta,2,\eta\rangle$ with
indeterminacy $\eta\pi_{2}S^{0}$. It follows that

$\langle\iota,\eta,2\rangle\eta=\iota\langle\eta,2,\eta\rangle=\iota 2\nu\neq
0\in\pi_{3}C_{\eta}.$

So $i_{*}(2\nu)=0$ and $\eta b^{\prime}=i_{*}(\nu\iota)$.

For the general case, note that any class $b\colon S^{n}\to X$ such that $\eta
b=0$ can be extended to a map $\bar{b}\colon\Sigma^{n}C_{\eta}\to X$. The
claim then follows from the commutativity of the following diagram

$\textstyle{\Sigma^{n}C_{\eta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{b}}$$\textstyle{\Sigma^{n}C_{\eta}\wedge
C_{\eta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{b}\wedge
C_{\eta}}$$\textstyle{\Sigma^{n+2}C_{\eta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Sigma^{2}\bar{b}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{X\wedge
C_{\eta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{\Sigma^{2}X}$

Then $b^{\prime}=(\bar{b}\wedge C_{\eta})_{*}\iota$ satisfies $\eta
b^{\prime}=i_{*}(\nu b)$. Now, suppose that $p_{*}\tilde{b}^{\prime}=b$. Then
$\tilde{b}^{\prime}-b^{\prime}\in\ker p_{*}=\mathop{\mathrm{im}}\nolimits
i_{*}$. Therefore, $\eta(\tilde{b}^{\prime}-b^{\prime})=0$ so,
$\eta\tilde{b}^{\prime}=i_{*}(\nu b)$ as well. ∎

###  2.6. Self-maps and their cofiber

It is well-known that $V(0)$ admits $v_{1}^{4}$ self-maps, i.e., maps
$\Sigma^{8}V(0)\to V(0)$ which induce multiplication by $v_{1}^{4}$ in
$K(1)$-homology for $K(1)$ the first Morava $K$-theory. The map on
$MU$-homology is given by multiplication by $x_{1}^{4}\in MU_{8}$. Under the
map from the Adams-Novikov spectral sequence of $V(0)$ to that of the ellpitic
spectral sequence of $tmf\wedge V(0)$, $x_{1}$ maps to $v_{1}$ on the
$E_{2}$-term. See the discussion surrounding (3.3). Any $v_{1}^{4}$ self-map
is detected by the same-named element. The spectral sequence inherits an
action of $v_{1}^{4}$ and the differentials are $v_{1}^{4}$-linear.

Recall that we let $Y$ be the spectrum $V(0)\wedge C_{\eta}$. In [DM81], Davis
and Mahowald show that there exist $v_{1}$ self-maps of $Y$, i.e., maps
$\Sigma^{2}Y\rightarrow Y$ which induce multiplication by $v_{1}$ in
$K(1)_{*}Y$. Any of these is detected by the element $v_{1}$ on the
$E_{2}$-term of elliptic spectral sequence for $tmf\wedge Y$ and the
differentials are $v_{1}$-linear.

In 6.40, we will be studying the $v_{1}$-multiplication in $tmf_{*}Y$. Some of
the answers will _depend on the choice of $v_{1}$-self map_, so we give a bit
of background here on this subject. This material can be found in [DM81].

In [DM81], the authors show that there are in fact $8$ $v_{1}$-self maps of
$Y$. They also show that a $v_{1}$-self map of $Y$ is detected in the Adams
spectral sequence by an element of
$\mathop{\mathrm{Ext}}\nolimits^{1,3}_{\mathcal{A}}(H^{*}(Y),H^{*}(Y))$, where
$\mathcal{A}$ denotes the Steenrod algebra at the prime $2$.

A class of
$\mathop{\mathrm{Ext}}\nolimits^{1,3}_{\mathcal{A}}(H^{*}(Y),H^{*}(Y))$ is
represented by a short sequence of $\mathcal{A}$-modules:

$0\rightarrow\Sigma^{2}H^{*}(Y)\rightarrow M\rightarrow H^{*}(Y)\rightarrow
0.$

Let $\mathcal{A}(1)$ be the sub-algebra of the Steenrod algebra generated by
$Sq^{1}$ and $Sq^{2}$. We know that
$\mathop{\mathrm{Ext}}\nolimits^{1,3}_{\mathcal{A}(1)}(H^{*}(Y),H^{*}(Y))\cong\mathbb{F}_{2}$
and its unique non-trivial class is represented by the short exact sequence of
$\mathcal{A}(1)$-module

$0\rightarrow\Sigma^{2}H^{*}(Y)\rightarrow A(1)\rightarrow H^{*}(Y)\rightarrow
0,$

where $A(1)$ is isomorphic to $\mathcal{A}(1)$ as an $\mathcal{A}(1)$-module,
thus the notation. Davis and Mahowald showed that a class of
$\mathop{\mathrm{Ext}}\nolimits^{1,3}_{\mathcal{A}}(H^{*}(Y),H^{*}(Y))$ which
detects a $v_{1}$-self map of $Y$ is sent to the unique non-trivial class of
$\mathop{\mathrm{Ext}}\nolimits^{1,3}_{\mathcal{A}(1)}(H^{*}(Y),H^{*}(Y))$
(via the map induced by the inclusion $\mathcal{A}(1)\subset\mathcal{A}$).

To put an $\mathcal{A}$-module structure on $A(1)$, it suffices to specify the
$Sq^{4}$ action. Indeed, the action of $Sq^{k}$, for $k\geq 8$ on $A(1)$ is
trivial for degree reasons. By the Adem relations, there must be a non-trivial
$Sq^{4}$ on the class of degree one of $A(1)$. There are possibilities for a
non-trivial action of $Sq^{4}$ on the classes of degrees zero and two, giving
rise to four different $\mathcal{A}$-module structures on $A(1)$. This
implies, in particular, that

$\mathop{\mathrm{Ext}}\nolimits^{1,3}_{\mathcal{A}}(H^{*}(Y),H^{*}(Y))\cong\mathbb{F}_{2}^{\oplus
3}.$

Computing the first three stems of
$\mathop{\mathrm{Ext}}\nolimits^{s,t}_{\mathcal{A}}(H^{*}(Y),H^{*}(Y))$, we
see that

$\mathop{\mathrm{Ext}}\nolimits^{s,s+2}_{\mathcal{A}}(H^{*}(Y),H^{*}(Y))\cong\begin{cases}\mathbb{F}_{2}&\mbox{if}\
s=2\\\ 0&\mbox{otherwise}.\end{cases}$

We deduce that there are eight homotopy classes of maps
$\Sigma^{2}Y\rightarrow Y$ detected in
$\mathop{\mathrm{Ext}}\nolimits^{1,3}_{\mathcal{A}}(H^{*}(Y),H^{*}(Y))$ and
mapping non-trivially to
$\mathop{\mathrm{Ext}}\nolimits^{1,3}_{\mathcal{A}(1)}(H^{*}(Y),H^{*}(Y))$.
These are the $v_{1}$ self-maps of $Y$.

The singular cohomology of the cofiber of each of the $v_{1}$-self map is
isomorphic to one of the four $A(1)$s as an $\mathcal{A}$-module. We denote
the four choices by $A_{1}[ij]$, with $i,j\in\\{0,1\\}$. Here, $A_{1}[ij]$
means that the cohomology has a non-trivial $Sq^{4}$ on the class of degree
$0$, respectively $2$ if $i=1$, respectively if $j=1$.

It is somewhat surprising that out of eight $v_{1}$-self-maps, there are only
four homotopy types which are distinguished by their cohomology, as is shown
[DM81]. We use the notation $A_{1}$, for short, when we mean any or all of the
four models.

##  3\. $tmf_{*}V(0)$: The $E_{2}$-page

From now on, we will be working exclusively with $2$-local spectra. We will
write $tmf$ for $tmf_{(2)}$ to simplify the notation. Furthermore, we will be
considering only elliptic spectral sequences for $M=tmf\wedge F$ for $F$ a
finite spectrum and so shorten our notation even more to

$E_{2}^{s,t}(F):=\mathop{\mathrm{Ext}}\nolimits^{s,t}_{\Lambda^{\prime}}(A^{\prime},A^{\prime}\otimes_{A}\mathcal{F}_{*}(F)).$

The map $S^{0}\xrightarrow{\times 2}S^{0}$ induces multiplication by $2$ on
$\mathcal{F}_{*}(S^{0})\cong A$, which is injective. Thus the cofiber sequence

$S^{0}\xrightarrow{2}S^{0}\rightarrow V(0)$

gives rise to a short exact sequence of $\Lambda^{\prime}$-comodules

(3.1) $\displaystyle 0\rightarrow A^{\prime}\xrightarrow{\times
2}A^{\prime}\rightarrow A^{\prime}\otimes_{A}\mathcal{F}_{*}(V(0))\rightarrow
0.$

It follows that $A^{\prime}\otimes_{A}\mathcal{F}_{*}(V(0))$ is isomorphic to
$A^{\prime}/(2)$ as a $\Lambda^{\prime}$-comodule. Since $(2)\subseteq
A^{\prime}$ is a $\Lambda^{\prime}$-invariant ideal, we have that

$\mathop{\mathrm{Ext}}\nolimits^{s,t}_{\Lambda^{\prime}}(A^{\prime},A^{\prime}/(2))\cong\mathop{\mathrm{Ext}}\nolimits^{s,t}_{\Lambda^{\prime}/(2)}(A^{\prime}/(2),A^{\prime}/(2)).$

See, for example, [Rav86, Proposition A1.2.16]. So, we have a spectral
sequence

(3.2)
$E_{2}^{s,t}(V(0))=\mathop{\mathrm{Ext}}\nolimits^{s,t}_{\Lambda^{\prime}/(2)}(A^{\prime}/(2),A^{\prime}/(2))\Longrightarrow\pi_{*}tmf\wedge
V(0).$

A computation of the cohomology of $(A^{\prime}/(2),\Lambda^{\prime}/(2))$ is
originally due to Hopkins and Mahowald and can be found in [DFHH14, Chapter
15, Section 7] and [Bau08, Section 7]. Let us describe the answer here and
introduce some notation.

Classical computations of modular forms yield

$\displaystyle\mathop{\mathrm{Ext}}\nolimits^{0,*}_{\Lambda^{\prime}}(A^{\prime},A^{\prime})\cong\mathbb{Z}_{(2)}[c_{4},c_{6},\Delta]/{(c_{4}^{3}-c_{6}^{2}-(12)^{3}\Delta)}$

where

$\displaystyle c_{4}$ $\displaystyle=a_{1}^{4}-24a_{1}a_{3}$ $\displaystyle
c_{6}$ $\displaystyle=-a_{1}^{6}+36a_{1}^{3}a_{3}-216a_{3}^{2}$
$\displaystyle\Delta$ $\displaystyle=a_{1}^{3}a_{3}^{3}-27a_{3}^{4}$

as well as

$\displaystyle\mathop{\mathrm{Ext}}\nolimits^{0,*}_{\Lambda^{\prime}/(2)}(A^{\prime}/(2),A^{\prime}/(2))\cong\mathbb{Z}/2[a_{1},\Delta].$

See, for example [Bau08] and [Sil86, III.1]. The map on
$\mathop{\mathrm{Ext}}\nolimits^{0,*}$ induced by the mod 2 reduction
$(A^{\prime},\Lambda^{\prime})\to(A^{\prime}/(2),\Lambda^{\prime}/(2))$ sends
$c_{4}\mapsto a_{1}^{4}$ and $c_{6}\mapsto a_{1}^{6}$.

There are also maps of Adams–Novikov Spectral Sequences, where $H$ and $h$ are
as in 2.3:

$\textstyle{\mathop{\mathrm{Ext}}\nolimits^{*,*}_{BP_{*}BP}(BP_{*},BP_{*}V(0))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{*}V(0)}$$\textstyle{\mathop{\mathrm{Ext}}\nolimits^{*,*}_{MU_{*}MU}(MU_{*},MU_{*}V(0))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{H}$$\textstyle{\pi_{*}V(0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{h}$$\textstyle{\mathop{\mathrm{Ext}}\nolimits^{*,*}_{\Lambda^{\prime}/(2)}(A^{\prime}/(2),A^{\prime}/(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{*}tmf\wedge
V(0)}$

Further,

$\mathop{\mathrm{Ext}}\nolimits^{0,*}_{BP_{*}BP}(BP_{*},BP_{*}V(0))\cong\mathbb{F}_{2}[v_{1}];$

see [Rav86, Thm 4.3.2].

So, we have
$a_{1}\in\mathop{\mathrm{Ext}}\nolimits^{0,2}_{\Lambda^{\prime}/(2)}(A^{\prime}/(2),A^{\prime}/(2))$,
$v_{1}\in\mathop{\mathrm{Ext}}\nolimits_{BP_{*}BP}^{0,2}(BP_{*},BP_{*}V(0))$
and
$x_{1}\in\mathop{\mathrm{Ext}}\nolimits_{MU_{*}MU}^{0,2}(MU_{*},MU_{*}V(0))$,
and

(3.3) $\displaystyle v_{1}\mapsfrom x_{1}\mapsto a_{1}.$

Note that $v_{1}$ detects either of the two classes in
$\pi_{2}V(0)\cong\mathbb{Z}/4$ which map to $\eta\in\pi_{1}V(0)$ under the
homomorphism $\pi_{2}V(0)\to\pi_{1}S^{0}$ in the long exact sequence in
homotopy. We fix a choice and call it $v_{1}\in\pi_{2}V(0)$. It follows that
$a_{1}$ survives to detect the image of $v_{1}\in\pi_{2}V(0)$ in
$\pi_{2}tmf\wedge V(0)$. From now on, in mod $2$ computations, we abuse
notation and denote all classes we have named $a_{1}$ by $v_{1}$.

Now we will present the $E_{2}$ page of (3.2) as computed in [DFHH14, p. 270],
[Sto14, Fig. 5] and [Bau08, p.26]. See Figure 2. Even if the elliptic spectral
sequence for $V(0)$ is not multiplicative, $E_{2}(V(0))$ is a ring and we can
completely describe the algebraic relations (which also follow from [Bau08]).
The ring structure will be used in our computation of $E_{2}(Y)$ below.

Recall that $\delta=\delta_{2}$ was defined in 2.17. In the theorem below,
$\kappa\in E_{2}^{2,16}(S^{0})$ is the unique non-zero element.

