# New infinite families in the stable homotopy groups of spheres

Prasit Bhattacharya  New Mexico State University<EMAIL_ADDRESS>,  Irina
Bobkova  Texas A&M University<EMAIL_ADDRESS>and  J.D. Quigley  University
of Virginia<EMAIL_ADDRESS>
###### Abstract.

We identify seven new $192$-periodic infinite families of elements in the
$2$-primary stable homotopy groups of spheres. Although their Hurewicz image
is trivial for topological modular forms, they remain nontrivial after
$\mathrm{T}(2)$\- as well as $\mathrm{K}(2)$-localization. We also obtain new
information about $2$-torsion and $2$-divisibility of some of the previously
known $192$-periodic infinite families in the stable stems.

##  1\. Introduction

Calculating the stable homotopy groups of spheres, or stable stems, has been
one of the central problems in algebraic topology for several decades, with a
plethora of applications in algebra and geometry. Since the sixties, there are
two main approaches to this problem: low-dimensional computations, which
attempt to give a complete description of the stable stems up to a finite
range using Adams or Adams–Novikov spectral sequences as the primary tools
[May64, MT67, Rav86, KM95, Isa19, IWX23a], and chromatic computations, which
attempt to pick out large-scale periodic patterns instead [Ada66, Smi77,
MRW77, Rav86].

The first large-scale phenomenon observed in stable stems is a result of Serre
[Ser53] which states that all positive dimensional stable stems are finite
abelian groups. This motivated the study of the stable stems one prime at a
time. In the sixties, Toda [Tod62] and Adams [Ada66] identified
$(2p-2)$-periodic families of $p$-torsion elements for primes $p>2$ and
$8$-periodic families when $p=2$ within the stable stems. A decade later,
Smith [Smi77] constructed $(2p^{2}-2)$-periodic families at $p>3$, and
Miller–Ravenel–Wilson [MRW77] constructed $(2p^{3}-2)$-periodic families at
$p>5$. These were the first few examples of what is called _periodic
phenomena_ in chromatic homotopy theory.

Chromatic homotopy theory implies that the $p$-local stable stems admit a
decreasing filtration indexed by natural numbers, where the $n$-th stratum is
called the _chromatic layer $n$_. The famous nilpotence and periodicity
results of [DHS88, HS98] combined with the chromatic convergence theorem
[Rav92] imply that each chromatic layer consists entirely of periodic infinite
families, each of which can be detected by some periodic self-map of some CW-
complex of ‘type $n$’.

Even before the advent of chromatic homotopy theory, Adams completely
identified chromatic layer $1$ [Ada66] (also see [DM89]) and famously
discovered a connection between the Bernoulli numbers and the stable stems.
Chromatic layer $2$ still remains an area of active research [SY95, GHMR05,
GHM04, Beh06, Beh12b], where the case $p=2$ is at the edge of our current
knowledge [Bea15, BO16, BG18, BHHM20, BMQ23, BBG+23, BGH22].

The $2$-local connective spectrum of topological modular forms (or ${\sf
tmf}$), constructed using derived algebraic geometry [Lur09, Beh20], is a
formidable tool for exploring chromatic height $2$ at the prime $2$. This is
because ${\sf tmf}$-homology groups are readily computable, while at the same
time the coefficient ring carries intricate patterns [Bau08, DFHH14, BR21]
reflecting patterns from the second chromatic layer. In the last decade, new
techniques involving ${\sf tmf}$ [BOSS19, BBT21, BBC23, Pha22] have led to
important and interesting results [BHHM20, Bob20, BE20, Pha23, BBB+21] closely
related to the study of chromatic layer $2$.

These developments also led to some of the first new $2$-local periodic
infinite families [BHHM20, BMQ23] since Adams’ work [Ada66]111 There are well-
known $2$-local infinite families in stable stems, namely the Mahowaldean
$\eta_{j}$ family [Mah77] and its variant [Bru81] (also see [Coh81, HK99]),
discovered after [Ada66]. However, these families are not periodic in the
sense above. in the sixties. The paper [BMQ23] completely calculates the image
of the Hurewicz homomorphism

${{\sf h}_{{\sf tmf}}:\pi_{*}\SS}$${{\sf tmf}_{*}}$

from the stable stems to the coefficients of ${\sf tmf}$, thereby identifying
several $192$-periodic infinite families and patterns in the chromatic layer
$2$.

Our first main result adds seven new 192-periodic families to this list:

######  Theorem 1.

_For each $m\in\\{23,47,71,74,95,119,167\\}$ and $k\in\mathbb{N}$, there
exists an element of order $2$ in dimension $m+192k$ of the stable stems whose
image is trivial under the ${\sf tmf}$-Hurewicz homomorphism. _

######  Remark 1.1.

A comparison of our work with known calculations [Isa19, IWX23a] suggest that
the elements ${\sf h}_{1}^{3}{\sf g}$, ${\sf h}_{1}^{2}\cdot(\Delta{\sf
h}_{1}{\sf g})$, ${\sf h}_{1}^{2}\cdot(\Delta^{2}{\sf h}_{1}{\sf g})$, ${\sf
d}_{0}{\sf g}^{3}$, $(\Delta{\sf h}_{1})^{3}{\sf g}$, $\Delta^{4}{\sf
h}_{1}^{3}{\sf g}$, and $\Delta^{6}{\sf h}_{1}^{3}{\sf g}$ in [IWX23b] detect
the elements in dimension $23$, $47$, $71$, $74$, $95$, $119$, and $167$ of 1,
respectively222We thank Bob Bruner and Dan Isaksen for these observations..

######  Remark 1.2.

Let $\eta\in\pi_{1}\SS$ denote the first Hopf map and let ${\sf ko}$ denote
the connective real $\mathrm{K}$-theory. Then $\eta^{3}$ is part of an
$8$-periodic infinite family in chromatic layer $1$ which is not detected in
the Hurewicz image of ${\sf ko}$. From the perspective where ${\sf tmf}$ is
the chromatic height $2$ analog of ${\sf ko}$, the $192$-periodic families in
1 can be regarded as height $2$ analogs of the $\eta^{3}$ family.

One subtlety of chromatic homotopy theory is the fact that there are two
different variants of the chromatic filtration: one closer to geometry defined
using telescopic localizations, and one which is computationally more
accessible defined using Bousfield localizations of Morava K-theories (see
[Rav92, Chapter 7]). Whether these two filtrations are isomorphic is the
subject of the famous telescope conjecture [Rav84, Conjecture 10.5] disproved
very recently by Burklund–Hahn–Levy–Schlank [BHLS23].