######  Theorem 3.4 (Figure 2).

The ring $E_{2}(V(0))$ is isomorphic to

$\mathbb{F}_{2}[v_{1},\Delta,\bar{\kappa},\eta,\nu,x,y]/(\sim)$

for elements

$\eta\in\mathop{\mathrm{Ext}}\nolimits^{1,2},\ \ \
\nu\in\mathop{\mathrm{Ext}}\nolimits^{1,3},\ \ \
\bar{\kappa}\in\mathop{\mathrm{Ext}}\nolimits^{0,24},\ \ \
\Delta\in\mathop{\mathrm{Ext}}\nolimits^{0,24}$

in the image of $E_{2}(S^{0})\to E_{2}(V(0))$, as well as elements

$v_{1}\in\mathop{\mathrm{Ext}}\nolimits^{0,2},\ \ \
x\in\mathop{\mathrm{Ext}}\nolimits^{1,8},\ \ \
y\in\mathop{\mathrm{Ext}}\nolimits^{1,15}$

in the image of $\delta_{2}\colon E_{2}(V(0))\to E_{2}(S^{0})$ where

$\delta_{2}(v_{1})=\eta,\ \ \delta_{2}(x)=\nu^{2},\ \ \delta_{2}(y)=\kappa.$

The relations $(\sim)$ is the ideal generated by

$\displaystyle(s=1)$ $\displaystyle v_{1}\nu$ $\displaystyle v_{1}^{2}x$
$\displaystyle v_{1}y$ $\displaystyle(s=2)$ $\displaystyle\nu\eta$
$\displaystyle\nu x-v_{1}\eta x$ $\displaystyle\eta y-v_{1}x^{2}$
$\displaystyle xy$ $\displaystyle y^{2}-\nu^{2}\Delta$ $\displaystyle(s=3)$
$\displaystyle\eta^{2}x-\nu^{3}$ $\displaystyle x^{3}-\nu^{2}y$
$\displaystyle(s=4)$ $\displaystyle\eta^{4}\Delta-v_{1}^{4}\bar{\kappa}\ .$

Furthermore, we have $\kappa=x^{2}$ and $\delta_{2}(\nu^{2}y)=4\bar{\kappa}$.

######  Remark 3.5.

The algebraic structure in 3.4 can also be deduced from the appendix of
[Bea17].

######  Remark 3.6.

The element $\Delta$ is detected by $v_{2}^{4}$ in the Bockstein spectral
sequence computation of [DFHH14, II.2.7].

Figure 2. The $E_{2}$-term of the elliptic spectral sequence for $tmf\wedge
V(0)$. A bullet $\bullet$ denotes $\mathbb{F}_{2}$ and a diamond $\diamond$
denotes a copy of $\mathbb{F}_{2}[v_{1}]$. The lines of slope 1 denote
multiplication by $\eta$, and the lines of slope 1/3 denote multiplication by
$\nu$. Horizontal lines are $v_{1}$-multiplications.

######  Remark 3.7.

Let $P$ denote the following pattern:

Then $E_{2}^{*,*}(V(0))$ can be summarized additively as

$E_{2}^{*,*}(V(0))=P[\bar{\kappa},\Delta]/(\Delta\eta^{4}-\bar{\kappa}v_{1}^{4}).$

##  4\. $tmf_{*}V(0)$: The differentials and extensions

We begin with an observation that $V(0)$ has a $v_{1}^{4}$ self map, hence all
differentials $d_{r}$ for $r\geq 3$ are $v_{1}^{4}$ linear. Since $\eta$,
$\nu$, $\bar{\kappa}$ and $\Delta^{8}$ are permanent cycles, all differentials
are linear with respect to multiplication by these elements. Note that there
are no even length differentials due to sparseness.

We will use the following methods when computing differentials in this
section.

  1. (1)

The map of spectral sequences induced by the map of spectra

$tmf\to tmf\wedge V(0)$

allows us to import a differential $d_{r}(a)=b$ from the spectral sequence for
$tmf$ if the images of $a$ and $b$ are both non-trivial on the $E_{r}$ page of
the spectral sequence for $tmf\wedge V(0)$. Note also that the elliptic
spectral for $tmf\wedge V(0)$ is a module over the elliptic spectral sequence
for $tmf$.

  2. (2)

The long exact sequence in homotopy groups associated to the fiber sequence

$tmf\xrightarrow{2}tmf\to tmf\wedge V(0)$

gives short exact sequences

$0\to(\pi_{i}tmf)/2\to\pi_{i}(tmf\wedge V(0))\to\ker_{2}(\pi_{i-1}tmf)\to 0$

where $\ker_{2}(\pi_{i-1}tmf)$ is the subgroup of elements of order $2$. This
allows us to compute the rank of $\pi_{i}(tmf\wedge V(0))$ and forces certain
differentials by various dimension count arguments.

  3. (3)

The Geometric Boundary Theorem, stated in 2.17.

###  4.1. The $d_{3}$-differentials

######  Lemma 4.1 (Figure 3).

The $d_{3}$-differentials are $\Delta$ and $v_{1}^{4}$-linear. They are
determined by this linearity, the differentials

$d_{3}(v_{1}^{2})=\eta^{3};\quad d_{3}(v_{1}^{3})=v_{1}\eta^{3},$

and the module structure over the elliptic spectral sequence for $tmf$.

###### Proof.

The two listed $d_{3}$-differentials occur in the Adams-Novikov spectral
sequence computing $\pi_{*}V(0)$ so happen here also by naturality. See, for
example, [Rav78, Theorem 5.13 (a)]. Since $\Delta$ is a $d_{3}$-cycle in the
elliptic spectral sequence computing $\pi_{*}tmf$ and the elliptic spectral
sequence for $V(0)$ is a module over this spectral sequence, the
$d_{3}$-differentials are $\Delta$-linear. For degree reasons (making use of
$\Delta$ and $\bar{\kappa}$-linearity), these determine all
$d_{3}$-differentials. ∎

Figure 3. The $d_{3}$-differentials Figure 4. $d_{5}$ and
$d_{7}$-differentials in stems 0-48. A $\circ$ denotes
$\mathbb{F}_{2}[v_{1}^{4}]$.

######  Remark 4.2.

On the $E_{5}$-page, all classes in filtrations $s\geq 3$ are
$v_{1}^{4}$-torsion. The $v_{1}^{4}$-free classes are concentrated in stems
$t-s\not\equiv 5,6,7\mod 8$.

###  4.2. The $d_{5}$-differentials

######  Lemma 4.3 (Figure 4).

The $d_{5}$-differentials are $\Delta^{2}$-linear. They are determined by this
linearity, the differential

$d_{5}(\Delta)=\bar{\kappa}\nu,$

and the module structure over the elliptic spectral sequence for $tmf$.

###### Proof.

The same differential occurs in the spectral sequence for $\pi_{*}tmf$. The
rest of the argument is as in the proof of 4.1. ∎

Figure 5. Differentials in stems $48$ to $96$

###  4.3. Higher differentials

Since all the classes in filtrations 4 and above are in the ideal generated by
$\bar{\kappa}$, the differentials that have sources in filtrations 0-3
generate the other differentials with respect to the module structure over the
elliptic spectral sequence for $tmf$ (denoted $E_{r}^{*,*}(S^{0})$). We focus
on these differentials in the narrative. See Figures 4, 5, 6 and 7.

######  Lemma 4.4.

The $d_{7}$-differentials are $\Delta^{4}$-linear and determined by

$\displaystyle d_{7}(\Delta\nu^{2}y)$
$\displaystyle=\bar{\kappa}^{2}\eta^{2}v_{1},$ $\displaystyle
d_{7}(\Delta^{3}\nu^{2}y)$
$\displaystyle=\Delta^{2}\bar{\kappa}^{2}\eta^{2}v_{1}\ ,$

and the module structure over the elliptic spectral sequence for $tmf$.

###### Proof.

First, note that $d_{7}(\Delta^{4})=\eta^{2}\bar{\kappa}=4\nu\bar{\kappa}$ in
the spectral sequence for $tmf$. Therefore, for any $a\in E_{7}(V(0))$

$d_{7}(\Delta^{4}a)=4\nu\bar{\kappa}a+\Delta^{4}d_{7}(a).$

Since $4E_{7}(V(0))=0$, we get $\Delta^{4}$-linearity.

We give a proof for the differential
$d_{7}(\Delta\nu^{2}y)=\bar{\kappa}^{2}\eta^{2}v_{1}$. The proof for the other
differential is similar. In the spectral sequence for $tmf$, we have

$d_{7}(\Delta 4\bar{\kappa})=\eta^{3}\bar{\kappa}^{2}.$

But, for $\delta_{2}\colon E_{2}^{s,t}(V(0))\to E_{2}^{s+1,t}(S^{0})$ the
connecting homomorphism, we have

$\delta_{2}(\Delta\nu^{2}y)=\Delta 4\bar{\kappa}$

and

$\delta_{2}(\bar{\kappa}^{2}\eta^{2}v_{1})=\bar{\kappa}^{2}\eta^{3}.$

The differential when $i=0$ then follows from 2.17.

Making use of the module structure over the spectral sequence for $tmf$, the
only other possible $d_{7}$-differential for degree reasons is on
$\Delta^{2}\nu^{2}y$. But this class is in fact a $d_{7}$-cycle since
$\Delta^{2}y$ is a $d_{7}$-cycle by sparseness. ∎

######  Lemma 4.5.

Using the module structure over the elliptic spectral sequence for $tmf$, the
$d_{9}$-differentials are determined by the following differentials with
$i=0,1$:

  1. (1)

$d_{9}(\Delta^{2+4i})=\Delta^{4i}\bar{\kappa}^{2}x$,

  2. (2)

$d_{9}(\Delta^{2+4i}x)=\Delta^{4i}\bar{\kappa}^{2}\kappa$,

  3. (3)

$d_{9}(\Delta^{3+4i}\eta)=\Delta^{1+4i}\bar{\kappa}^{2}\epsilon$

  4. (4)

$d_{9}(\Delta^{3+4i}\epsilon)=\Delta^{1+4i}\kappa\bar{\kappa}^{2}\eta$

  5. (5)

$d_{9}(\Delta^{2+4i}v_{1})=\Delta^{4i}\bar{\kappa}^{2}v_{1}x$

  6. (6)

$d_{9}(\Delta^{2+4i}v_{1}x)=\Delta^{4i}\bar{\kappa}^{2}\eta y$

  7. (7)

$d_{9}(\Delta^{3+4i}v_{1})=\Delta^{1+4i}\bar{\kappa}^{2}v_{1}x$

  8. (8)

$d_{9}(\Delta^{3+4i}v_{1}x)=\Delta^{1+4i}\bar{\kappa}^{2}\eta y$

###### Proof.

We prove the claim for $i=0$. To prove $i=1$, one uses exactly the same
arguments in later stems.

In order to show (1), note that $\Delta^{2}$ cannot support any $d_{r}$ for
$r<9$ by sparseness. Then we have the differential from the elliptic spectral
sequence for $tmf$

$d_{9}(\Delta^{2}\eta)=\bar{\kappa}^{2}\epsilon$

and this differential becomes $\eta$ divisible in the spectral sequence for
$tmf\wedge V(0)$. For (2), we use the same argument with the differential
$d_{9}(\Delta^{2}\epsilon)=\Delta\bar{\kappa}^{2}\kappa\eta$ from the elliptic
spectral sequence for $tmf$.

The differentials (3) and (4) are the images of the same differentials in the
elliptic spectral sequence for $tmf$. The differentials (5)–(8) are proved
using 2.17. For example, the differential
$d_{9}(\Delta^{2}\eta)=\bar{\kappa}^{2}\epsilon$ and the facts that
$\delta(v_{1})=\eta$ and $\delta(v_{1}x)=\epsilon$ together imply (5). The
others are similar.

It remains to argue that there are no other generating $d_{9}$-differentials.
As noted above, it suffices to determine this on classes in filtration less
than four. Combining a comparison with the spectral sequence for $tmf$ and
sparseness, we see that the only question is whether or not the classes
$\Delta^{4}x$ and $\Delta^{4}v_{1}x$ support non-trivial $d_{9}$s. However,
the possible targets are the sources of $\bar{\kappa}$-multiples of the
$d_{11}$-differentials (1) and (3) of 4.6 shown below, which settles the
question. ∎

Figure 6. Differentials in stems $96$ to $140$

Figure 7. Differentials in stems $140$ to $192$

######  Lemma 4.6.

Using the module structure over the elliptic spectral sequence for $tmf$, the
$d_{11}$-differentials are determined by the following differentials with
$i=0,1$:

  1. (1)

$d_{11}(\Delta^{2+4i}\kappa)=\Delta^{4i}\bar{\kappa}^{3}\eta$

  2. (2)

$d_{11}(\Delta^{3+4i}\kappa\eta)=\Delta^{1+4i}\bar{\kappa}^{3}\eta^{2}$

  3. (3)

$d_{11}(\Delta^{2+4i}y)=\Delta^{4i}\bar{\kappa}^{3}v_{1}$

  4. (4)

$d_{11}(\Delta^{3+4i}\kappa v_{1})=\Delta^{1+4i}v_{1}\bar{\kappa}^{3}\eta$

  5. (5)

$d_{11}(\Delta^{5}v_{1})=\Delta^{3}\bar{\kappa}^{2}\nu^{3}$

###### Proof.

The differentials (1) and (2) are images of the same differentials in the
spectral sequence for $tmf$. The differentials (3) and (4) follow from (1) and
(2) respectively using 2.17. The differential (5) follows from the fact that
$\pi_{121}(tmf\wedge V(0))$ does not contain $v_{1}^{4}$-torsion, which can be
verified by comparing with $\pi_{*}tmf$ using the long exact sequence on
$\pi_{*}$.

Sparseness and multiplicative structure guarantees that these are all the
generating $d_{11}$-differentials, except for a possible $d_{11}$ on
$\Delta^{7}\eta^{2}v_{1}$. However, the possible target is the source of the
$\bar{\kappa}$-multiple of the $d_{13}$ below. ∎

######  Lemma 4.7.

The $d_{13}$-differentials are determined by

$d_{13}(\Delta^{4}y)=\Delta^{2}\bar{\kappa}^{3}\eta^{2}.$

There are no $d_{15}$-differentials and the $d_{17}$-differentials are
determined by

$d_{17}(\Delta^{4})=\bar{\kappa}^{4}y.$

The $d_{19}$-differentials are determined by

$d_{19}(\Delta^{7}\nu^{3})=\bar{\kappa}^{5}\Delta^{3}v_{1}\eta^{2}.$

###### Proof.

The first and second differentials follow from the facts that

$\displaystyle\pi_{110}(tmf\wedge V(0))=\mathbb{Z}/2\ \ \text{and}\ \
\pi_{95}(tmf\wedge V(0))=0$

respectively. The $d_{19}$-differential follows from the fact that the there
is no $v_{1}^{4}$-torsion in $\pi_{177}(tmf\wedge V(0))$.