The spectrum $\mathrm{TMF}\simeq(\Delta^{8})^{-1}{\sf tmf}$, obtained from
${\sf tmf}$ by inverting the periodicity generator $\Delta$ in degree $192$,
is a $\mathrm{K}(2)$-local spectrum. Here $\mathrm{K}(2)$ is the second Morava
K-theory. The failure of the telescope conjecture [BHLS23] implies that the
natural map from the height $2$ telescopic localization to the
$\mathrm{K}(2)$-localization of the sphere spectrum

${\iota:\SS_{\mathrm{T}(2)}}$${\SS_{\mathrm{K}(2)}}$

is not an equivalence. But the chromatic height $2$ elements in the Hurewicz
image of ${\sf tmf}$ do not see this difference as they lift to both
$\mathrm{T}(2)$-local and $\mathrm{K}(2)$-local stable stems. This is because
the unit map of $\mathrm{TMF}$

(1)
${\upiota_{\mathrm{TMF}}:\SS}$${\SS_{\mathrm{T}(2)}}$${\SS_{\mathrm{K}(2)}}$${\mathrm{TMF}}$$\scriptstyle{\iota}$

factors through $\iota$. This argument does not apply to elements listed in 1
because they are trivial in the Hurewicz image of ${\sf tmf}_{*}$. However, we
can still show that:

######  Theorem 2 (3.6 and 3.14).

_All elements listed in 1 have nonzero images in the $\mathrm{K}(2)$-local and
$\mathrm{T}(2)$-local stable stems at $p=2$._

Although our infinite families do not contradict the telescope conjecture,
they still have significant geometric implications. The groundbreaking work of
Kervaire and Milnor [KM63] directly relates the stable stems to the
classification of smooth structures on homotopy spheres. In odd dimensions,
the work of Kervaire and Milnor [KM63], Browder [Bro69], Hill, Hopkins, and
Ravenel [HHR16], and Wang and Xu [WX17] implies that exotic spheres exist in
every odd dimension except for $1$, $3$, $5$, and $61$. The results of Adams
[Ada66] and Toda [Tod62] imply that exotic spheres exist in at least one
quarter of the even dimensions, while the results in [BHHM20, BMQ23] imply
that exotic spheres exist in over half of the even dimensions. Wang and Xu
[WX17, Conjecture 1.17] have conjectured that exotic spheres exist in all
dimensions except for a small number of low-dimensional exceptions.

1 also has implications for exotic spheres. Following Schultz [Sch85, p. 246],
an exotic sphere is called _very exotic_ if it does not bound a parallelizable
manifold. Very exotic spheres are more mysterious than exotic spheres which
bound parallelizable manifolds; for instance, the latter are always known to
admit Riemannian metrics of positive Ricci curvature [Wra97], while only one
very exotic sphere is known to admit such a metric.

In even dimensions, every exotic sphere is a very exotic sphere, but most of
the known odd-dimensional exotic spheres are not “very exotic.” The results of
[BHHM20, BMQ23] imply that very exotic spheres exist in at least $37$ of the
$96$ odd congruence classes of dimensions modulo $192$. The $6$ odd dimensions
in 1 are not covered by those results, so we obtain:

######  Corollary.

Very exotic spheres exist in at least $43$ of the $96$ odd congruence classes
of dimensions modulo $192$.

###  1.1. Methodology

We consider a type $2$ spectrum $\mathrm{A}_{1}$ which is constructed using
three cofiber sequences

(2)
${\SS}$${\SS}$${\mathrm{M}}$${\Sigma\SS,}$$\scriptstyle{2}$$\scriptstyle{{\sf
p}_{1}}$ (3)
${\Sigma\mathrm{M}}$${\mathrm{M}}$${\mathrm{Y}}$${\Sigma^{2}\mathrm{M},}$$\scriptstyle{\eta}$$\scriptstyle{{\sf
p}_{2}}$ (4)
${\Sigma^{2}\mathrm{Y}}$${\mathrm{Y}}$${\mathrm{A}_{1}}$${\Sigma^{3}\mathrm{Y},}$$\scriptstyle{v}$$\scriptstyle{{\sf
p}_{3}}$

where $v$ is a choice of a $v_{1}^{1}$-self-map of $\mathrm{Y}$. Our starting
point is the recent work of Viet-Cuong Pham [Pha23], which shows that the
${\sf tmf}$-Hurewicz homomorphism

(5) ${{\sf h}_{{\sf tmf}}:\pi_{*}\mathrm{A}_{1}}$${{\sf
tmf}_{*}\mathrm{A}_{1}}$

is a surjection. We then study the long exact sequences associated to the
cofiber sequences (2), (3) and (4) using our knowledge of ${\sf tmf}_{*}$
[Bau08, DFHH14, BR21], as well as ${\sf tmf}_{*}\mathrm{M}$, ${\sf
tmf}_{*}\mathrm{Y}$, and ${\sf tmf}_{*}\mathrm{A}_{1}$ [BBPX22, Pha23].

By combining this study with our complete knowledge of the Hurewicz image in
${\sf tmf}_{*}$ [BMQ23], we identify seven new infinite families of elements
in $\pi_{*}\SS$ (listed in 1) which are in the image of the pinch map

(6) ${{\sf
p}:\mathrm{A}_{1}}$${\Sigma^{3}\mathrm{Y}}$${\Sigma^{5}\mathrm{M}}$${\Sigma^{6}\SS}$$\scriptstyle{{\sf
p}_{3}}$$\scriptstyle{{\sf p}_{2}}$$\scriptstyle{{\sf p}_{1}}$

in stable homotopy. We then use the $v_{2}^{32}$-self-map of $\mathrm{A}_{1}$
[BEM17] to show that these infinite families have nontrivial images in the
$\mathrm{T}(2)$-local stable stems. Finally, using results and techniques from
[Pha23, Lau04, BMQ23] we argue that these infinite families have nontrivial
images in the $\mathrm{K}(2)$-local stable stems as well.

The $192$-periodic elements in the stable stems in the Hurewicz image of ${\sf
tmf}$ were all shown to have order at most $8$ [BMQ23]. The ${\sf
tmf}$-homology calculations of Section 2 lead us to new information about the
$2$-torsion and $2$-divisibility of some of these infinite families. We deduce
from LABEL:Table:A1lifts that some of these elements are in the image of
$p_{1}$, and some map to nonzero elements under $i_{1}$, where $p_{1}$ and
$i_{1}$ are the maps in the long exact sequence

${\dots}$${\pi_{*}(\SS)}$${\pi_{*}(\SS)}$${\pi_{*}(\mathrm{M})}$${\pi_{*-1}(\SS)}$${\dots}$$\scriptstyle{\cdot
2}$$\scriptstyle{i_{1}}$$\scriptstyle{p_{1}}$$\scriptstyle{\cdot 2}$

associated to (2). Since nonzero elements in the image of $p_{1}$ are
$2$-torsion and elements with nontrivial image under $i_{1}$ are not
$2$-divisible, we get:

######  Theorem 3.

_An element in the stable stems is simple $2$-torsion if its ${\sf
tmf}$-Hurewicz image is_ $\Delta^{8k}x$, $k\geq 0$, where

$x\in\\{\kappa\nu,\ \overline{\kappa}^{2}\eta^{2},\
\eta\Delta\overline{\kappa}^{2},\ 4\Delta^{2}\overline{\kappa},\
\overline{\kappa}^{4},\ \eta^{2}\Delta^{2}\overline{\kappa}^{2},\
2\Delta^{4}\cdot 2\overline{\kappa},\ 4\Delta^{6}\overline{\kappa}\\},$

_and not $2$-divisible if it maps to $\Delta^{8k}x$, $k\geq 0$, where_

$x\in\\{\nu^{2},\ \nu^{3},\ \overline{\kappa}\nu,\ q\eta,\
\overline{\kappa}^{2}\eta^{2},\ \eta^{2}\Delta^{2}\nu,\ \nu\Delta^{2}\nu,\
\nu\Delta^{2}\nu^{2},\ \nu\Delta^{2}\nu^{3},\ 4\Delta^{2}\overline{\kappa},\
\overline{\kappa}^{4},\ \eta\Delta\overline{\kappa}^{3},\ $
$\quad\eta^{2}\Delta^{2}\overline{\kappa}^{2},\ \overline{\kappa}^{5},\
\nu^{3}\Delta^{4},\ \eta\Delta\overline{\kappa}^{4},\
2\Delta^{4}\overline{\kappa},\ \eta\Delta\overline{\kappa}^{5},\
\eta^{2}\Delta^{2}\kappa,\ \nu\Delta^{6}\nu^{2},\
\nu\Delta^{6}\eta\epsilon\\}.$

### Organization of the paper

In Section 2, we perform some technical ${\sf tmf}$-homology calculations
which are necessary in Section 3 to prove 1 and 2.