There are no $d_{15}$ differentials and no other $d_{17}$ and $d_{19}$ for
degree reasons. The only argument needed beyond sparseness and multiplicative
structure to show that there are no other $d_{13}$-differentials is as
follows. There are possible $d_{13}$s on $\Delta^{3}\nu^{3}$ and
$\Delta^{7}\nu^{3}$. These classes are in the image of the $tmf$ spectral
sequence. For $tmf$, $d_{13}(\Delta^{3}\nu^{3})=2\bar{\kappa}^{4}$ and the
target maps to zero in the spectral sequence for $tmf\wedge V(0)$ and
similarly for $\Delta^{7}\nu^{3}$. ∎

######  Warning 4.8.

The $d_{13}$ differential above is in fact equivalent to the $2$-extension in
$\pi_{110}tmf$. For those familiar with names, this corresponds to
$2\kappa_{4}=\eta_{1}\bar{\kappa}^{3}$. For a recent detailed treatment of
this extension, see [BR, Chapter 9].

######  Lemma 4.9.

There are no $d_{21}$-differentials. The $d_{23}$-differentials are determined
by:

  1. (1)

$d_{23}(\Delta^{5}\eta)=\bar{\kappa}^{6}$

  2. (2)

$d_{23}(\Delta^{6}\eta^{2})=\bar{\kappa}^{6}\Delta\eta$

  3. (3)

$d_{23}(\Delta^{6}\eta v_{1})=\bar{\kappa}^{6}\Delta v_{1}$

  4. (4)

$d_{23}(\Delta^{7}\eta^{2}v_{1})=\bar{\kappa}^{6}\Delta^{2}\eta v_{1}$

###### Proof.

The differentials (1) and (2) occur in the elliptic spectral sequence for
$tmf$. The differential (3) is the geometric boundary of (2) as in 2.17. The
last differential is forced by the fact that the $v_{1}^{4}$-torsion in
$\pi_{171}(tmf\wedge V(0))$ is trivial. There are no $d_{21}$ or other
$d_{23}$-differentials for degree reasons. ∎

The following is now immediate.

######  Lemma 4.10.

The spectral sequence collapses at $E_{24}$ with a horizontal vanishing line
at $s=22$, i.e., $E_{\infty}^{s,t}(V(0))=0$ for $s\geq 22$.

###  4.4. Exotic extensions

We list the exotic extensions that do occur. All other possibilities can be
ruled out using algebraic structure and duality. We bring to the attention of
the reader the precise meaning of exotic extensions given in 2.18. Note also
that all exotic $2$-extensions are deduced from Lemma 2.19. We do not discuss
$2$-extensions further but include them in our figures.

Figure 8. Exotic extensions in the elliptic spectral sequence for $tmf\wedge
V(0)$ in stems $0$ to $96$. This records
$tmf_{*}V(0)\cong\widetilde{tmf}_{*+1}\mathbb{R}P^{2}$.

######  Lemma 4.11 (Figure 8).

In stems $0$ to $45$, there are exotic extensions:

  1. (1)

$[\Delta\eta]\nu=\bar{\kappa}\epsilon$

  2. (2)

$[\Delta\epsilon]\nu=\kappa\bar{\kappa}\eta$

  3. (3)

$[\Delta\kappa\eta]\nu=\bar{\kappa}^{2}\eta^{2}$

  4. (4)

$[\Delta v_{1}]\nu=\bar{\kappa}v_{1}x$

  5. (5)

$[\Delta v_{1}x]\nu=\kappa\bar{\kappa}v_{1}$

  6. (6)

$[\Delta\kappa v_{1}]\nu=\bar{\kappa}^{2}\eta v_{1}$

  7. (7)

$[y\nu^{2}]\nu=\bar{\kappa}v_{1}\eta^{2}$

###### Proof.

The first three extensions are between elements from $\pi_{*}tmf$, see
[Bau08]. The next three are forced by the fact that the connecting
homomorphism in the long exact sequence on homotopy groups is a map of
$\pi_{*}S^{0}$-modules, the geometric boundary theorem, and the fact that
under the map

$\delta\colon E_{2}^{s,t}(V(0))\to E_{2}^{s+1,t}(S^{0})$

we have $\delta(v_{1})=\eta$ (and so $\delta(xv_{1})=\epsilon$, $\delta(\kappa
v_{1})=\eta\kappa$, etc.).

The last extension follows from duality and the fact that there is a $\nu$
multiplication between stems $147$ and $150$ (already present on the
$E_{2}$-page). ∎

######  Lemma 4.12 (Figure 8).

In stems $46$ to $96$, there are exotic extensions:

  1. (1)

$[\Delta^{2}\eta^{2}]\nu=\Delta\bar{\kappa}\nu^{3}$

  2. (2)

$[\Delta^{2}\nu]\eta=\Delta\bar{\kappa}\epsilon$

  3. (3)

$[\Delta^{2}v_{1}\eta]\nu=\Delta\bar{\kappa}x\nu$

  4. (4)

$[\Delta^{2}x\nu]\eta=\Delta\bar{\kappa}\kappa\eta$

  5. (5)

$[\Delta^{2}x\nu]\nu=\Delta\bar{\kappa}\kappa\eta v_{1}$

  6. (6)

$[\Delta^{2}\kappa\nu]\nu=\Delta\bar{\kappa}^{2}\eta^{2}v_{1}$

  7. (7)

$[\Delta^{2}y\nu^{2}]\nu=\Delta^{2}\bar{\kappa}v_{1}\eta^{2}$

###### Proof.

The first two extensions (1) and (2) are multiplicative relations that hold in
$\pi_{*}tmf$. Extension (3) follows from (1) and 2.17. Extension (4) is dual
to the algebraic $\eta$ multiplication from stem $112$ to $113$, and similarly
for (5). The extension (6) involves classes in the image of $i_{*}$ and this
extension happens in $tmf_{*}$. Finally, (7) is dual to the algebraic $\nu$
multiplication from stem $99$ to $102$. ∎

######  Remark 4.13.

Looking at the charts in [Bau08], one might have expected extensions
$[\Delta^{2}\kappa\nu]\eta=\Delta\bar{\kappa}^{2}\eta^{2}$ and, by the
Geometric Boundary Theorem, $[\Delta^{2}y\nu]\eta=\Delta\bar{\kappa}^{2}\eta
v_{1}$. However, these are not exotic extensions according to 2.18.

We also note that $[\Delta^{2}c_{4}]\nu\neq[\Delta\bar{\kappa}\kappa\eta]$ and
$[\Delta^{3}c_{4}v_{1}]\nu\neq[\Delta\bar{\kappa}^{3}\eta]$. The first comes
from the fact in $\pi_{*}tmf$, there is no such extension. (This can be seen,
for example, from the Adams Spectral Sequence of $tmf$.) The second follows
from the fact that the target has a non-trivial $\bar{\kappa}$-multiple and
$\bar{\kappa}\nu=0$.

Figure 9. Exotic extensions in the elliptic spectral sequence of $tmf\wedge
V(0)$ in stems 96 to 192, recording
$tmf_{*}V(0)\cong\widetilde{tmf}_{*+1}\mathbb{R}P^{2}$.

######  Lemma 4.14 (Figure 9).

In stems $97$ to $144$, there are exotic extensions:

  1. (1)

$[\Delta^{4}\eta]\nu=\bar{\kappa}^{5}$

  2. (2)

$[\Delta^{4}\nu]\eta=\bar{\kappa}^{5}$

  3. (3)

$[\Delta^{4}\bar{\kappa}\epsilon]\eta=\Delta\bar{\kappa}^{5}\eta$

  4. (4)

$[\Delta^{5}\eta^{2}]\nu=\Delta\bar{\kappa}^{5}\eta$

  5. (5)

$[\Delta^{5}\epsilon]\nu=\Delta^{4}\kappa\bar{\kappa}\eta$

  6. (6)

$[\Delta^{5}\kappa\eta]\nu=\Delta^{4}\bar{\kappa}^{2}\eta^{2}$

  7. (7)

$[\Delta^{4}\bar{\kappa}xv_{1}]\eta=\Delta\bar{\kappa}^{5}v_{1}$ (from
$(125,5)$ to $(126,20)$)

  8. (8)

$[\Delta^{5}xv_{1}]\nu=\Delta^{4}\kappa\bar{\kappa}v_{1}$ (from $(129,1)$ to
$(132,6)$)

  9. (9)

$[\Delta^{5}\kappa v_{1}]\nu=\Delta^{4}\bar{\kappa}^{2}\eta v_{1}$ (from
$(136,2)$ to $(139,9)$)

  10. (10)

$[\Delta^{5}\epsilon v_{1}]\eta=\Delta^{2}\bar{\kappa}^{4}v_{1}\eta$ (from
$(130,2)$ to $(131,17)$)

  11. (11)

$[\Delta^{4}v_{1}]\nu=\Delta^{3}\bar{\kappa}\nu^{3}$ (from $(98,0)$ to
$(101,7)$)

  12. (12)

$[\Delta^{5}\kappa\eta]\eta=\Delta^{3}\bar{\kappa}^{3}\eta^{2}v_{1}$ (from
$(135,3)$ to $(136,14)$)

  13. (13)

$[\Delta^{4}y\nu^{2}]\nu=[\Delta^{4}\bar{\kappa}v_{1}\eta^{2}]$ (from
$(117,3)$ to $(120,6)$)

###### Proof.

Extensions (1)–(6) follow from studying $i_{*}\colon tmf_{*}\to tmf_{*}V(0)$.
Note that (4) is missing from the [Bau08] charts, but can be obtained from the
classical Adams Spectral Sequence for $tmf$. See [DFHH14, Chapter 13] or [BR,
Chapter 9]. Extensions (7), (8) and (9) follow from (3), (5) and (6),
respectively, using 2.17.

For (11), note that by 2.17, $[\Delta^{4}v_{1}]$ has geometric boundary
$[\Delta^{4}\eta]$. Since $[\Delta^{4}\eta]\nu\neq 0$,
$[\Delta^{4}v_{1}]\nu\neq 0$ and this extension is the only choice. For (12),
use 2.14 and the algebraic $\eta$ multiplication between $\pi_{35}tmf\wedge
V(0)$ and $\pi_{36}tmf\wedge V(0)$. A similar argument applies for (13). ∎

######  Remark 4.15.

There is no exotic $\nu$-extension on $[\Delta^{5}c_{4}]$ since the potential
target is not annihilated by $\bar{\kappa}$.

######  Lemma 4.16 (Figure 9).

In stems $145$ to $191$, there are exotic extensions:

  1. (1)

$[\Delta^{6}\nu]\eta=[\Delta^{5}\bar{\kappa}\epsilon]$ (from $(147,1)$ to
$(148,6)$)

  2. (2)

$[\Delta^{6}\kappa\nu]\eta=[\Delta^{5}\bar{\kappa}^{2}\eta^{2}]$ (from
$(161,3)$ to $(162,10)$)

  3. (3)

$[\Delta^{5}\bar{\kappa}\kappa\eta]\eta=[\Delta^{3}\bar{\kappa}^{4}\eta^{2}v_{1}]$
(from $(155,7)$ to $(156,18)$)

  4. (4)

$[\Delta^{6}y\nu]\eta=[\Delta^{5}\bar{\kappa}^{2}v_{1}\eta]$ (from $(162,2)$
to $(163,9)$)

  5. (5)

$[\Delta^{5}\bar{\kappa}\epsilon v_{1}]\eta=[\Delta^{2}\bar{\kappa}^{5}\eta
v_{1}]$ (from $(150,6)$ to $(151,21)$)

  6. (6)

$[\Delta^{6}\nu^{3}]\nu=\Delta^{3}\bar{\kappa}^{4}v_{1}\eta^{2}$ (from
$(153,3)$ to $(156,18)$)

  7. (7)

$[\Delta^{6}\epsilon v_{1}]\eta=\Delta^{5}\bar{\kappa}\kappa\eta$ (from
$(154,2)$ to $(155,7)$)

  8. (8)

$[\Delta^{6}\epsilon v_{1}]\nu=\Delta^{5}\bar{\kappa}\kappa\nu$ (from
$(154,2)$ to $(157,7)$)

  9. (9)

$[\Delta^{6}\kappa\nu]\nu=\Delta^{5}\bar{\kappa}^{2}v_{1}\eta^{2}$ (from
$(161,3)$ to $(164,10)$)

  10. (10)

$[\Delta^{6}y\nu^{2}]\nu=[\Delta^{6}\bar{\kappa}v_{1}\eta^{2}]$ (from
$(165,3)$ to $(168,6)$)

###### Proof.

The first two extensions occur in $tmf_{*}$. The third is also an extension in
$tmf_{*}$, namely
$[\Delta^{5}\bar{\kappa}\kappa\eta]\eta=[\Delta^{4}2\bar{\kappa}^{3}]$, but
the image of the class $[\Delta^{4}2\bar{\kappa}^{3}]$ is detected by
$[\Delta^{3}\bar{\kappa}^{4}\eta^{2}v_{1}]$ in $tmf_{*}V(0)$. The extension
(4) follow from (2) and 2.17. This result also implies (5) from the extensions
$[\Delta^{5}\bar{\kappa}\nu^{3}]\eta=[\Delta^{2}\bar{\kappa}^{5}\eta^{2}]$ in
$tmf_{*}$. All the extensions (6)–(10) follow from 2.10 and 2.14 and the data
for algebraic multiplications in the range $3\leq t-s\leq 20$. ∎

##  5\. $tmf_{*}Y$: The $E_{2}$-page

Let $C_{\eta}$ be the cofiber of the Hopf map $\eta$, so that there is an
exact triangle

(5.1) $S^{1}\xrightarrow{\eta}S^{0}\to C_{\eta}\to S^{2}.$

We define $Y\simeq V(0)\wedge C_{\eta}$ and study its elliptic spectral
sequence. Recall that if $F$ is a finite spectrum, then we abbreviate

$\mathcal{F}_{*}(F):=\pi_{*}(tmf\wedge X(4)\wedge F).$

We first describe $\mathcal{F}_{*}(C_{\eta})$. Since $\pi_{*}(tmf\wedge
X(4))\cong A$ is concentrated in even degrees, the cofiber sequence (5.1)
induces a short exact sequence on $tmf\wedge X(4)$-homology

$0\rightarrow
A\rightarrow\mathcal{F}_{*}(C_{\eta})\rightarrow\Sigma^{2}A\rightarrow 0.$

This splits as a sequence of $A$-modules so that

$\mathcal{F}_{*}(C_{\eta})\cong A\oplus\Sigma^{2}A.$

Multiplication by $2$ on $C_{\eta}$ induces multiplication by $2$ on
$tmf\wedge X(4)$-homology, which is injective because
$\mathcal{F}_{*}(C_{\eta})$ is torsion-free. Thus the cofiber sequence

$C_{\eta}\xrightarrow{2}C_{\eta}\rightarrow Y$

induces a short exact sequence in $tmf\wedge X(4)$-homology

$0\rightarrow\mathcal{F}_{*}(C_{\eta})\rightarrow\mathcal{F}_{*}(C_{\eta})\rightarrow\mathcal{F}_{*}(Y)\rightarrow
0,$

and it follows that

(5.2) $\mathcal{F}_{*}(Y)\cong A/(2)\oplus\Sigma^{2}A/(2)$

as an $A/(2)$-module.