While reading this paper, the reader may find [DFHH14, Part I, Ch. 12]
convenient for looking up the homotopy groups of ${\sf tmf}$, where the
generators in the Hurewicz image are marked with colored dots. We refer to
[BBPX22] for explicit descriptions of ${\sf tmf}_{*}\mathrm{M}$ and ${\sf
tmf}_{*}\mathrm{Y}$.

### Acknowledgments

The authors are indebted to Mark Behrens for many stimulating conversations.
We also thank Guozhen Wang and Zhouli Xu for some helpful discussions, Bob
Bruner and Dan Isaksen for their predictions in Remark 1.1, and Bert Guillou
for pointing out a consequential typo.

The research for this paper has been supported by the National Science
Foundation through the grants DMS-2005627, DMS-2039316, DMS-2135884, and
DMS-2414922 (formerly DMS-2203785 and DMS-2314082).

##  2\. ${\sf tmf}$-homology calculations

Using our knowledge of ${\sf tmf}_{*}$ [Bau08, BR21], ${\sf
tmf}_{*}\mathrm{M}$, ${\sf tmf}_{*}\mathrm{Y}$, and ${\sf
tmf}_{*}\mathrm{A}_{1}$ [BBPX22], we will compute the maps $i_{k}$ and $p_{k}$
in the long exact sequences

(7) ${\cdots}$${{\sf tmf}_{k}\mathrm{Y}}$${{\sf tmf}_{k}\mathrm{A}_{1}}$${{\sf
tmf}_{k-3}\mathrm{Y}}$${\cdots,}$$\scriptstyle{i_{3}}$$\scriptstyle{p_{3}}$$\scriptstyle{v_{*}}$
(8) ${\cdots}$${{\sf tmf}_{k-3}\mathrm{M}}$${{\sf tmf}_{k-3}\mathrm{Y}}$${{\sf
tmf}_{k-5}\mathrm{M}}$${\cdots,}$$\scriptstyle{i_{2}}$$\scriptstyle{p_{2}}$$\scriptstyle{\eta_{*}}$
(9) ${\cdots}$${{\sf tmf}_{k-5}}$${{\sf tmf}_{k-5}\mathrm{M}}$${{\sf
tmf}_{k-6}}$${\cdots}$$\scriptstyle{i_{1}}$$\scriptstyle{p_{1}}$$\scriptstyle{2}$

associated to the cofiber sequences (2), (3) and (4), respectively. This is
the technical core of the paper and requires careful bookkeeping using
Adams–Novikov spectral sequences.

In our arguments, we ignore $v_{1}$-periodic classes for reasons we will now
explain.

###  2.1. Suppression of $v_{1}$-periodic families

Note that the element $c_{4}\in{\sf tmf}_{*}$ maps to $v_{1}^{4}\in{\sf
ko}_{*}$ under the map

${{\sf tmf}}$${{\sf ko}}$${{\sf k}(1)}$

where ${\sf k}(1)$ is the connective height $1$ Morava K-theory (see [DM10,
BR21]). We define

$v_{1}^{-1}{\sf tmf}:={\operatorname{colim}}\ \\{{\sf
tmf}\overset{c_{4}}{\longrightarrow}\Sigma^{-8}{\sf
tmf}\overset{c_{4}}{\longrightarrow}\Sigma^{-16}{\sf
tmf}\overset{c_{4}}{\longrightarrow}\dots\\}\simeq\mathrm{KO}[j^{-1}],$

where $j^{-1}=\Delta/c_{4}^{3}$ (see [Lau04, Corollary 3]).

######  Definition 2.1.

For any spectrum $\mathrm{X}$, define the $v_{1}$-torsion part of ${\sf
tmf}_{*}(\mathrm{X})$ as the kernel

${\sf tmf}_{*}(\mathrm{X})^{\sf tor}:=\operatorname{ker}\left(\leavevmode\hbox
to110.11pt{\vbox to16.17pt{\pgfpicture\makeatletter\hbox{\hskip
55.05301pt\lower-8.13399pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
21.95555pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-17.65001pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\ell:{\sf
tmf}_{*}\mathrm{X}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
21.95555pt\hfil&\hfil\hskip 45.09744pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${v_{1}^{-1}{\sf
tmf}_{*}\mathrm{X}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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of the $v_{1}$-localization map.

######  Definition 2.2.

For any spectrum $\mathrm{X}$, define the $v_{1}$-periodic part of ${\sf
tmf}_{*}\mathrm{X}$ as the cokernel

${\sf tmf}_{*}(\mathrm{X})^{\sf
per}:=\operatorname{coker}\left(\leavevmode\hbox to107.33pt{\vbox
to19.14pt{\pgfpicture\makeatletter\hbox{\hskip 53.66338pt\lower-9.62067pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-53.66338pt}{-9.52083pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
24.56892pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-20.26338pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{\sf
tmf}_{*}(\mathrm{X})^{\sf tor}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{\sf
tmf}_{*}\mathrm{X}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right)$

of the natural inclusion map.

The homotopy groups ${\sf tmf}_{*}$, ${\sf tmf}_{*}\mathrm{M}$ and ${\sf
tmf}_{*}\mathrm{Y}$ have non-trivial $v_{1}$-torsion and $v_{1}$-periodic
parts, whereas ${\sf tmf}_{*}\mathrm{A}_{1}$ consists of only $v_{1}$-torsion
because $\mathrm{A}_{1}$ is a type 2 spectrum. Since ${\sf
tmf}_{*}\mathrm{A}_{1}$ is our starting point, it turns out that we can ignore
the $v_{1}$-periodic parts of ${\sf tmf}_{*}$, ${\sf tmf}_{*}\mathrm{M}$ and
${\sf tmf}_{*}\mathrm{Y}$ in our calculations. The following lemmas make this
precise.

######  Lemma 2.3.

For any nonzero element $a\in{\sf tmf}_{*}\mathrm{A}_{1}$ we have

  1. (a)

$p_{3}(a)\in{\sf tmf}_{*}(\mathrm{Y})^{\sf tor}$,

  2. (b)

$p_{2}(p_{3}(a))\in{\sf tmf}_{*}(\mathrm{M})^{\sf tor}$, and

  3. (c)

$p_{1}(p_{2}(p_{3}(a))\in{\sf tmf}_{*}(\SS)^{\sf tor}$.

###### Proof.

Since $\mathrm{A}_{1}$ is a type $2$ spectrum, it follows that $v_{1}^{-1}{\sf
tmf}_{*}\mathrm{A}_{1}=0$. Therefore, from the commutative diagram

${\dots}$${{\sf tmf}_{k}\mathrm{Y}}$${{\sf tmf}_{k}\mathrm{A}_{1}}$${{\sf
tmf}_{k-3}\mathrm{Y}}$${\dots}$${\dots}$${v_{1}^{-1}{\sf
tmf}_{k}\mathrm{Y}}$${v_{1}^{-1}{\sf tmf}_{k}\mathrm{A}_{1}}$${v_{1}^{-1}{\sf
tmf}_{k-3}\mathrm{Y}}$${\dots}$$\scriptstyle{i_{3}}$$\scriptstyle{\ell}$$\scriptstyle{p_{3}}$$\scriptstyle{\ell}$$\scriptstyle{\ell}$$\scriptstyle{i_{3}}$$\scriptstyle{p_{3}}$

of long exact sequences, we get

$\ell(p_{3}(a))=p_{3}(\ell(a))=p_{3}(0)=0$

which means $p_{3}(a)\in{\sf tmf}_{*}(\mathrm{Y})^{\sf tor}$.

For part (b), we consider the diagram

${\dots}$${{\sf tmf}_{k-3}\mathrm{M}}$${{\sf tmf}_{k-3}\mathrm{Y}}$${{\sf
tmf}_{k-5}\mathrm{M}}$${\dots}$${\dots}$${v_{1}^{-1}{\sf
tmf}_{k-3}\mathrm{M}}$${v_{1}^{-1}{\sf tmf}_{k-3}\mathrm{Y}}$${v_{1}^{-1}{\sf
tmf}_{k-5}\mathrm{M}}$${\dots}$$\scriptstyle{i_{2}}$$\scriptstyle{\ell}$$\scriptstyle{p_{2}}$$\scriptstyle{\ell}$$\scriptstyle{\ell}$$\scriptstyle{i_{2}}$$\scriptstyle{p_{2}}$

of long exact sequences, and observe

$\ell(p_{2}(p_{3}(a)))=p_{2}(\ell(p_{3}(a)))=p_{2}(0)=0$

which implies $p_{2}(p_{3}(a))\in{\sf tmf}_{*}(\mathrm{M})^{\sf tor}$.

A similar study of a commutative diagram for the cofiber sequence (2) proves
(c). ∎

######  Remark 2.4 (Exactness of $v_{1}$-periodic part).

Direct calculations show that the long exact sequences in ${\sf tmf}$-homology
associated to the cofiber sequences (2), (3), and (4) give rise to long exact
sequences on $v_{1}$-periodic parts. The authors are unaware if this is a part
of a general pattern, i.e., whether ${\sf tmf}_{*}(-)^{\sf per}$ is a homology
theory.

######  Lemma 2.5.

Suppose $a\in{\sf tmf}_{*}\mathrm{A}_{1}$ such that $p_{2}(p_{3}(a))=0$ in
${\sf tmf}_{*}\mathrm{M}$. Then there exists a class

$m_{0}\in{\sf tmf}_{*}(\mathrm{M})^{{\sf tor}}$

such that $i_{2}(m_{0})=p_{3}(a)$.

###### Proof.

The map $\eta:\Sigma\mathrm{M}\longrightarrow\mathrm{M}$ induces

${\eta^{\sf per}_{*}:{\sf tmf}_{*-1}(\mathrm{M})^{\sf per}}$${{\sf
tmf}_{*}(\mathrm{M})^{\sf per},}$

and we have a commutative diagram

${\dots}$${{\sf tmf}_{*-1}\mathrm{M}}$${{\sf tmf}_{*}\mathrm{M}}$${{\sf
tmf}_{*}\mathrm{Y}}$${\dots}$${\dots}$${{\sf tmf}_{*-1}(\mathrm{M})^{\sf
per}}$${{\sf tmf}_{*}(\mathrm{M})^{\sf per}}$${{\sf tmf}_{*}(\mathrm{Y})^{\sf
per}}$${\dots}$$\scriptstyle{\eta_{*}}$$\scriptstyle{\uppi_{1}}$$\scriptstyle{\uppi_{2}}$$\scriptstyle{i_{2}}$$\scriptstyle{\uppi_{3}}$$\scriptstyle{\eta_{*}^{\sf
per}}$$\scriptstyle{i_{2}}$

in which the vertical maps are surjections.

If $p_{2}(p_{3}(a))=0$ then there exists $m\in{\sf tmf}_{*}\mathrm{M}$ such
that $i_{2}(m)=p_{3}(a)$. By Lemma 2.3

$i_{2}(\uppi_{2}(m))=\uppi_{3}(i_{2}(m))=\uppi_{3}(p_{2}(a))=0,$

therefore, by Remark 2.4, $\uppi_{2}(m)=\eta_{*}^{\sf per}(m^{\prime})$ for
some $m^{\prime}\in{\sf tmf}_{*-1}$. Let $m^{\prime\prime}\in{\sf
tmf}_{*-1}\mathrm{M}$ be a lift of $m^{\prime}$ along $\uppi_{1}$. It is easy
to see that

$m_{0}=m-\eta_{*}(m^{\prime\prime})\in{\sf tmf}_{*}(\mathrm{M})^{\sf tor}$

and
$i_{2}(m_{0})=i_{2}(m-\eta_{*}(m^{\prime\prime}))=i_{2}(m)-i_{2}(\eta_{*}(m^{\prime\prime}))=i_{2}(m)=p_{3}(a)$.
∎

A similar argument leads to the following result.

######  Lemma 2.6.

Suppose $a\in{\sf tmf}_{*}\mathrm{A}_{1}$ such that $p_{1}(p_{2}(p_{3}(a)))=0$
in ${\sf tmf}_{*}$. Then there exists a class

$s\in{\sf tmf}_{*}^{{\sf tor}}$

such that $i_{1}(s)=p_{2}(p_{3}(a))$.

###  2.2. From ${\sf tmf}_{*}\mathrm{Y}$ to ${\sf tmf}_{*}\mathrm{M}$

An element $y\in{\sf tmf}_{k-3}\mathrm{Y}$ is in the image of $p_{3}$ for some
version of $\mathrm{A}_{1}$ if and only if

$v_{1}\cdot y=0\in{\sf tmf}_{k-1}\mathrm{Y}$

for a choice of $v_{1}$. The action of $v_{1}$-self-maps have been identified
on all generators of ${\sf tmf}_{*}\mathrm{Y}$ [BBPX22, Figs. 22, 23]. Thus,
the image of $p_{3}$ is easily determined; we list these elements in the
leftmost column of LABEL:Table:A1lifts using the following notations.

######  Notation 2.7.

Let ${\sf s}_{\sf i,j}$, ${\sf m}_{\sf i,j}$ and ${\sf y}_{\sf i,j}$ denote
elements of ${\sf tmf}_{*}$, ${\sf tmf}_{*}\mathrm{M}$, and ${\sf
tmf}_{*}\mathrm{Y}$, respectively, which are detected in filtration $({\sf
j,j+i})$ of the Adams–Novikov spectral sequence (11). In the bidegrees that we
are interested in, there is only one element which is $v_{1}$-torsion and
nonzero, thus ${\sf s}_{\sf i,j}$, ${\sf m}_{\sf i,j}$, and ${\sf y}_{\sf
i,j}$ represents unique elements up to higher Adams–Novikov filtration.

Next, we determine the effect of the map $p_{2}$ on the classes in ${\sf
img}(p_{3})\subset{\sf tmf}_{*-3}\mathrm{Y}$ using the long exact sequence (8)

${\cdots}$${{\sf tmf}_{k-3}\mathrm{M}}$${{\sf tmf}_{k-3}\mathrm{Y}}$${{\sf
tmf}_{k-5}\mathrm{M}}$${\cdots.}$$\scriptstyle{i_{2}}$$\scriptstyle{p_{2}}$$\scriptstyle{\eta_{*}}$

By Lemma 2.3 and Lemma 2.5, it suffices to study the short exact sequence

(10) ${\mathrm{C}_{k-3}^{\sf tor}(\mathrm{M})}$${{\sf
tmf}_{k-3}(\mathrm{Y})^{\sf tor}}$${\mathrm{K}_{k-5}^{\sf
tor}(\mathrm{M}),}$$\scriptstyle{p_{2}}$

where $\mathrm{C}_{k-3}^{\sf tor}(\mathrm{M})$ is the cokernel and
$\mathrm{K}_{k-5}^{\sf tor}(\mathrm{M})$ is the kernel of $\eta_{*}$
restricted to $v_{1}$-torsion in (8). We employ some standard techniques in
our analysis which are listed below.