Likewise, since $\mathcal{F}_{*}(V(0))$ is concentrated in even degrees, the
induced map on $tmf\wedge X(4)$-homology of the cofiber sequence

$\Sigma V(0)\xrightarrow{\eta}V(0)\rightarrow Y$

is trivial. It follows that there is a short exact sequence of
$\Lambda$-comodules

$0\rightarrow
A/(2)\rightarrow\mathcal{F}_{*}(Y)\rightarrow\Sigma^{2}A/(2)\rightarrow 0.$

This short exact sequence of $A$-modules splits because of (5.2). Tensoring it
with $A^{\prime}$ over $A$, we obtain a short exact sequence of
$\Lambda^{\prime}$-comodules, which splits as a sequence of
$A^{\prime}$-modules

(5.3) $0\rightarrow A^{\prime}/(2)\rightarrow
A^{\prime}\otimes_{A}\mathcal{F}_{*}(Y)\rightarrow\Sigma^{2}A^{\prime}/(2)\rightarrow
0.$

As $\mathcal{F}_{*}(Y)$ is $2$-torsion, (5.3) is a short exact sequence of
$A^{\prime}/(2)$-module, and hence splits as such. Therefore, applying
$\mathop{\mathrm{Ext}}\nolimits^{*,*}_{\Lambda^{\prime}}(A^{\prime},-)$ to
(5.3), we get a long exact sequence of
$\mathop{\mathrm{Ext}}\nolimits^{*,*}_{\Lambda^{\prime}}(A^{\prime},A^{\prime}/(2))$-modules.
See, for example, [Bro10, p.110, (3.3)]. Its connecting homomorphism

(5.4)
$\delta:\mathop{\mathrm{Ext}}\nolimits^{s,t}_{\Lambda^{\prime}}(A^{\prime},A^{\prime}/(2))\rightarrow\mathrm{Ext}^{s+1,t+2}_{\Lambda^{\prime}}(A^{\prime},A^{\prime}/(2))$

is given by multiplication with
$\eta\in\mathop{\mathrm{Ext}}\nolimits^{1,2}_{\Lambda^{\prime}}(A^{\prime},A^{\prime}/(2))$.
Here, as is often the case, we denote by $\eta$ the class in
$\mathop{\mathrm{Ext}}\nolimits$ which detects the same-named homotopy class.

We present the effect of the connecting homomorphism separately for the
$v_{1}$-power torsion and for the $v_{1}$-free classes of $E_{2}(V(0))$ in
Figure 10 and Figure 11, respectively.

Figure 10. The connecting homomorphism (5.4) for the $v_{1}$-power torsion
classes Figure 11. The connecting homomorphism (5.4) for the $v_{1}$-free
classes

In Figure 11 a $\circ$ denotes a copy of $\mathbb{F}_{2}[v_{1}]$, and a line
of slope 1 denotes, as usual, multiplication by $\eta$. Note that we have
$\bar{\kappa}v_{1}^{4}=\Delta\eta^{4}$, hence $\bar{\kappa}v_{1}^{4}=0$ in
$E_{2}(Y)$, while $v_{1}$ itself is not nilpotent and $\Delta^{i}$ is not
$v_{1}$ torsion. For our purposes, we need to determine completely the action
of $v_{1}$ on $E_{2}(Y)$. The class
$v_{1}\in\mathop{\mathrm{Ext}}\nolimits_{\Lambda^{\prime}}^{0}(A^{\prime},A^{\prime}/(2))$
is detected by the primitive $a_{1}\in A^{\prime}/(2)$ (with respect to the
$\Lambda^{\prime}$-comodule structure).

Since $a_{1}\in A^{\prime}/(2)$ is a primitive, multiplication by $a_{1}\in
A^{\prime}/(2)$ induces the following diagram of $\Lambda^{\prime}$-comodules

(5.15)

We let

$\mathcal{M}:=A^{\prime}/(2,a_{1})\otimes_{A/(2)}\mathcal{F}_{*}(Y)$

The middle vertical short exact sequence induces a long exact sequence

(5.16)
$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{s,t}_{2}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v_{1}}$$\textstyle{E^{s,t+2}_{2}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathop{\mathrm{Ext}}\nolimits^{s,t+2}_{\Lambda^{\prime}}(A^{\prime},\mathcal{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{2}^{s+1,t+2}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v_{1}}$$\textstyle{\ldots}$

so that we can determine the action of $v_{1}$ on $E_{2}(Y)$ by computing

$\mathop{\mathrm{Ext}}\nolimits^{*,*}_{\Lambda^{\prime}}(A^{\prime},\mathcal{M})\cong\mathop{\mathrm{Ext}}\nolimits^{*,*}_{\Lambda^{\prime}/(2,a_{1})}(A^{\prime}/(2,a_{1}),A^{\prime}/(2,a_{1})\otimes_{A/(2)}\mathcal{F}_{*}(Y)).$

The cohomology ring
$\mathop{\mathrm{Ext}}\nolimits_{\Lambda^{\prime}/(2,a_{1})}^{*,*}(A^{\prime}/(2,a_{1}),A^{\prime}/(2,a_{1}))$
is computed in [Bau08, Section 7]. With our notation,

$\mathop{\mathrm{Ext}}\nolimits_{\Lambda^{\prime}/(2,a_{1})}^{*,*}(A^{\prime}/(2,a_{1}),A^{\prime}/(2,a_{1}))\cong\mathbb{F}_{2}[\eta,\nu,\bar{\kappa},v_{2}]/(v_{2}\eta^{3}-\nu^{3},\eta\nu).$

The bottom short exact sequence of the above diagram (5.15) splits as a
sequence of $A^{\prime}/(2,a_{1})$-modules. However, it does not split as a
one of $\Lambda^{\prime}/(2,a_{1})$-comodules, rather it represents the
element
$\eta\in\mathop{\mathrm{Ext}}\nolimits_{\Lambda^{\prime}/(2,a_{1})}^{1,2}(A^{\prime}/(2,a_{1}),A^{\prime}/(2,a_{1}))$.
Therefore, the connecting homomorphism

(5.17)
$\mathop{\mathrm{Ext}}\nolimits_{\Lambda^{\prime}/(2,a_{1})}^{s,t}(A^{\prime}/(2,a_{1}),A^{\prime}/(2,a_{1}))\rightarrow\mathop{\mathrm{Ext}}\nolimits_{\Lambda^{\prime}/(2,a_{1})}^{s+1,t+2}(A^{\prime}/(2,a_{1}),A^{\prime}/(2,a_{1}))$

of the induced long exact sequence in
$\mathop{\mathrm{Ext}}\nolimits_{\Lambda^{\prime}/(2,a_{1})}^{*,*}(A^{\prime}/(2,a_{1}),-)$
is given by multiplication by $\eta$. We obtain:

######  Lemma 5.18.

As a module over the ring
$\mathbb{F}_{2}[\eta,\nu,\bar{\kappa},v_{2}]/(v_{2}\eta^{3}-\nu^{3},\eta\nu)$,
the cohomology group

$\mathop{\mathrm{Ext}}\nolimits^{*,*}_{\Lambda^{\prime}/(2,a_{1})}(A^{\prime}/(2,a_{1}),A^{\prime}/(2,a_{1})\otimes_{A/(2)}\mathcal{F}_{*}(Y))$

is generated by $a[0,0]\in\mathop{\mathrm{Ext}}\nolimits^{0,0}$ and
$a[5,1]\in\mathop{\mathrm{Ext}}\nolimits^{1,6}$ with the relations

$\displaystyle\eta a[0,0]$ $\displaystyle=0,$ $\displaystyle\eta
a[5,1]=\nu^{2}a[0,0].$

###### Proof.

By the description of the connecting homomorphism (5.17), we see that

$\mathop{\mathrm{Ext}}\nolimits^{*,*}_{\Lambda^{\prime}/(2,a_{1})}(A^{\prime}/(2,a_{1}),A^{\prime}/(2,a_{1})\otimes_{A/(2)}\mathcal{F}_{*}(Y))\cong\mathbb{F}_{2}[\nu,\bar{\kappa},v_{2}]/(\nu^{3})\\{a[0,0],a[5,0]\\}$

as an $\mathbb{F}_{2}[\nu,\bar{\kappa},v_{2}]/(\nu^{3})$-module. Next, we
determine the action of $\eta$. We see easily that $\eta a[0,0]=0$. To
calculate $\eta a[5,1]$, we remark that

$\displaystyle\nu^{2}a[0,0]$ $\displaystyle=\langle\eta,\nu,\eta\rangle
a[0,0]=\eta\langle\nu,\eta,a[0,0]\rangle,$

where the first equality comes from the Massey product
$\nu^{2}=\langle\eta,\nu,\eta\rangle$ and the second is a shuffle. As
$\nu^{2}a[0,0]\neq 0$, $\langle\nu,\eta,a[0,0]\rangle$ is not trivial and must
be equal to $a[5,1]$ by sparseness. Hence, $\nu^{2}a[0,0]=\eta a[5,1]$. ∎

$0$$6$$12$$18$$24$$0$$1$$2$$3$$4$ Figure 12.
$\mathop{\mathrm{Ext}}\nolimits^{s,t}_{\Lambda^{\prime}/(2,a_{1})}(A^{\prime}/(2,a_{1}),A^{\prime}/(2,a_{1})\otimes_{A/(2)}\mathcal{F}_{*}(Y))$
depicted in the coordinate $(t-s,s)$.

######  Proposition 5.19 (Figure 13).

As a module over $E_{2}(V(0))$, $E_{2}(Y)$ is generated by classes

$\displaystyle a[0,0],a[5,1],a[17,3]$

The submodule generated by $a[0,0]$ is isomorphic to $E_{2}(V(0))/\eta$. There
are Massey products

$a[5,1]=\langle\nu,\eta,a[0,0]\rangle,\ \ a[17,3]=\langle\eta
x^{2},\eta,a[0,0]\rangle$

and these classes are subject to the following relations. On the new classes,
we have $v_{1}$ mulitiplications

$\displaystyle v_{1}a[5,1]=xa[0,0]\ \ \ \ v_{1}a[17,3]=x^{2}a[5,1],$

$\eta$ and $\nu$ multiplications

$\displaystyle\eta a[5,1]=\nu^{2}a[0,0],\ \ \ \eta a[17,3]=\nu
a[17,3]=ya[17,3]=0$

as well as

$\displaystyle\nu^{2}ya[5,1]=v_{1}^{3}\bar{\kappa}a[0,0]\ .$

###### Proof.

Using the description of $E_{2}(V(0))$, the effect of the connecting
homomorphism $\delta$ of (5.4) is straightforward to compute. The cokernel is
simply $E_{2}V(0)/\eta$ as an $E_{2}(V(0))$-module. Using the multiplication
on $E_{2}(V(0))$, the kernel is generated by classes $a[5,1]$ and $a[17,3]$
defined as

$\displaystyle a[5,1]=p_{*}(\nu)\ \ \ a[17,3]=p_{*}(\eta x^{2}),$

where $p_{*}$ is induced by the map
$A^{\prime}\otimes_{A}\mathcal{F}_{*}(Y)\rightarrow\Sigma^{2}A^{\prime}/(2)$
of (5.3).

Inspecting the long exact sequence (5.16) and the structure of

$\mathop{\mathrm{Ext}}\nolimits^{*,*}_{\Lambda^{\prime}/(2,a_{1})}(A^{\prime}/(2,a_{1}),A^{\prime}/(2,a_{1})\otimes_{A/(2)}\mathcal{F}_{*}(Y)),$

we see that $v_{1}a[5,1]=xa[0,0]$ (else the latter
$\mathop{\mathrm{Ext}}\nolimits^{1,8}$-term would be nonzero and contain the
image of $xa[0,0]$). That $v_{1}a[17,3]=x^{2}a[5,1]$ follows from the fact
that $v_{1}\eta x^{2}=x^{2}\nu$ in $E_{2}(V(0))$ and the definition these
classes as images of $\delta$.

By the same argument used in Lemma 5.18, we deduce the $\eta$-multiplication
on $a[5,1]$. The relations $\eta a[17,3]=\nu a[17,3]=ya[17,3]=0$ follows for
degree reasons.

It remains to verify that $\nu^{2}ya[5,1]=v_{1}^{3}\bar{\kappa}a[0,0]$. A
juggling of Massey products gives

$y\nu^{2}\langle\nu,\eta,a[0,0]\rangle=\langle y\nu^{2},\nu,\eta\rangle
a[0,0].$

The relation $\nu^{2}ya[5,1]=v_{1}^{3}\bar{\kappa}a[0,0]$ then follows by
Lemma 5.20 below and the fact that $\eta a[0,0]=0$. ∎

Figure 13. $E_{2}(Y)$ as a module over $E_{2}(V(0))$. The dashed lines are
$x$-multiplications and dotted lines $y$-multiplications. Other structure
lines are as in Figure 2.