######  Technique 1 (Vanishing $\mathrm{K}$).

If $\mathrm{K}_{k-5}^{\sf tor}(\mathrm{M})=0$ in (10). Then

$p_{2}(y)=0$

for any $y\in{\sf tmf}_{k-3}(\mathrm{Y})^{\sf tor}$.

######  Application 1.

We employ 1 to conclude that the following elements map to zero under the map
$p_{2}:{\sf tmf}_{k-3}\mathrm{Y}\to{\sf tmf}_{k-5}\mathrm{M}$:

  * •

${\sf y}_{3,1}$

  * •

${\sf y}_{6,2}$

  * •

${\sf y}_{14,2}$

  * •

${\sf y}_{18,2}$

  * •

${\sf y}_{21,3}$

  * •

${\sf y}_{29,5}$

  * •

${\sf y}_{34,6}$

  * •

${\sf y}_{39,7}$

  * •

${\sf y}_{40,6}$

  * •

${\sf y}_{45,9}$

  * •

${\sf y}_{51,1}$

  * •

${\sf y}_{54,2}$

  * •

${\sf y}_{60,10}$

  * •

${\sf y}_{60,12}$

  * •

${\sf y}_{65,7}$

  * •

${\sf y}_{65,13}$

  * •

${\sf y}_{66,2}$

  * •

${\sf y}_{69,3}$

  * •

${\sf y}_{75,13}$

  * •

${\sf y}_{76,10}$

  * •

${\sf y}_{80,16}$

  * •

${\sf y}_{81,11}$

  * •

${\sf y}_{85,17}$

  * •

${\sf y}_{86,12}$

  * •

${\sf y}_{90,14}$

  * •

${\sf y}_{91,13}$

  * •

${\sf y}_{96,14}$

  * •

${\sf y}_{97,9}$

  * •

${\sf y}_{101,15}$

  * •

${\sf y}_{105,21}$

  * •

${\sf y}_{106,16}$

  * •

${\sf y}_{111,17}$

  * •

${\sf y}_{112,12}$

  * •

${\sf y}_{117,3}$

  * •

${\sf y}_{117,13}$

  * •

${\sf y}_{123,11}$

  * •

${\sf y}_{132,16}$

  * •

${\sf y}_{137,17}$

  * •

${\sf y}_{142,18}$

  * •

${\sf y}_{143,15}$

  * •

${\sf y}_{148,18}$

  * •

${\sf y}_{161,7}$

  * •

${\sf y}_{165,3}$

  * •

${\sf y}_{168,22}$

######  Technique 2 (Vanishing $\mathrm{C}$).

Suppose $y\in{\sf tmf}_{k-3}(\mathrm{Y})^{\sf tor}$ is a nonzero element and
$\mathrm{C}_{k-3}^{\sf tor}(\mathrm{M})=0$, then

$p_{2}(y)\neq 0$

in (10). Further, if $\mathrm{rank}_{\mathbb{F}_{2}}(\mathrm{K}^{\sf
tor}_{k-5}(\mathrm{M}))=1$ then the image of $y$ is the unique nonzero element
of $\mathrm{K}_{k-5}^{\sf tor}(\mathrm{M})$.

######  Application 2.

We employ 2 to determine

  * •

$p_{2}({\sf y}_{8,2})={\sf m}_{6,2}$

  * •

$p_{2}({\sf y}_{11,3})={\sf m}_{9,3}$

  * •

$p_{2}({\sf y}_{23,3})={\sf m}_{21,5}$

  * •

$p_{2}({\sf y}_{26,4})={\sf m}_{24,6}$

  * •

$p_{2}({\sf y}_{44,8})={\sf m}_{42,8}$

  * •

$p_{2}({\sf y}_{59,3})={\sf m}_{57,3}$

  * •

$p_{2}({\sf y}_{62,2})={\sf m}_{60,12}$

  * •

$p_{2}({\sf y}_{68,2})\neq 0$

  * •

$p_{2}({\sf y}_{74,4})={\sf m}_{72,6}$

  * •

$p_{2}({\sf y}_{77,5})={\sf m}_{75,13}$

  * •

$p_{2}({\sf y}_{82,6})={\sf m}_{80,16}$

  * •

$p_{2}({\sf y}_{83,3})={\sf m}_{81,3}$

  * •

$p_{2}({\sf y}_{87,7})={\sf m}_{85,13}$

  * •

$p_{2}({\sf y}_{88,6})={\sf m}_{86,12}$

  * •

$p_{2}({\sf y}_{92,8})={\sf m}_{90,10}$

  * •

$p_{2}({\sf y}_{93,3})={\sf m}_{91,9}$

  * •

$p_{2}({\sf y}_{98,4})={\sf m}_{96,6}$

  * •

$p_{2}({\sf y}_{108,10})={\sf m}_{106,16}$

  * •

$p_{2}({\sf y}_{113,7})\neq 0$

  * •

$p_{2}({\sf y}_{119,3})={\sf m}_{117,3}$

  * •

$p_{2}({\sf y}_{127,15})={\sf m}_{125,21}$

  * •

$p_{2}({\sf y}_{133,11})\neq 0$

  * •

$p_{2}({\sf y}_{155,3})={\sf m}_{153,3}$

  * •

$p_{2}({\sf y}_{158,16})={\sf m}_{156,18}$

  * •

$p_{2}({\sf y}_{167,3})={\sf m}_{165,3}$

  * •

$p_{2}({\sf y}_{170,4})={\sf m}_{168,6}$.

######  Technique 3 (Action of ${\sf tmf}_{*}$).

The maps $i_{2}$ and $p_{2}$ in (8) and (10) are ${\sf tmf}_{*}$-linear, i.e.,

  1. (1)

$p_{2}(t\cdot y)=t\cdot p_{2}(y)$,

  2. (2)

$i_{2}(t\cdot m)=t\cdot i_{2}(m)$

for all $t\in{\sf tmf}_{*}$, $m\in{\sf tmf}_{*}\mathrm{M}$, and $y\in{\sf
tmf}_{*}\mathrm{Y}$.

######  Application 3.

We use 3 to show that

  * •

$p_{2}({\sf y}_{102,10})=p_{2}(\overline{\kappa}\cdot{\sf
y}_{82,6})=\overline{\kappa}\cdot p_{2}({\sf
y}_{82,6})=\overline{\kappa}\cdot{\sf m}_{80,16}={\sf m}_{100,20}$ which
forces $p_{2}({\sf y}_{102,2})=0$,

  * •

$p_{2}({\sf y}_{118,8})=p_{2}(\overline{\kappa}\cdot{\sf
y}_{98,4})=\overline{\kappa}\cdot p_{2}({\sf
y}_{98,4})=\overline{\kappa}\cdot{\sf m}_{96,6}={\sf m}_{116,10}$,

  * •

$p_{2}({\sf y}_{138,12})=p_{2}(\overline{\kappa}\cdot{\sf
y}_{118,8})=\overline{\kappa}\cdot p_{2}({\sf
y}_{118,8})=\overline{\kappa}\cdot{\sf m}_{116,10}={\sf m}_{136,14}$,

  * •

$p_{2}({\sf y}_{153,15})=p_{2}(\overline{\kappa}\cdot{\sf
y}_{133,11})=\overline{\kappa}\cdot p_{2}({\sf
y}_{133,11})=\overline{\kappa}\cdot{\sf m}_{131,17}={\sf m}_{151,21}$ which
forces $p_{2}({\sf y}_{153,11})=0$.

The next few techniques use the fact that the ${\sf tmf}$-homology of
$\mathrm{Y}$ and $\mathrm{M}$ are calculated in [BBPX22] using the
Adams–Novikov spectral sequence

(11)
${{{}^{(-)}\mathrm{E}}_{2}^{s,t}:=\mathrm{Ext}_{\Gamma}^{s,t}(\mathrm{A},\pi_{*}({\sf
tmf}\wedge\mathrm{X}(4)\wedge(-)))}$${{\sf tmf}_{t-s}(-),}$

where the spectrum $\mathrm{X}(4)$ and the Hopf algebroid
$(\mathrm{A},\Gamma)$ are as described in [BBPX22, $\mathsection$2.1].

######  Definition 2.8.

We say an element $x\in{\sf tmf}_{*}(\mathrm{X})$ has Adams–Novikov filtration
$s$, denoted $\mathrm{AF}(x)=s$, if it is detected by an element

$\widehat{x}\in{{}^{\mathrm{X}}\mathrm{E}}_{2}^{s,*+s}$

in the $\mathrm{E}_{2}$-page of (11).

######  Technique 4 (Analysis of $\mathrm{E}_{2}$-pages).

Corresponding to the cofiber sequence (3), there is a long exact sequence

(12)
${\dots}$${{{}^{\mathrm{M}}\mathrm{E}}_{2}^{s,t}}$${{{}^{\mathrm{Y}}\mathrm{E}}_{2}^{s,t}}$${{{}^{\mathrm{M}}\mathrm{E}}_{2}^{s,t-2}}$${\dots}$$\scriptstyle{\widehat{i}_{2}}$$\scriptstyle{{\widehat{p}}_{2}}$$\scriptstyle{\widehat{\eta}}$

of $\mathrm{E}_{2}$-pages of Adams–Novikov spectral sequences. Suppose
$y\in{\sf tmf}_{k-3}\mathrm{Y}$ is detected by
$\widehat{y}\in{{}^{\mathrm{Y}}\mathrm{E}}_{2}^{s,k-3+s}$.

  1. (1)

If $m\in{\sf tmf}_{k-3}\mathrm{M}$ is detected by
$\widehat{m}\in{{}^{\mathrm{M}}\mathrm{E}}_{2}^{s,k-3+s}$ such that

    1. (a)

${\widehat{i}}_{2}(\widehat{m})=\widehat{y}$ and

    2. (b)

$\widehat{m}$ is a permanent cycle,

then $i_{2}(m)=y$.

  2. (2)

If $m\in{\sf tmf}_{k-5}\mathrm{M}$ is detected by
$\widehat{m}\in{{}^{\mathrm{M}}\mathrm{E}}_{2}^{s,k-5+s}$ such that

    1. (a)

${\widehat{p}}_{2}(y)=\widehat{m}$ and

    2. (b)

$\widehat{m}$ is a permanent cycle,

then $p_{2}(y)=m$.

######  Application 4.

We use 4 to determine

  * •

$i_{2}({\sf m}_{20,4})={\sf y}_{20,4}$ which forces $p_{2}({\sf
y}_{20,2})={\sf m}_{18,2}$,

  * •

$p_{2}({\sf y}_{68,2})={\sf m}_{66,2}$,

  * •

$i_{2}({\sf m}_{103,1})={\sf y}_{103,1}$ which forces $p_{2}({\sf
y}_{103,7})={\sf m}_{101,2}$,

  * •

$i_{2}({\sf m}_{150,2})={\sf y}_{150,2}$.

######  Remark 2.9.

In 2.7, the Adams filtration of elements ${\sf s}_{\sf i,j}$, ${\sf m}_{\sf
i,j}$ and ${\sf y}_{\sf i,j}$ equals ${\sf j}$.

Our next technique follows from the fact that maps of spectra cannot decrease
Adams–Novikov filtration.

######  Technique 5 (Adams–Novikov filtration argument).

Suppose $y\in{\sf tmf}_{k-3}\mathrm{Y}$ is a nonzero element and $m\in{\sf
tmf}_{*}\mathrm{M}$.

  1. (1)

If $\mathrm{AF}(y)>\mathrm{AF}(m)$, then $p_{2}(y)\neq m$.

  2. (2)

If $\mathrm{AF}(y)<\mathrm{AF}(m)$, then $i_{2}(m)\neq y$.

######  Application 5.

We use 5 to conclude that

  * •

$i_{2}({\sf m}_{35,5})\neq{\sf y}_{35,3}$ and $p_{2}({\sf y}_{35,3})\neq{\sf
m}_{33,1}$ which forces $p_{2}({\sf y}_{35,3})={\sf m}_{33,3}$,

  * •

$i_{2}({\sf m}_{45,5})\neq{\sf y}_{45,3}$ which forces $p_{2}({\sf
y}_{45,3})={\sf m}_{43,9}$ and $p_{2}({\sf y}_{45,9})=0$,

  * •

$i_{2}({\sf m}_{55,9})\neq{\sf y}_{55,7}$ which forces $p_{2}({\sf
y}_{55,7})={\sf m}_{53,7}$,

  * •

$i_{2}({\sf y}_{56,6})\neq{\sf m}_{54,2}$ which forces $p_{2}({\sf
y}_{56,2})={\sf m}_{54,2}$,

  * •

$p_{2}({\sf y}_{57,11})\neq{\sf m}_{55,9}$ which forces $p_{2}({\sf
y}_{57,11})=0$,

  * •

$p_{2}({\sf y}_{71,9})\neq{\sf m}_{69,3}$ which forces $p_{2}({\sf
y}_{71,9})=0$ and $p_{2}({\sf y}_{71,3})={\sf m}_{69,3}$,

  * •

$p_{2}({\sf y}_{107,11})\neq{\sf m}_{105,3}$ which forces $p_{2}({\sf
y}_{107,3})={\sf m}_{105,3}$ and $p_{2}({\sf y}_{107,11})={\sf m}_{105,17}$,

  * •

$p_{2}({\sf y}_{113,7})\neq{\sf m}_{111,3}$ which forces $p_{2}({\sf
y}_{113,7})={\sf m}_{111,13}$,

  * •

$p_{2}({\sf y}_{122,14})\neq{\sf m}_{120,6}$, which forces $p_{2}({\sf
y}_{122,14})=0$ and $p_{2}({\sf y}_{122,4})={\sf m}_{120,6}$,

  * •

$p_{2}({\sf y}_{133,11})\neq{\sf m}_{133,7}$ which forces $p_{2}({\sf
y}_{133,11})={\sf m}_{131,17}$.

######  Technique 6 (Geometric boundary theorem [Beh12a, Lemma A.4.1 (5)]).

Consider the maps of the Adams–Novikov spectral sequences

${{{}^{\Sigma\mathrm{M}}\mathrm{E}}_{r}^{s,*+s}}$${{{}^{\mathrm{M}}\mathrm{E}}_{r}^{s,*+s}}$${{{}^{\mathrm{Y}}\mathrm{E}}_{r}^{s,*+s}}$${{{}^{\Sigma^{2}\mathrm{M}}\mathrm{E}}_{r}^{s,*+s}}$$\scriptstyle{\widehat{\eta}}$$\scriptstyle{\widehat{i}_{2}}$$\scriptstyle{\widehat{p}_{2}}$

induced by (3), and suppose
$\widehat{m}\in{{}^{\mathrm{M}}\mathrm{E}}_{r}^{s,*+s}$ such that

  * •

$d_{r}(\widehat{m})=\widehat{\eta}(\widehat{m}^{\prime})$,

  * •

$\widehat{i}_{2}(\widehat{m})=\widehat{y}$ is a nonzero permanent cycle.