######  Lemma 5.20.

In
$\mathop{\mathrm{Ext}}\nolimits_{\Lambda^{\prime}}^{*,*}(A^{\prime},A^{\prime}/(2))$,
there is the following Massey product

$\bar{\kappa}v_{1}^{3}\in\langle y\nu^{2},\nu,\eta\rangle$

with indeterminacy

$\eta\mathop{\mathrm{Ext}}\nolimits^{3,28}_{\Lambda^{\prime}}(A^{\prime},A^{\prime}/(2))+y\nu^{2}\mathop{\mathrm{Ext}}\nolimits^{1,6}_{\Lambda^{\prime}}(A^{\prime},A^{\prime}/(2)).$

###### Proof.

By [Bau08, Formula 7.9],
$\bar{\kappa}v_{1}^{2}=\langle\eta,\kappa\eta,x\rangle$ and

$\bar{\kappa}v_{1}^{3}=v_{1}\langle\eta,\kappa\eta,x\rangle\subset\langle
v_{1}\eta,\kappa\eta,x\rangle\subset\langle\eta,v_{1}\kappa\eta,x\rangle=\langle\eta,\eta^{2}y,x\rangle$

and the indeterminacy
$\eta\mathop{\mathrm{Ext}}\nolimits^{3,28}_{\Lambda^{\prime}}(A^{\prime},A^{\prime}/(2))$
does not contain $\bar{\kappa}v_{1}^{3}$. Here, we used the relation
$v_{1}\kappa\eta=\eta^{2}y$. So
$\bar{\kappa}v_{1}^{3}=\langle\eta,\eta^{2}y,x\rangle$ and it follows that

$\displaystyle\bar{\kappa}v_{1}^{4}$
$\displaystyle=v_{1}\langle\eta,\eta^{2}y,x\rangle=\langle
v_{1}\eta,y,\eta^{2}x\rangle=\langle v_{1}\eta,y,\nu^{3}\rangle\subset\langle
v_{1}\eta,y\nu,\nu^{2}\rangle$

and the indeterminacy $\eta v_{1}\mathop{\mathrm{Ext}}\nolimits^{3,28}$ does
not contain $\bar{\kappa}v_{1}^{4}$. So

$\bar{\kappa}v_{1}^{4}=\langle v_{1}\eta,y\nu,\nu^{2}\rangle=\langle
v_{1}\eta,\nu,\nu^{2}y\rangle=v_{1}\langle\eta,\nu,y\nu^{2}\rangle.$

As $v_{1}$ acts injectively on
$\mathop{\mathrm{Ext}}\nolimits_{\Lambda^{\prime}}^{4,30}(A^{\prime},A^{\prime}/(2))$,
so $\bar{\kappa}v_{1}^{3}=\langle\eta,\nu,y\nu^{2}\rangle=\langle
y\nu^{2},\nu,\eta\rangle$. ∎

######  Remark 5.21.

In $E_{2}^{s,t}(Y)$, there is at most one non-zero element in any bi-degree
$(s,t)$ with filtration $s>0$. There is also a unique non-zero element in bi-
degree $(0,0)$. So, for $s>0$ or $(s,t)=(0,0)$, we often denote by
$a[t-s,s]\in E_{r}^{s,t}$ the non-zero element, if it exists. Furthermore,
when $s=0$ and $t>0$, we let $a[t,0]$ denote the element of $E_{2}^{0,t}(Y)$
which is divisible by the largest power of $\Delta$. For example,
$E_{2}^{0,52}(Y)\cong\mathbb{F}_{2}\\{v_{1}^{26}a[0,0],v_{1}^{14}\Delta
a[0,0],v_{1}^{2}\Delta^{2}a[0,0]\\}$ and $a[52,0]=v_{1}^{2}\Delta^{2}a[0,0]$.

Although Proposition 5.19 gives us a very compact description of $E_{2}(Y)$,
the elliptic spectral sequence of $tmf\wedge Y$ is not a module over the
elliptic spectral sequence of $tmf\wedge V(0)$ as the latter is not even a
multiplicative spectral sequence. However, the elliptic spectral sequence of
$tmf\wedge Y$ is a module over the elliptic spectral sequence of $tmf$. In
fact, we get even more structure than that from the fact that $Y$ has
$v_{1}$-self maps. As explained in Section 2.6, we have:

######  Lemma 5.22 ($v_{1}$-linearity).

The differentials in the elliptic spectral sequence for $tmf\wedge Y$ are
$v_{1}$-linear.

We state the following “intermediate” result for convenience of reference in
the computations below. The module structure of the elliptic spectral sequence
spectral sequence of $tmf\wedge Y$ over that of $tmf$ is richer than what is
stated here but that information can be read off of Proposition 5.19.

######  Corollary 5.23.

As a module over

$\mathbb{F}_{2}[v_{1},\nu,\bar{\kappa},\Delta]/(v_{1}\nu,\nu^{3},v_{1}^{4}\bar{\kappa})$

$E_{2}(Y)$ is generated by

$a[0,0],\ a[5,1],\ a[12,2],\ a[15,1],\ a[17,3],\ a[20,2]$

subject to the relations generated by

$\displaystyle v_{1}^{3}a[5,1]=v_{1}^{2}a[12,2]=v_{1}a[15,1]=\nu a[12,2]=\nu
a[17,3]=0$

and

$\displaystyle\nu^{2}a[15,1]=v_{1}^{2}a[17,3],\ \ \ \
\nu^{2}a[20,2]=v_{1}^{3}\bar{\kappa}a[0,0]\ .$

Furthermore, the differentials are
$\mathbb{F}_{2}[v_{1},\nu,\bar{\kappa},\Delta^{8}]/(v_{1}\nu,\nu^{3},v_{1}^{4}\bar{\kappa})$-linear.

###### Proof.

This follows from the results of this section and the fact that $\Delta^{8}$
is a permanent cycle in the elliptic spectral sequence spectral sequence of
$tmf$. ∎

##  6\. $tmf_{*}Y$: The differentials and extensions

Our approach to computing the differentials of the elliptic spectral sequence
for $\pi_{*}(tmf\wedge Y)$ is based largely on the analysis of the action of
$\bar{\kappa}$. Since $\bar{\kappa}$ is a permanent cycle in the elliptic
spectral sequence for $tmf$, $\bar{\kappa}$ acts on the spectral sequence for
$tmf\wedge Y$ and differentials are linear with respect to this action.

######  Lemma 6.1.

The $E_{r}$-term of the elliptic spectral sequence for $Y$ has the following
properties:

  1. (1)

All classes in filtration greater than $(r-1)$ are $\bar{\kappa}$-free.

  2. (2)

All classes in filtration greater than or equal to 4 are divisible by
$\bar{\kappa}$.

###### Proof.

We prove these two properties by induction on $r\geq 2$. For $r=2$, this
follows from Proposition 5.19. Suppose now that $r>2$. Let $a$ be a
$d_{r-1}$-cycle and $[a]\in E_{r}^{s,t}$ the corresponding class. Suppose that
$a$ lives in filtration $s$ with $s>(r-1)$. We have that $\bar{\kappa}[a]=0$
if and only if there exists $b\in E_{r-1}$ such that
$d_{r-1}(b)=\bar{\kappa}a$. Then, $b$ must live in filtration $(4+s)-(r-1)>4$.
By the second property, $b$ is divisible by $\bar{\kappa}$, i.e., there exists
$c\in E_{r-1}$ such that $\bar{\kappa}c=b$. As a consequence of the
$\bar{\kappa}$-linearity, $\bar{\kappa}d_{r-1}(c)=d_{r-1}(b)=\bar{\kappa}a$,
and so $\bar{\kappa}(d_{r-1}(c)-a)=0$. Since $(d_{r-1}(c)-a)\in E_{r-1}$ lives
in filtration $s$ greater than $r-2$, it is $\bar{\kappa}$-free by the second
property. It follows that $d_{r-1}(c)=a$, and so $[a]=0$. Therefore, the
$E_{r}$-term has the first property.

For the second property, suppose that $a$ lives in filtration greater than or
equal to 4. By the second property for $E_{r-1}$, there exists $b\in E_{r-1}$
such that $\bar{\kappa}b=a$. It suffices to prove that $b$ is a
$d_{r-1}$-cycle. Suppose that $d_{r-1}(b)=c$. The latter implies that $c$
lives in filtration greater that $(r-2)$, hence is $\bar{\kappa}$-free by the
first property. Since $a$ is a $d_{r-1}$-cycle by assumption, we have, by
$\bar{\kappa}$-linearity, that

$0=d_{r-1}(a)=d_{r-1}(\bar{\kappa}b)=\bar{\kappa}c.$

This means that $c=0$ and so $b$ is a $d_{r-1}$-cycle, as required. ∎

######  Terminology.

For convenience, we will call all $\bar{\kappa}$-multiples of a class which
has filtration less than four the $\bar{\kappa}$-family of that class. By part
$(2)$ of the above lemma, at any term of the spectral sequence, every class
belongs to some $\bar{\kappa}$-family. The following corollary tells us how
these $\bar{\kappa}$-families are organized.

######  Corollary 6.2.

  1. (1)

At any term of the spectral sequence, all non-zero $\bar{\kappa}$-power
torsion classes survive to the $E_{\infty}$-term.

  2. (2)

Every $\bar{\kappa}$-free family consisting of permanent cycles is truncated
by one and only one other $\bar{\kappa}$-free family.

###### Proof.

For part $(1)$, let $a\in E_{r}$ be a non-zero $\bar{\kappa}$-power torsion
class. By part (1) of 6.1, $a$ is in filtration less than or equal to $r-1$.
It follows that $a$ cannot be hit by any differential from the $E_{r}$-term
onwards. Moreover, by part (1) of 6.1 again, the possible targets of
$d_{r^{\prime}}(a)$, $r^{\prime}\geq r$ are $\bar{\kappa}$-free classes. Since
$a\in E_{r}$ is $\bar{\kappa}$-power torsion, it is a permanent cycle, by
$\bar{\kappa}$-linearity. Therefore, $a$ persists to the $E_{\infty}$-term.

For part $(2)$, let $a$ be a permanent cycle of filtration striclty less than
four which is $\bar{\kappa}$-free at the $E_{2}$-term. Then the
$\bar{\kappa}$-family of $a$ consists of permanent cycles. Since
$\bar{\kappa}$ is nilpotent at the $E_{\infty}$-term of the elliptic spectral
sequence for $tmf$, some $\bar{\kappa}$-multiple of $a$ must be hit by a
differential. Suppose that $a$ is $\bar{\kappa}$-free at the $E_{r}$-term and
that $\bar{\kappa}^{l}a$ is the smallest $\bar{\kappa}$-multiple of $a$ that
is hit by a differential, say $d_{r}(b)=\bar{\kappa}^{l}a$. Since $a$ is
$\bar{\kappa}$-free at the $E_{r}$-term, so is $b$. It follows that the
$\bar{\kappa}$-multiples of $b$ truncate those of $\bar{\kappa}^{l}a$ by
differentials $d_{r}$, i.e., $d_{r}(\bar{\kappa}^{n}b)=\bar{\kappa}^{l+n}a$.
So, all the classes $\bar{\kappa}^{k}a$ for $k\leq l-1$ are non-zero
$\bar{\kappa}$-power torsion classes on the $E_{r+1}$-term, hence are
essential by part $(1)$.

Finally, we claim that $b$ has filtration less than four so that the
$\bar{\kappa}$-family of $b$ truncates the $\bar{\kappa}$-family of $a$. If
$b$ had filtration greater than or equal to $4$, then $b$ would be divisible
by $\bar{\kappa}$, i.e., there would exist $c\in E_{r}$ such that
$\bar{\kappa}c=b$, by 6.1 part $(2)$. By $\bar{\kappa}$-linearity, we have
that $\bar{\kappa}^{l}a=d_{r}(b)=\bar{\kappa}d_{r}(c)$, and so
$\bar{\kappa}(\bar{\kappa}^{l-1}a-d_{r}(c))=0$. This means that
$d_{r}(c)=\kappa^{l-1}a$ because $d_{r}(c)-\bar{\kappa}^{l-1}a$ has filtration
at least $r$ so that it is $\bar{\kappa}$-free, by 6.1 part $(1)$. This
contradicts the minimality of $\ell$, so $b$ has filtration less than four. ∎

######  Slogan 6.3.

The $\bar{\kappa}$-free families at the $E_{r}$-page come in pairs. The first
member of the pair is a family consisting of permanent cycles. The second
member is a family which eventually supports differentials (i.e., possibly at
a later page) to truncate the first family.

######  Corollary 6.4.

At the $E_{r}$-term, we have:

  1. (1)

The homomorphism $E_{r}^{s,t}\rightarrow E_{r}^{s,t+192}$ induced by
multiplication by $\Delta^{8}$ is an injection for all $s$ and $t$,

  2. (2)

If $a$ is a class of the $E_{2}$-term such that $\Delta^{8}a$ is a
$d_{r}$-cycle, then $a$ is also a $d_{r}$-cycle.

###### Proof.

We prove part $(1)$ by induction on $r\geq 2$. For $r=2$, this can be seen
from the explicit structure of the $E_{2}$-term. Suppose the
$E_{r^{\prime}}$-term has these properties for $r^{\prime}<r$. Let us prove
part (1) for $E_{r}$. Let $a\in E_{r-1}$ represent a class of $E_{r}$. If
$\Delta^{8}[a]=0\in E_{r}.$ This means that there exists $b\in E_{r-1}$ such
that $d_{r-1}(b)=\Delta^{8}a$. It is obvious that $b$ lives in stem at least
$192$, hence there exists $c\in E_{r-1}$ such that $b=\Delta^{8}c$, by the
induction hypothesis. It follows that $\Delta^{8}(d_{r-1}(c)-a)=0$, and so
$d_{r-1}(c)=a$ because of part (1) of the induction hypothesis. Thus $[a]=0\in
E_{r}$, as needed.

For part (2), by induction, suppose that $a$ is a $d_{r-1}$-cycle. We need to
prove that $a$ is a $d_{r}$-cycle. In effect, if $d_{r}(a)=b$, then

$0=d_{r}(\Delta^{8}a)=\Delta^{8}d_{r}(a)=\Delta^{8}b.$

By part (1), $b=0$, and so $d_{r}(a)=0$, as needed. ∎

Finally, we will also use the following result to establish the differentials.

######  Lemma 6.5 (Vanishing line).

The spectral sequence for $\pi_{*}tmf\wedge Y$ degenerates at the
$E_{24}$-term and has a horizontal vanishing line at $s=24$, i.e.,
$E_{24}^{s,t}=E_{\infty}^{s,t}=0$ for $s\geq 24$.

###### Proof.

We know that $\bar{\kappa}^{6}$ is hit by a differential $d_{23}$ in the
elliptic spectral sequence for $tmf$, see [Bau08]. This means that at the
$E_{24}$-term of the elliptic spectral sequence for $tmf\wedge Y$, all the
classes are annihilated by $\bar{\kappa}^{6}$, hence are $\bar{\kappa}$-power
torsion. Therefore, by 6.1, all the classes in the $E_{24}$-term are in
filtrations less than $24$, meaning that the spectral sequence has the
horizontal vanishing line at $s=24$, i.e., $E^{s,t}_{r}=0$ for $s\geq 24$ and
$r\geq 24$. ∎

######  Remark 6.6.

The cofiber sequence

$V(0)\xrightarrow{i}Y\xrightarrow{p}\Sigma^{2}V(0)\xrightarrow{\eta}\Sigma
V(0)$

gives rise to maps of spectral sequences

$i_{*}\colon E_{2}^{s,t}(V(0))\to E_{2}^{s,t}(Y),\ \ \ \ p_{*}\colon
E_{2}^{s,t}(Y)\to E_{2}^{s,t-2}(V(0))$

as well as a long exact sequence

(6.7) $\displaystyle\ldots\to
tmf_{*+1}V(0)\xrightarrow{\eta}tmf_{*}V(0)\xrightarrow{i_{*}}tmf_{*}Y\xrightarrow{p_{*}}tmf_{*-1}V(0)\to\ldots$

###  6.1. The $d_{3}$, $d_{5}$ and $d_{7}$-differentials

Note that for $r$ even, $E_{r}(Y)\cong E_{r+1}(Y)$ since the spectral sequence
is concentrated in bi-degrees $(s,t)$ with $t$ even. The differentials in this
section are depicted in Figures 14, 15, 16 and 17.

######  Proposition 6.8.

There is no non-trivial $d_{3}$-differential, and so $E_{3}(Y)\cong E_{5}(Y)$.

###### Proof.

Since $\Delta$ is a $d_{3}$-cycle in the elliptic spectral sequence of $tmf$,
the $d_{3}$-differentials are
$\mathbb{F}_{2}[v_{1},\nu,\bar{\kappa},\Delta]/(v_{1}\nu,\nu^{3},v_{1}^{4}\bar{\kappa})$-linear.
All the generators listed in 5.23 are $d_{3}$-cycles for degree reasons. ∎

We then get the following result for degree reasons.

######  Corollary 6.9.

The classes in stems $t-s<24$ are permanent cycles.

######  Lemma 6.10.

The $d_{5}$-differentials are linear with respect to
$\bar{\kappa},\nu,v_{1},\Delta^{2}$ and are determined by

$\displaystyle d_{5}(\Delta)$ $\displaystyle=\nu\bar{\kappa},$ $\displaystyle
d_{5}(\Delta a[5,1])$ $\displaystyle=\nu\bar{\kappa}a[5,1]$ $\displaystyle
d_{5}(\Delta a[15,1])$ $\displaystyle=\nu\bar{\kappa}a[15,1],$ $\displaystyle
d_{5}(\Delta a[20,2])$ $\displaystyle=\nu\bar{\kappa}a[20,2]$

under multiplication by elements of
$\mathbb{F}_{2}[\Delta^{2},\bar{\kappa},\nu,v_{1}]/(v_{1}\nu,\nu^{3},\bar{\kappa}v_{1}^{4})$.

###### Proof.

For linearity, we only need to prove the $\Delta^{2}$-linearity. Note that
$d_{5}(\Delta)=\nu\bar{\kappa}$ in the elliptic spectral sequence of $tmf$. By
Leibniz rule and the fact that $E_{2}(Y)$ is $2$-torsion,

$d_{5}(\Delta^{2}x)=2\Delta
d_{5}(\Delta)x+\Delta^{2}d_{5}(c)=\Delta^{2}d_{5}(x).$

Using the module structure over the elliptic spectral sequence of $tmf$, we
get

$d_{5}(\Delta a[5,1])=d_{5}(\Delta)a[5,1]+\Delta
d_{5}(a[5,1])=\nu\bar{\kappa}a[5,1].$

The other arguments are similar. ∎

######  Lemma 6.11.

There are no non-trivial $d_{7}$-differentials.

###### Proof.

This is an immediate consequence of sparseness. ∎

The following observation will be crucial for our computation and is motivated
by 6.3.

######  Corollary 6.12 (Figure 14).

The $\bar{\kappa}$-free families on the $E_{9}$-term of the elliptic spectral
sequence of $tmf\wedge Y$ in stems $0\leq t-s<48$ are given by the following
$24$ classes

$\displaystyle a[0,0]$ $\displaystyle a[2,0]=v_{1}a[0,0]$ $\displaystyle
a[4,0]=v_{1}^{2}a[0,0]$ $\displaystyle a[5,1]$ $\displaystyle
a[7,1]=v_{1}a[5,1]$ $\displaystyle a[9,1]=v_{1}^{2}a[5,1]$ $\displaystyle
a[12,2]$ $\displaystyle a[14,2]=v_{1}a[12,2]$ $\displaystyle a[15,1]$
$\displaystyle a[17,3]$ $\displaystyle a[19,3]=v_{1}a[17,3]$ $\displaystyle
a[20,2]$ $\displaystyle a[26,0]=\Delta v_{1}a[0,0]$ $\displaystyle
a[28,0]=\Delta v_{1}^{2}a[0,0]$ $\displaystyle a[30,0]=\Delta v_{1}^{3}a[0,0]$
$\displaystyle a[30,2]=\Delta\nu^{2}a[0,0]$ $\displaystyle a[31,1]=\Delta
v_{1}a[5,1]$ $\displaystyle a[33,1]=\Delta v_{1}^{2}a[5,1]$ $\displaystyle
a[35,3]=\Delta\nu^{2}a[5,1]$ $\displaystyle a[36,2]=\Delta a[12,2]$
$\displaystyle a[38,2]=\Delta v_{1}a[12,2]$ $\displaystyle a[41,3]=\Delta
a[17,3]$ $\displaystyle a[43,3]=\Delta v_{1}a[17,3]$ $\displaystyle
a[45,3]=\Delta v_{1}^{2}a[17,3]$

All $\bar{\kappa}$-free families at $E_{9}$ are given by these classes and
their $\Delta^{2}$-multiples. All the elements in filtrations four and above
are $\bar{\kappa}$-multiples of these generators.

Figure 14. $d_{5}$-differentials in stems 0 to 48 and $\bar{\kappa}$-free
generators at $E_{9}$

The generators of the $\bar{\kappa}$-free families in stems $0\leq t-s<48$ are
presented in Figure 14. The $\bar{\kappa}$-free generators in the range $0\leq
t-s<192$ are given by multiples of these with $\Delta^{2},\Delta^{4}$ and
$\Delta^{6}$ and all other $\bar{\kappa}$-free generators are multiples of
these with the powers of $\Delta^{8}$. By 6.2, each $\bar{\kappa}$-free family
consisting of permanent cycles is truncated by exactly one other
$\bar{\kappa}$-free family. Thus, using the $\Delta^{8}$-linearity and 6.4, we
see that the $24\times 4$ $\bar{\kappa}$-free generators in the range $0\leq
t-s<192$ organize themselves as follows. Exactly half of them are permanent
cycles and the other half are not. The $\bar{\kappa}$-family of each non-
permanent $\bar{\kappa}$-free generator supports a differential that hits the
$\bar{\kappa}$-family of exactly one of the other permanent generators. Note
that the truncation must begin in stems less than four by 6.2. This allows us
to determine longer differentials before settling shorter ones.

All $24$ $\bar{\kappa}$-free generators in the range $0\leq t-s<48$ are
permanent cycles due to sparseness and in the next section we will find their
“partners”.

###  6.2. The $d_{9}$-differentials

To analyze the $d_{9}$-differentials, we make the following observation,
which, in some sense, is a very small part of the geometric boundary theorem
as in [Beh12, Appendix 4].

######  Lemma 6.13.

Let $a\in E_{r}^{s,t}(Y)$ so that $p_{*}(a)\in E_{r}^{s,t-2}(V(0))$. Suppose
$p_{*}(a)$ persists to the $E_{r^{\prime}}$ term for some $r^{\prime}\geq r$
and that there is a non-trivial differential, $d_{r^{\prime}}(p_{*}a)\neq 0$.
Then $d_{r^{\prime\prime}}(a)\neq 0$ for $r^{\prime\prime}\leq r^{\prime}$.

###### Proof.

This is a straight forward application of naturality. Our assumptions imply
that $a$ cannot be hit by a differential $d_{r^{\prime\prime}}$ for
$r^{\prime\prime}\leq r^{\prime}$ and, furthermore, that if it persists to the
$E_{r^{\prime}}$ term, that $d_{r^{\prime}}(a)=b^{\prime}$ for $b^{\prime}$
such that $p_{*}b^{\prime}=b$. ∎

The differentials below are depicted in Figures 14, 15, 16 and 17.

######  Lemma 6.14 (Figures 15, 16 and 17).

There are $d_{9}$-differentials, for $i=0,1$,

  1. (1)

$d_{9}(\Delta^{4i+2}a[0,0])=\bar{\kappa}^{2}\Delta^{4i}v_{1}a[5,1]$

  2. (2)

$d_{9}(\Delta^{4i+2}a[5,1])=\bar{\kappa}^{2}\Delta^{4i}a[12,2]$

  3. (3)

$d_{9}(\Delta^{4i+3}v_{1}a[0,0])=\bar{\kappa}^{2}\Delta^{4i+1}v_{1}^{2}a[5,1]$

  4. (4)

$d_{9}(\Delta^{4i+2}a[17,3])=\bar{\kappa}^{3}\Delta^{4i}v_{1}^{2}a[0,0]$

  5. (5)

$d_{9}(\Delta^{4i+3}a[17,3])=\bar{\kappa}^{3}\Delta^{4i+1}v_{1}^{2}a[0,0]$

  6. (6)

$d_{9}(\Delta^{4i+3}a[12,2])=\bar{\kappa}^{2}\Delta^{4i+1}v_{1}a[17,3]$

  7. (7)

$d_{9}(\Delta^{4i+2}a[12,2])=\bar{\kappa}^{2}\Delta^{4i}v_{1}a[17,3]$

  8. (8)

$d_{9}(\Delta^{4i+3}v_{1}a[5,1])=\bar{\kappa}^{2}\Delta^{4i+1}\ v_{1}a[12,2]$

  9. (9)

$d_{9}(\Delta^{4i+2}v_{1}a[0,0])=\bar{\kappa}^{2}\Delta^{4i}v_{1}^{2}a[5,1]$

  10. (10)

$d_{9}(\Delta^{4i+2}v_{1}a[5,1])=\bar{\kappa}^{2}\Delta^{4i}v_{1}a[12,2]$

  11. (11)

$d_{9}(\Delta^{4i+3}v_{1}a[17,3])=\bar{\kappa}^{3}\Delta^{4i+1}v_{1}^{3}a[0,0].$

  12. (12)

$d_{9}(\Delta^{4i+3}v_{1}a[12,2])=\bar{\kappa}^{2}\Delta^{4i+1}v_{1}^{2}a[17,3]$

###### Proof.

Let $i=0$. The differentials (1) and (3) are the image of a differential in
$E_{2}(V(0))$ under $i_{*}$. The second differential (2) follows
$v_{1}$-linearity and from the fact that
$d_{9}(\Delta^{4i+2}x)=\bar{\kappa}^{2}\Delta^{4i}\kappa$ in $E_{2}(V(0))$,
$i_{*}(x)=v_{1}a[5,1]$ and $i_{*}(\kappa)=v_{1}a[12,2]$.

For (4), we use 6.13. In $E_{*}(V(0))$, we have
$d_{11}(\Delta^{2}\eta\kappa)=\eta^{2}\bar{\kappa}^{3}$. Since
$p_{*}(\Delta^{2}a[17,3])=\Delta^{2}\eta\kappa$, $\Delta^{2}a[17,3]$ supports
a differential of length at most $11$. This $d_{9}$ is the only choice. The
argument for (5) is the same, with one more power of $\Delta$.

For (6), note that $p_{*}(\Delta^{3}a[12,2])=\Delta^{3}v_{1}\eta x$. Since
$d_{9}(\eta v_{1}x)=\nu\kappa\bar{\kappa}^{2}\Delta$, the class
$\Delta^{3}a[12,2]$ supports a differential of length at most $9$. This is the
only choice.

The arguments (1)–(6) when $i=1$ are the same as those for $i=0$.

For (7)–(8), note that from our computation above,
$tmf_{59}Y\cong\mathbb{Z}/2$. This forces (7) when $i=0$. Arguing in a similar
way, $tmf_{79}Y=0$, $tmf_{155}Y\cong\mathbb{Z}/2$ and $tmf_{175}Y=0$ imply the
other $d_{9}$s.

The $d_{9}$-differentials $(9)$-$(12)$ follow from those of $(1)$, $(2)$,
$(5)$, $(6)$, respectively, by $v_{1}$-linearity. ∎

######  Remark 6.15.

It turns out these are all the $d_{9}$-differentials. For degree reasons,
there can be very few other $d_{9}$s. The class $\Delta^{5}v_{1}a[0,0]$ is the
image of a $d_{9}$-cycle in $E_{9}(V(0))$ so does not support a $d_{9}$.

The only other possible $d_{9}$ differentials for degree reasons are

  * •

A non-trivial $d_{9}$ on $\Delta^{5}a[17,3]$. This does not happen since it
implies a non-trivial $d_{9}$ on $v_{1}\Delta^{5}a[17,3]=\Delta^{4}a[43,3]$,
but this family has already been paired: it is truncated by
$\Delta^{6}a[36,2]$.

  * •

A nontrivial $d_{9}$ on $\Delta^{4}a[17,3]$, truncating the
$\bar{\kappa}$-family of $\Delta^{2}a[4,0]$. We will see below that this does
not happen, but at this point, we leave this undecided.

Figure 15. $d_{5}$ and $d_{9}$ differentials in stems $46$ to $86$

###  6.3. Higher differentials

We begin our analysis using 6.3. The reader should remember that we only need
to analyze the generators of the $\bar{\kappa}$-free families, which are in
filtration less than four. All differentials discussed in this section are
depicted in Figures 19 and 20.