Then $\widehat{p}_{2}(\widehat{y})=\widehat{m}^{\prime}$.

######  Application 6.

We use 6 in the following arguments:

  * •

Since $d_{5}({\sf m}_{50,6})=\widehat{\eta}({\sf m}_{48,6})$ and
$\widehat{i}_{2}({\sf m}_{50,6})={\sf y}_{50,6}$ is a nonzero permanent cycle,
we get $p_{2}({\sf y}_{50,6})={\sf m}_{48,6}$. Consequently,
$p_{2}({\sf{y}_{50,4}})=0$.

  * •

Since $d_{5}({\sf m}_{70,10})=\widehat{\eta}({\sf m}_{68,10})$ and
$\widehat{i}_{2}({\sf m}_{70,10})={\sf y}_{70,10}$ is a permanent cycle, we
get $p_{2}({\sf y}_{70,10})={\sf m}_{48,10}$. Consequently, $p_{2}({\sf
y}_{70,8})=0$. Alternatively, this case follows from the previous case using
$\bar{\kappa}$-linearity.

  * •

Since $d_{5}({\sf m}_{128,14})=\widehat{\eta}({\sf m}_{126,20})$ and
$\widehat{i}_{2}({\sf m}_{128,14})={\sf y}_{128,14}$ is a permanent cycle, we
get $p_{2}({\sf y}_{128,14})={\sf m}_{126,20}$. Alternatively, this follows
using $\bar{\kappa}$-linearity from the fact that $p_{2}({\sf
y}_{108,10})={\sf m}_{106,16}$ which was established earlier using 2.

###  2.3.  From ${\sf tmf}_{*}\mathrm{M}$ to ${\sf tmf}_{*}$

All the techniques above have analogs corresponding to the cofiber sequence
(2). We use them to study the short exact sequence

(13) ${\mathrm{C}_{k-3}^{\sf tor}}$${{\sf tmf}_{k-3}(\mathrm{M})^{\sf
tor}}$${\mathrm{K}_{k-4}^{\sf tor},}$

where $\mathrm{C}_{k-3}^{\sf tor}$ is the cokernel and $\mathrm{K}_{k-4}^{\sf
tor}$ is the kernel of the multiplication by $2$ map restricted to
$v_{1}$-torsion in (9).

######  Application 7.

We use the analog of 1 to determine

  * •

$i_{1}({\sf s}_{6,2})={\sf m}_{6,2}$

  * •

$i_{1}({\sf s}_{9,3})={\sf m}_{9,3}$

  * •

$i_{1}({\sf s}_{21,5})={\sf m}_{21,5}$

  * •

$i_{1}({\sf s}_{24,0})={\sf m}_{24,0}$

  * •

$i_{1}({\sf s}_{48,0})={\sf m}_{48,6}$

  * •

$i_{1}({\sf s}_{53,7})={\sf m}_{53,7}$

  * •

$i_{1}({\sf s}_{57,3})={\sf m}_{57,3}$

  * •

$i_{1}({\sf s}_{60,12})={\sf m}_{60,12}$

  * •

$i_{1}({\sf s}_{68,4})={\sf m}_{68,10}$

  * •

$i_{1}({\sf s}_{72,0})={\sf m}_{72,6}$

  * •

$i_{1}({\sf s}_{75,3})={\sf m}_{75,13}$

  * •

$i_{1}({\sf s}_{80,16})={\sf m}_{80,16}$

  * •

$i_{1}({\sf s}_{85,13})={\sf m}_{85,13}$

  * •

$i_{1}({\sf s}_{90,10})={\sf m}_{90,10}$

  * •

$i_{1}({\sf s}_{96,0})={\sf m}_{96,6}$

  * •

$i_{1}({\sf s}_{100,20})={\sf m}_{100,20}$

  * •

$p_{1}({\sf m}_{105,3})=0$

  * •

$p_{1}({\sf m}_{105,17})=0$

  * •

$i_{1}({\sf s}_{116,4})={\sf m}_{116,10}$

  * •

$i_{1}({\sf s}_{120,0})={\sf m}_{120,6}$

  * •

$i_{1}({\sf s}_{153,3})={\sf m}_{153,3}$

  * •

$i_{1}({\sf s}_{168,0})={\sf m}_{168,6}$.

######  Application 8.

We use the analog of 2 to determine

  * •

$p_{1}({\sf m}_{18,2})={\sf s}_{17,2}$

  * •

$p_{1}({\sf m}_{43,9})={\sf s}_{42,11}$

  * •

$p_{1}({\sf m}_{69,3})={\sf s}_{68,4}$

  * •

$p_{1}({\sf m}_{81,3})={\sf s}_{80,16}$

  * •

$p_{1}({\sf m}_{86,12})={\sf s}_{85,13}$

  * •

$p_{1}({\sf m}_{91,9})={\sf s}_{90,10}$

  * •

$p_{1}({\sf m}_{101,7})={\sf s}_{100,20}$

  * •

$p_{1}({\sf m}_{106,16})={\sf s}_{105,17}$

  * •

$p_{1}({\sf m}_{126,20})={\sf m}_{125,21}$

  * •

$p_{1}({\sf m}_{131,17})={\sf s}_{130,18}$

  * •

$p_{1}({\sf m}_{151,21})={\sf s}_{150,22}$

  * •

$p_{1}({\sf m}_{165,3})={\sf s}_{164,4}$.

######  Application 9.

We use the analog of 3 to deduce that

  * •

$p_{2}({\sf m}_{111,13})=p_{2}(\overline{\kappa}\cdot{\sf
m}_{91,9})=\overline{\kappa}\cdot{\sf s}_{90,10}={\sf s}_{110,14}$,

  * •

$i_{1}({\sf s}_{136,8})=i_{1}(\overline{\kappa}\cdot{\sf
s}_{116,4})=\overline{\kappa}\cdot{\sf m}_{116,10}={\sf m}_{136,14}$,

  * •

$i_{1}({\sf s}_{156,12})=i_{1}(\overline{\kappa}\cdot{\sf
s}_{136,8})=\overline{\kappa}\cdot{\sf m}_{136,14}={\sf m}_{156,18}$.

######  Application 10.

We use the analog of 4 to argue that

  * •

$i_{2}({\sf s}_{54,2})={\sf m}_{54,2}$.

######  Application 11.

The analog of 5 is used to deduce that

  * •

$i_{1}({\sf s}_{9,3})\neq{\sf m}_{9,1}$ which forces $i_{1}({\sf
s}_{9,3})={\sf m}_{9,3}$,

  * •

$p_{1}({\sf m}_{33,3})\neq{\sf s}_{32,2}$ which forces $i_{1}({\sf
s}_{33,3})={\sf m}_{33,3}$,

  * •

$i_{1}({\sf s}_{42,10})\neq{\sf m}_{42,8}$ which forces $i_{1}({\sf
s}_{42,10})={\sf m}_{42,10}$,

  * •

$i_{1}({\sf s}_{60,12})\neq{\sf m}_{60,7}$ which forces $p_{1}({\sf
m}_{60,7})={\sf s}_{59,7}$,

  * •

$p_{1}({\sf m}_{66,8})\neq{\sf s}_{65,3}$ and $i_{1}({\sf s}_{66,10})\neq{\sf
m}_{66,8}$ which forces $p_{1}({\sf m}_{66,8})={\sf s}_{65,9}$, which along
with $i_{1}({\sf s}_{66,10})\neq{\sf m}_{66,2}$ forces $p_{1}({\sf
m}_{66,2})={\sf s}_{65,3}$,

  * •

$i_{1}({\sf s}_{105,11})\neq{\sf m}_{105,3}$ which forces $i_{1}({\sf
s}_{105,11})={\sf m}_{105,11}$ and consequently $i_{1}({\sf s}_{105,3})={\sf
m}_{105,3}$,

  * •

$i_{1}({\sf s}_{117,5})\neq{\sf m}_{117,3}$ which forces $p_{1}({\sf
m}_{117,3})={\sf s}_{116,4}$,

  * •

$p_{2}({\sf m}_{125,21})\neq{\sf s}_{124,6}$ which forces $i_{1}({\sf
s}_{125,21})={\sf m}_{125,21}$.