######  Lemma 6.16.

There are differentials

  1. (1)

$d_{19}(\Delta^{4}a[5,1])=\bar{\kappa}^{5}a[0,0]$

  2. (2)

$d_{19}(\Delta^{5}v_{1}a[5,1])=\bar{\kappa}^{5}\Delta v_{1}a[0,0]$

  3. (3)

$d_{19}(\Delta^{4}a[36,2]=\bar{\kappa}^{5}a[31,1]$

  4. (4)

$d_{19}(\Delta^{4}a[41,3])=\bar{\kappa}^{5}a[36,2]$

  5. (5)

$d_{19}(\Delta^{4}a[26,0])=\bar{\kappa}^{4}a[41,3]$

###### Proof.

For (1), since the element $\bar{\kappa}^{4}\in\pi_{80}(tmf\wedge V(0))$ is
not divisible by $\eta$ and $\bar{\kappa}^{5}\in\pi_{100}(tmf\wedge V(0))$ is
divisible by $\eta$, the $\bar{\kappa}$-family of $a[0,0]$ in the elliptic
spectral sequence for $tmf\wedge Y$ must be truncated at
$\bar{\kappa}^{5}a[0,0]$. Remembering that the source has to have filtration
less than four, the only possibility is this differential.

Inspection then show that the differentials (2)-(4) are the only possibilities
to satisfy 6.3. ∎

######  Lemma 6.17.

There are differentials

  1. (1)

$d_{17}(\Delta^{4}a[0,0])=\bar{\kappa}^{4}a[15,1]$

  2. (2)

$d_{17}(\Delta^{4}a[15,1])=\bar{\kappa}^{4}a[30,2]$

###### Proof.

For (1), note that in $\pi_{*}(tmf\wedge V(0))$, $\bar{\kappa}^{3}y$ is not
divisible by $\eta$ and $\bar{\kappa}^{4}y=0$. The class $y$ maps to $a[15,1]$
under $i_{*}$ so it follows that the $\bar{\kappa}$-family of $a[15,1]$ is
truncated at $\bar{\kappa}^{4}a[15,1]$. The only possibility is this
differential.

For (2), using the long exact sequence, we obtain that $\pi_{111}(tmf\wedge
Y)=\mathbb{Z}/2$. By part 6.16 (3), the class $\bar{\kappa}^{4}a[31,1]\in
E_{5}^{17,128}$ survives the spectral sequence and so detects the unique non-
trivial class of $\pi_{111}(tmf\wedge Y)$. This implies that the class
$\Delta^{4}a[15,1]\in E_{5}^{1,112}$ must support a differential. Taking into
account the $d_{9}$ differentials proves (2). ∎

######  Lemma 6.18.

There is a differential $d_{23}(\Delta^{4}a[30,2])=\bar{\kappa}^{6}a[5,1]$.

###### Proof.

By inspection, taking into account the $d_{9}$s, the only generators that can
be paired with $a[5,1]$ are $\Delta^{4}a[30,2]$ and $\Delta^{4}a[30,0]$.
However, it cannot be $\Delta^{4}a[30,0]$ because such a differential would
have length $25$, contradicting 6.5. ∎

######  Lemma 6.19.

For $i=0,1$, there are differentials:

  1. (1)

$d_{11}(\Delta^{4i+2}a[15,1])=\bar{\kappa}^{3}\Delta^{4i}a[2,0]$

  2. (2)

$d_{11}(\Delta^{4i+2}a[28,0])=\bar{\kappa}^{2}\Delta^{4i}a[35,3]$

###### Proof.

In (1), for both $i=0,1$, these are the image of differentials in the spectral
sequence $E_{*}(V(0))$. Both source and targets survive to $E_{11}(Y)$ and so
these two differentials occur.

For (2), the long exact sequence shows that $\pi_{75}(tmf\wedge
Y)=\mathbb{Z}/2$. 6.17 (1) implies that the class $\bar{\kappa}^{3}a[15,1]\in
E_{7}^{13,88}$ survives the spectral sequence and detects the unique non-
trivial element of the $\pi_{75}(tmf\wedge Y)$. On the other hand, the class
$\bar{\kappa}^{2}\Delta\nu^{2}a[5,1]\in E_{7}^{11,86}$ is a permanent cycle.
Thus, it must be hit by a differential and this is the possibility.

For $i=1$, by taking into account the $d_{9}$-differentials and the
$d_{17}$-differential 6.17 (2), we see that $\Delta^{4}a[35,3]$ is a permanent
cycle, which is $\bar{\kappa}$-free at the $E_{11}$-term. By inspection, the
only class which can truncate its $\kappa$-family is $\Delta^{6}a[28,0]$ by
the indicated $d_{11}$-differential. ∎

######  Lemma 6.20.

There are differentials:

  1. (1)

$d_{13}(\Delta^{2}a[30,2])=\bar{\kappa}^{3}a[17,3]$

  2. (2)

$d_{13}(\Delta^{2}a[33,1])=\bar{\kappa}^{3}a[20,2]$

###### Proof.

For (1), it follows from (6.7) that $\pi_{78}(tmf\wedge Y)\cong\mathbb{Z}/2$.
By sparseness, either $\Delta^{2}a[30,2]$ or $\Delta^{2}a[30,0]$ is a
permanent cycle detecting the non-zero element of $\pi_{78}(tmf\wedge Y)$.
Suppose that

$\Delta^{2}a[30,2]=\Delta^{3}\nu^{2}a[0,0]$

is a permanent cycle detecting a class $\alpha\in\pi_{78}(tmf\wedge Y)$. At
$E_{2}$, $\Delta^{3}\nu^{2}a[0,0]$ is in the image of $i_{*}\colon
E_{2}(V(0))\to E_{2}(Y)$ and so $p_{*}(\Delta^{3}\nu^{2}a[0,0])=0$. However,
since $\pi_{78}(tmf\wedge V(0))=0$, $p_{*}\alpha\neq 0$ in $\pi_{76}(tmf\wedge
V(0))$ and so is detected by a non-zero class in filtration $s>2$, but such a
class does not exist. We conclude that $\Delta^{2}a[30,0]$ is a permanent
cycle and that $\Delta^{2}a[30,2]$ supports the stated differential. For (2),
by inspection, only $\Delta^{2}a[33,1]$ and $\Delta^{4}a[5,1]$ can support
differentials truncating the $\bar{\kappa}$-family of $a[20,2]$. But
$\Delta^{4}a[5,1]$ is already paired with $a[0,0]$. ∎

Figure 16. $d_{5}$ and $d_{9}$ differentials in stems $86$ to $160$ Figure 17.
$d_{5}$ and $d_{9}$ differentials in stems $160$ to $194$

######  Proposition 6.21.

The following classes are $\bar{\kappa}$-free permanent cycles:

$\displaystyle(A):$
$\displaystyle\begin{array}[]{ccccc}\Delta^{2}a[4,0]&\Delta^{2}a[9,1]&\Delta^{2}a[14,2]&\Delta^{2}a[19,3]&\Delta^{2}a[20,2]\\\
\Delta^{2}a[30,0]&\Delta^{2}a[35,3]&\Delta^{2}a[45,3]&\Delta^{4}a[17,3]&\Delta^{4}a[20,2]\end{array}$

and the following classes are not permanent cycles:

$\displaystyle(B):$
$\displaystyle\begin{array}[]{ccccc}\Delta^{6}a[4,0]&\Delta^{6}a[9,1]&\Delta^{6}a[14,2]&\Delta^{6}a[19,3]&\Delta^{6}a[20,2]\\\
\Delta^{6}a[30,0]&\Delta^{6}a[30,2]&\Delta^{6}a[33,1]&\Delta^{6}a[35,3]&\Delta^{6}a[45,3]\end{array}$

Consequently, in the elliptic spectral sequence for $tmf\wedge Y$, each
generator in (B) truncates some $\bar{\kappa}$-multiple of one and only one
generator in (A).

###### Proof.

These are the remaining generators of $\bar{\kappa}$-free families. No class
in (B) can be a permanent cycle because the $\bar{\kappa}$-family of a class
of (B) cannot be truncated. This means that all the $10$ classes of (B) are
non-permanent cycles, and so all the $10$ classes of (A) are permanent cycles.
∎

######  Lemma 6.22.

We have the following differentials:

  1. (1)

$d_{19}(\Delta^{6}a[4,0])=\bar{\kappa}^{4}\Delta^{2}a[19,3]$

  2. (2)

$d_{19}(\Delta^{6}a[9,1])=\bar{\kappa}^{5}\Delta^{2}a[4,0]$

  3. (3)

$d_{19}(\Delta^{6}a[14,2])=\bar{\kappa}^{5}\Delta^{2}a[9,1]$

  4. (4)

$d_{19}(\Delta^{6}a[19,3])=\bar{\kappa}^{5}\Delta^{2}a[14,2]$

  5. (5)

$d_{17}(\Delta^{6}a[20,2])=\bar{\kappa}^{4}\Delta^{2}a[35,3]$

  6. (6)

$d_{13}(\Delta^{6}a[33,1])=\bar{\kappa}^{3}\Delta^{4}a[20,2]$

  7. (7)

$d_{17}(\Delta^{6}a[35,3])=\bar{\kappa}^{5}\Delta^{2}a[30,0]$

  8. (8)

$d_{23}(\Delta^{6}a[45,3])=\bar{\kappa}^{6}\Delta^{2}a[20,2]$

###### Proof.

Taking into account the differentials shown above, these are only possible
pairings remaining between the classes in (B) which are the sources in (1)–(8)
and classes of (A).∎

######  Remark 6.23.

There are only two generators in (B) left living in the same topological
degree, namely $\Delta^{6}a[30,0]$ and $\Delta^{6}a[30,2]$. Each of these
supports a differential truncating the $\bar{\kappa}$-families of either
$\Delta^{4}a[17,3]$ or $\Delta^{2}a[45,3]$ and one differential determines the
other.

Determining the last differential pattern turns out to be unfortunately tricky
(as far as we know). A crucial step towards settling the last differentials is
to establish the following extension in the $E_{\infty}$-term of the elliptic
spectral sequence for $tmf\wedge Y$.

######  Proposition 6.24.

There is an exotic extension

$\nu^{2}(\nu\Delta^{6}a[0,0])=\bar{\kappa}^{2}\Delta^{4}a[17,3].$

To prove this extension, we need some intermediate results.

######  Lemma 6.25.

In
$\mathop{\mathrm{Ext}}\nolimits_{\Lambda^{\prime}}^{*,*}(A^{\prime},A^{\prime}/(2,a_{1})\otimes\mathcal{F}_{*}(Y))$,
there is a Massey product

$\langle\eta,\nu,\Delta^{4}a[12,2]\rangle=\Delta^{4}a[17,3].$

###### Proof.

Since $\Delta^{4}a[12,2]=\eta\Delta^{4}a[11,1]$ (see Figure 12), we have that

$\langle\eta,\nu,\Delta^{4}a[12,2]\rangle=\langle\eta,\nu,\eta\Delta^{4}a[11,1]\rangle\supseteq\langle\eta,\nu,\eta\rangle\Delta^{4}a[11,1]=\nu^{2}a[11,1]=a[17,1].$

The indeterminacy is zero since

$\eta\mathop{\mathrm{Ext}}\nolimits_{\Lambda^{\prime}}^{2,114}(A^{\prime},A^{\prime}/(2,a_{1})\otimes\mathcal{F}_{*}(Y))+\mathop{\mathrm{Ext}}\nolimits_{\Lambda^{\prime}}^{1,6}(A^{\prime},A^{\prime}/(2))\Delta^{4}a[12,2]=0.\qed$

######  Proposition 6.26.

In
$\mathop{\mathrm{Ext}}\nolimits_{\Lambda^{\prime}}^{*,*}(A^{\prime},\mathcal{F}_{*}(Y))$,
there is a Massey product

$\langle\eta,\nu,\Delta^{4}a[12,2]\rangle=\Delta^{4}a[17,3].$

###### Proof.

Let
$f_{*}\colon\mathop{\mathrm{Ext}}\nolimits_{\Lambda^{\prime}}^{*,*}(A^{\prime},\mathcal{F}_{*}(Y))\rightarrow\mathop{\mathrm{Ext}}\nolimits_{\Lambda^{\prime}}^{*,*}(A^{\prime},A^{\prime}/(2,a_{1})\otimes\mathcal{F}_{*}(Y))$
be the map induced by the $\Lambda$-comodule homomorphism
$\mathcal{F}_{*}(Y)\rightarrow A^{\prime}/(2,a_{1})\otimes\mathcal{F}_{*}(Y)$.
By naturality of Massey products, we have that

$f_{*}(\langle\eta,\nu,\Delta^{4}a[12,2]\rangle)\subseteq\langle\eta,\nu,f_{*}(\Delta^{4}a[12,2])\rangle.$

Further, $f_{*}(\Delta^{4}a[12,2])=\Delta^{4}a[12,2]$. By Lemma 6.25, the
above equation gives

$f_{*}(\langle\eta,\nu,\Delta^{4}a[12,2]\rangle)=\Delta^{4}a[17,3].$

The pre-image of $\Delta^{4}a[17,3]$ is the same-named class. The
indeterminacy is zero. ∎

######  Lemma 6.27.

There is an element of $\pi_{108}(tmf\wedge Y)$ detected by
$\Delta^{4}a[12,2]$ and annihilated by $\bar{\kappa}^{2}$.

###### Proof.

We have already determined $E_{\infty}(Y)$ in stems $t-s=108,148$. We see that
there is a short exact sequence

$0\rightarrow\mathbb{Z}/2\\{\bar{\kappa}^{2}\Delta^{2}a[20,2]\\}\rightarrow
G\rightarrow\mathbb{Z}/2\\{\Delta^{4}a[12,2]\\}\rightarrow 0$

where $G\subseteq\pi_{108}(tmf\wedge Y)$ is the subgroup of elements detected
in positive filtration. At the $E_{\infty}$-term in stem $t-s=148$, the only
non-zero class in positive filtration is $\bar{\kappa}^{4}\Delta^{2}a[20,2]$.
In particular, $\bar{\kappa}^{2}\Delta^{4}a[12,2]=0$. So, one of the classes
detected by $\Delta^{4}a[12,2]$ satisfies the claim. ∎

We will denote also by $\Delta^{4}a[12,2]$ the element in $\pi_{108}(tmf\wedge
Y)$, which is detected by $\Delta^{4}a[12,2]$ and is annihilated by
$\bar{\kappa}^{2}$.

######  Proposition 6.28.

There are the following relations in $\pi_{*}(tmf\wedge Y)$:

  1. (1)

$\nu^{2}[\nu\Delta^{6}a[0,0]]\neq 0$

  2. (2)

$\eta[\nu\Delta^{6}a[0,0]]=0$

###### Proof.

The class detected by $\nu\Delta^{6}a[0,0]$ lifts to $\pi_{*}(tmf\wedge V(0))$
and there is a lift detected by $\nu\Delta^{6}$. But in $\pi_{*}(tmf\wedge
V(0))$, $\nu^{2}[\nu\Delta^{6}]$ is not divisible by $\eta$. ∎

Now, we use the truncated spectral sequences of Section 2.2, applied to the
elliptic spectral sequence of $tmf\wedge Y$. As in Section 2.2, let

$\mathrm{sk}_{16}(tmf\wedge Y)=X_{0}/X_{17}$

for $X_{n}$ the $n$th term of the $X(4)$-Adams tower of $tmf\wedge Y$. Then
$E_{r,<17}^{*,*}(Y)$ as in Section 2.2 is a spectral sequence computing
$\pi_{*}\mathrm{sk}_{16}(tmf\wedge Y)$, and it satisfies
$E_{r,<17}^{s,*}(Y)=0$ for $s\geq 17$. Furthermore, we have a map of spectral
sequences

$T_{r}^{s,t}\colon E_{r}^{s,t}(Y)\to E_{r,<17}^{s,t}(Y).$

######  Proposition 6.29.

In $\pi_{*}(\mathrm{sk}_{16}(tmf\wedge Y))$, we have

$\langle\eta,\nu,\Delta^{4}a[12,2]\rangle=\Delta^{4}a[17,3].$

###### Proof.

In $\pi_{*}(tmf\wedge Y)$, the product $\nu\Delta^{4}a[12,2]$, if not trivial,
is detected in filtration $17$. It follows that $\nu\Delta^{4}a[12,2]$ is
equal to zero in $\pi_{*}(\mathrm{sk}_{16}(tmf\wedge Y))$. Thus, the Toda
bracket $\langle\eta,\nu,\Delta^{4}a[12,2]\rangle$ can be formed. Proposition
6.26 means that in $E_{2,<17}^{s,t}(Y)$, there is Massey product

$\langle\eta,\nu,\Delta^{4}a[12,2]\rangle=\Delta^{4}a[17,3].$

The conditions of the Moss Convergence Theorem [Mos70] are satisfied, so the
Toda bracket $\langle\eta,\nu,\Delta^{4}a[12,2]\rangle$ contains
$\Delta^{4}a[17,3]$ and the indeterminacy is zero. ∎

######  Proposition 6.30.

In the elliptic spectral sequence for $tmf\wedge Y$, there is an exotic
extension

$\eta a[152,2]=\bar{\kappa}^{2}\Delta^{4}a[17,3].$

###### Proof.

Since $\bar{\kappa}^{2}\Delta^{4}a[17,3]$ lives in filtration $s=11$, it
suffices to prove that extension in the $E_{\infty}$-term of the spectral
sequence for $\mathrm{sk}_{16}(tmf\wedge Y)$. The above proposition and the
choice of $\Delta^{4}a[12,2]$ imply that

$\bar{\kappa}^{2}\Delta^{4}a[17,3]=\langle\eta,\nu,\Delta^{4}a[12,2]\rangle\bar{\kappa}^{2}=\eta\langle\nu,\Delta^{4}a[12,2],\bar{\kappa}^{2}\rangle.$

Since $\bar{\kappa}^{2}\Delta^{4}a[17,3]\neq 0$ at $E_{\infty}$,
$\langle\nu,\Delta^{4}a[12,2],\bar{\kappa}^{2}\rangle$ must be non-trivial,
and it must be detected by a class which is not in the kernel of $\eta$. This
forces $\langle\nu,\Delta^{4}a[12,2],\bar{\kappa}^{2}\rangle$ to be detected
by $a[152,2]$, and so $\eta a[152,2]$ is detected by
$\bar{\kappa}^{2}\Delta^{4}a[17,3]$. ∎

###### Proof of Proposition 6.24.

Let $\beta=[\nu\Delta^{6}a[0,0]]$. By Proposition 6.28, $\eta\beta=0$ and we
can form the Toda bracket $\langle\nu,\eta,\beta\rangle$. Then

$\eta\langle\nu,\eta,\beta\rangle=\langle\eta,\nu,\eta\rangle\beta=\nu^{2}\beta$

On the other hand, $\nu^{2}\beta\neq 0$ by Proposition 6.28. It follows that
$\langle\nu,\eta,\beta\rangle\neq 0$. We see that it must be detected by
$a[152,2]$. So $\eta a[152,2]=\nu^{2}\beta$ and Proposition 6.30 implies that
$\nu^{2}\beta$ is detected by $\bar{\kappa}^{2}\Delta^{4}a[17,3]$. ∎

######  Lemma 6.31.

There are differentials:

  1. (1)

$d_{13}(\Delta^{6}a[30,2])=\bar{\kappa}^{3}\Delta^{4}a[17,3]$

  2. (2)

$d_{19}(\Delta^{6}a[30,0])=\bar{\kappa}^{4}\Delta^{2}a[45,3]$

###### Proof.

Let

$tmf\wedge Y\leftarrow(tmf\wedge Y)_{1}\leftarrow(tmf\wedge
Y)_{2}\leftarrow\ldots$

be the Adams tower associated to the $X(4)$-based resolution of $tmf\wedge Y$.
We consider its $1$-co-truncated tower and the induced map of spectral
sequences

$cT_{r}^{s,t}\colon E_{r,\geq 1}^{s,t}\to E_{r}^{s,t}.$

By Lemma 2.2, $cT_{r}^{s,t}$ is surjective for $s\geq 1$.

Let $a=\nu^{2}\Delta^{6}a[0,0]\in E_{2}^{2,150+2}$. This is a permanent cycle
representing a unique non-zero element of $\pi_{150}(tmf\wedge Y)$, which in
this proof we denote by $\alpha$. Since $a$ has positive filtration, there is
a class $\bar{a}\in E_{2,\geq 1}^{2,150+2}$ such that $cT_{2}(\bar{a})=a$ and
the surjectivity of $cT_{\infty}$ guarantees that we can choose $\bar{a}$ to
be a permanent cycle. It then detects classes
$\bar{\alpha}\in\pi_{150}((tmf\wedge Y)_{1})$ that map to $\alpha$.

Since $\nu\alpha$ is detected by $b=\bar{\kappa}^{2}\Delta^{4}a[17,3]\in
E_{\infty}^{11,153+11}$ (Proposition 6.24), $\nu\bar{\alpha}$ must be detected
in $E_{\infty}^{s,153+s}(cT_{1})$ for $3\leq s\leq 11$. Since
$E_{2}^{s,153+s}(cT_{1})=0$ for $3\leq s\leq 10$ (this is true for
$E_{2}^{*,*}$), $\nu\bar{\alpha}$ must be detected by a lift $\bar{b}$ of $b$.

The relation $\bar{\kappa}\nu=0\in\pi_{*}tmf$ implies that
$\bar{\kappa}\nu\bar{\alpha}=0\in\pi_{*}((tmf\wedge Y)_{1})$. This implies
that $d_{r}(\bar{c})=\bar{\kappa}\bar{b}$ for some non-trivial element
$\bar{c}\in E_{r,\geq 1}^{15-r,174+(15-r)}$. As $E_{2,\geq 1}^{0,*}=0$,
$\bar{c}$ must live in filtration $1\leq s\leq 13$, and hence so does
$cT_{r}(\bar{c})$. In particular, $cT_{r}(\bar{c})\neq\Delta^{6}a[30,0]$.
However, we find that

$\displaystyle d_{r}(cT_{r}(\bar{c}))$
$\displaystyle=cT_{r}(\bar{\kappa}\bar{b})=\bar{\kappa}\cdot
cT_{r}(\bar{b})=\bar{\kappa}^{3}\Delta^{4}a[17,3].$

The only way for this to make sense is if $cT_{r}(\bar{c})$ is equal to
$\Delta^{6}a[30,2]$ and this is the desired differential (1).

This differential then determines (2) as noted in 6.23. ∎

######  Remark 6.32.

From, this discussion, we also learn that there is a non-trivial class in
$i_{*}\pi_{150}V(0)$ which is detected by $a[153,11]$.

###  6.4. Exotic extensions

In this section we resolve the exotic $2$, $\eta$, $\nu$ and $v_{1}$
extensions in the elliptic spectral sequence of $tmf\wedge Y$. The extensions
are depicted in Figures 21 and 22.

We begin with the exotic $\eta$-extensions, which are few. To determine them,
we use the following strategies. First, the long exact sequence

$\displaystyle\ldots\to
tmf_{*+1}V(0)\xrightarrow{\eta}tmf_{*}V(0)\xrightarrow{i_{*}}tmf_{*}Y\xrightarrow{p_{*}}tmf_{*-1}V(0)\to\ldots$

We use the following basic, but useful facts.

######  Lemma 6.33.

For $a\in tmf_{*}Y$ and $b\in tmf_{*}V(0)$,

  1. (1)

if $a=i_{*}b$, then $\eta a=i_{*}\eta b=0$,

  2. (2)

$p_{*}\eta a=\eta p_{*}a=0$, and

  3. (3)

$v_{1}\eta a=\eta v_{1}a$.

###### Proof.

These are easy consequences of the long exact sequence on homotopy groups
combined with the fact that composition as well as the smash product induces
the $\pi_{*}S^{0}$-module structure in the stable homotopy category. ∎

Note further that 2.10 as described in 2.14 gives a way to relate extensions
in different stems between the $v_{1}$-power torsion classes. We also use
Lemma 2.19 and 2.21

A stem-by-stem analysis using the above techniques then allows us to determine
that the only non-trivial exotic $\eta$-extensions are as follows:

######  Lemma 6.34.

In the elliptic spectral sequence of $Y$, there are exotic extensions

  1. (1)

$\eta[\Delta^{2}\nu a[5,1]]=\bar{\kappa}^{2}a[17,3]$

  2. (2)

$\eta[\Delta^{4}\nu a[5,1]]=\bar{\kappa}^{5}a[5,1]$

  3. (3)

$\eta[\Delta^{6}\nu a[5,1]]=\bar{\kappa}^{2}[\Delta^{4}a[17,3]]$

  4. (4)

$\eta[\Delta^{6}\nu a[20,2]]=\bar{\kappa}^{5}[\Delta^{2}a[20,2]]$

There are no other exotic $\eta$-extensions.

###### Proof.

The first extension (1) follows from 2.21. The extension (2) and (4) follow
from duality: (2) from $\eta[\Delta^{2}a[20,2]]=[\Delta^{2}v_{1}^{2}a[17,3]]$
and (4) from $\eta a[5,1]=\nu^{2}a[0,0]$. Finally, (3) is Proposition 6.30.

All possible exotic $\eta$-extensions are shown not to occur using 6.33,
duality and 2.21. In particular, the possible $\eta$-extensions with source in
stems $52\leq t-s\leq 57$ are shown not to occur using 2.21 and
$v_{1}$-linearity. ∎

Now, we turn to the exotic $2$-extensions.

######  Theorem 6.35.

There are no exotic $2$-extensions in the elliptic spectral sequence for $Y$
and, consequently,

$2(\pi_{*}tmf\wedge Y)=0.$

###### Proof.

Since we have a cofiber sequence

$tmf\wedge C_{\eta}\xrightarrow{2}tmf\wedge C_{\eta}\xrightarrow{j}tmf\wedge
Y\xrightarrow{q}\Sigma tmf\wedge C_{\eta},$

we can apply Lemma 2.19 with $X=tmf\wedge C_{\eta}$, $i=j$ and $p=q$. From
this, we deduce that if $a^{\prime}\in\pi_{*}tmf\wedge Y$ is in the image of
$j_{*}$, then it has order $2$ and that if $q_{*}(a^{\prime})=a$, then
$2a^{\prime}=j_{*}(\eta a)$. It follows that if $2a^{\prime}\neq 0$, then
$2a^{\prime}$ is divisible by $\eta$.

This leaves one possible extension in stem 57. But such a $2$-extension would
lead, by duality, to a $2$-extension in stem 116. However, there are no
$\eta$-divisible classes in that stem. Since the $E_{2}$-term was $2$-torsion
and there are no exotic $2$-extensions, $\pi_{*}tmf\wedge Y$ is annihilated by
$2$. ∎

Next, we turn to the $\nu$ extensions.

######  Remark 6.36.

We will use without mention that $\bar{\kappa}\nu=0$ in $tmf_{*}$-modules.
This allows us to eliminate many possible exotic $\nu$-extensions.

######  Lemma 6.37.

In the elliptic spectral sequence of $Y$, there are exotic extensions

  1. (1)

$\nu a[26,0]=a[29,5]$

  2. (2)

$\nu a[41,3]=a[44,8]$

  3. (3)

$\nu a[52,0]=a[55,7]$

  4. (4)

$\nu a[54,2]=\bar{\kappa}^{2}a[17,3]$

  5. (5)

$\nu a[67,3]=\bar{\kappa}^{2}a[30,0]$

  6. (6)

$\nu a[98,0]=a[101,15]$

  7. (7)

$\nu a[102,2]=\bar{\kappa}^{5}a[5,1]$

  8. (8)

$\nu a[103,1]=a[106,16]$

  9. (9)

$\nu a[124,0]=a[127,15]$

  10. (10)

$\nu a[129,1]=a[132,16]$

  11. (11)

$\nu a[150,2]=a[153,11]$

  12. (12)

$\nu a[155,3]=a[158,16]$

  13. (13)

$\nu a[165,3]=a[168,22]$

###### Proof.

The extensions (1) and (6) follow from the extensions $\nu a[26,0]=a[29,5]$
and $a[98,0]=a[101,7]$, respectively, in $\pi_{*}tmf\wedge V(0)$ by applying
$i_{*}$. The extensions (2), (3), (5) and (9) follow from examining the effect
of $p_{*}$ and the extensions $\nu a[39,3]=a[42,10]$, $\nu a[50,2]=a[53,7]$,
$\nu a[65,3]=a[68,10]$ and $\nu a[122,2]=a[125,21]$ in $\pi_{*}tmf\wedge
V(0)$, respectively.

Extensions (4), (7), (12) and (13) are obtained by duality from algebraic
extensions. The extensions (10) and (8) follow by duality from (2) and (5).

The extension (11) is proved in Proposition 6.24. ∎

######  Lemma 6.38.

In the elliptic spectral sequence of $Y$, there are exotic extensions

  1. (1)

$\nu a[57,1]=\bar{\kappa}^{2}a[20,2]$

  2. (2)

$\nu a[62,2]=\bar{\kappa}a[45,3]$

Dually, we have

  3. (3)

$\nu a[108,2]=a[111,17]$

  4. (4)

$\nu a[113,3]=a[116,18]$

Together with 6.37, there are no other non-trivial exotic $\nu$-extensions.

To prove 6.38, we use the $tmf$-based Atiyah–Hirzebruch spectral sequence for