###  2.4. Summary Table

We summarize our calculations in LABEL:Table:A1lifts as follows. The leftmost
column lists the image of $p_{3}$ in ${\sf tmf}_{*}\mathrm{Y}$. We determine
their image in column $2$ and indicate the technique used, among 1 through 6,
in column $3$.

We calculate the image under $p_{1}$ of nonzero elements in column $2$ and
record them in column $4$. If the image is zero, we identify a $v_{1}$-torsion
element which is its lift along $i_{1}$ and record it in column $5$. We
indicate the technique in column $6$.

Note that the elements listed in columns $4$ and $5$ are elements of ${\sf
tmf}_{*}$. We record their familiar names from [DFHH14] in column $7$.

Table 1. Detecting elements in ${\sf tmf}_{*}$ ${\sf img}(p_{3})$ | ${\sf img}(p_{2})$ | (T) | ${\sf img}(p_{1})$ | $i_{1}^{-1}(-)$ | (T) | name in ${\sf tmf}_{*}$
---|---|---|---|---|---|---
${\sf y}_{3,1}$ | $0$ | (1) |  |  |  |
${\sf y}_{6,2}$ | $0$ | (1) |  |  |  |
${\sf y}_{8,2}$ | ${\sf m}_{6,2}$ | (2) | $0$ | ${\sf s}_{6,2}$ | (1) | $\nu^{2}$
${\sf y}_{11,3}$ | ${\sf m}_{9,3}$ | (2) | $0$ | ${\sf s}_{9,3}$ | (5) | $\nu^{3}$
${\sf y}_{14,2}$ | $0$ | (1) |  |  |  |
${\sf y}_{18,2}$ | $0$ | (1) |  |  |  |
${\sf y}_{20,2}$ | ${\sf m}_{18,2}$ | (4) | ${\sf s}_{17,2}$ |  | (2) | $\kappa\nu$
${\sf y}_{21,3}$ | $0$ | (1) |  |  |  |
${\sf y}_{23,3}$ | ${\sf m}_{21,5}$ | (2) | $0$ | ${\sf s}_{21,5}$ | (1) | $\overline{\kappa}\nu$
${\sf y}_{26,4}$ | ${\sf m}_{24,6}$ | (2) | $0$ | ${\sf s}_{24,0}$ | (1) | $8\Delta$
${\sf y}_{29,5}$ | $0$ | (1) |  |  |  |
${\sf y}_{34,6}$ | $0$ | (1) |  |  |  |
${\sf y}_{35,3}$ | ${\sf m}_{33,3}$ | (5) | 0 | ${\sf s}_{33,3}$ | (5) | $q\eta$
${\sf y}_{39,7}$ | $0$ | (1) |  |  |  |
${\sf y}_{40,6}$ | $0$ | (1) |  |  |  |
${\sf y}_{44,8}$ | ${\sf m}_{42,10}$ | (2) | $0$ | ${\sf s}_{42,10}$ | (5) | $\overline{\kappa}^{2}\eta^{2}$
${\sf y}_{45,3}$ | ${\sf m}_{43,9}$ | (5) | ${\sf s}_{42,10}$ |  | (2) | $\overline{\kappa}^{2}\eta^{2}$
${\sf y}_{45,9}$ | $0$ | (1) |  |  |  |
${\sf y}_{50,4}$ | $0$ | (6) |  |  |  |
${\sf y}_{50,6}$ | ${\sf m}_{48,6}$ | (6) | $0$ | ${\sf s}_{48,0}$ | (1) | $4\Delta^{2}$
${\sf y}_{51,1}$ | $0$ | (1) |  |  |  |
${\sf y}_{54,2}$ | $0$ | (1) |  |  |  |
${\sf y}_{55,7}$ | ${\sf m}_{53,7}$ | (5) | $0$ | ${\sf s}_{53,7}$ | (1) | $\eta^{2}\Delta^{2}\nu$
${\sf y}_{56,2}$ | ${\sf m}_{54,2}$ | (5) | 0 | ${\sf s}_{54,2}$ | (4) | $\nu\Delta^{2}\nu$
${\sf y}_{57,11}$ | $0$ | (5) |  |  |  |
${\sf y}_{59,3}$ | ${\sf m}_{57,3}$ | (2) | 0 | ${\sf s}_{57,3}$ | (1) | $\nu\Delta^{2}\nu^{2}$
${\sf y}_{60,10}$ | $0$ | (1) |  |  |  |
${\sf y}_{60,12}$ | $0$ | (1) |  |  |  |
${\sf y}_{62,2}$ | ${\sf m}_{60,12}$ | (2) |  | ${\sf s}_{60,12}$ | (1) | $\nu\Delta^{2}\nu^{3}$
${\sf y}_{65,7}$ | $0$ | (1) |  |  |  |
${\sf y}_{65,13}$ | $0$ | (1) |  |  |  |
${\sf y}_{66,2}$ | $0$ | (1) |  |  |  |
${\sf y}_{68,2}$ | ${\sf m}_{66,2}$ | (2,4) | ${\sf s}_{65,3}$ |  | (5) | $\eta\Delta\bar{\kappa}^{2}$
${\sf y}_{69,3}$ | $0$ | (1) |  |  |  |
${\sf y}_{70,8}$ | $0$ | (6) |  |  |  |
${\sf y}_{70,10}$ | ${\sf m}_{68,10}$ | (6) | 0 | ${\sf s}_{68,4}$ | (1) | $4\Delta^{2}\bar{\kappa}$
${\sf y}_{71,3}$ | ${\sf m}_{69,3}$ | (5) | ${\sf s}_{68,4}$ | 0 | (2) | $4\Delta^{2}\bar{\kappa}$
${\sf y}_{71,9}$ | $0$ | (5) |  |  |  |
${\sf y}_{74,4}$ | ${\sf m}_{72,6}$ | (2) | $0$ | ${\sf s}_{72,0}$ | (1) | $8\Delta^{3}$
${\sf y}_{75,13}$ | $0$ | (1) |  |  |  |
${\sf y}_{76,10}$ | $0$ | (1) |  |  |  |
${\sf y}_{77,5}$ | ${\sf m}_{75,13}$ | (2) | $0$ | ${\sf s}_{75,3}$ | (1) | $(\eta\Delta)^{3}$
${\sf y}_{80,16}$ | $0$ | (1) |  |  |  |
${\sf y}_{81,11}$ | $0$ | (1) |  |  |  |
${\sf y}_{82,6}$ | ${\sf m}_{80,16}$ | (2) | $0$ | ${\sf s}_{80,16}$ | (1) | $\overline{\kappa}^{4}$
${\sf y}_{83,3}$ | ${\sf m}_{81,3}$ | (2) | ${\sf s}_{80,16}$ |  | (2) | $\overline{\kappa}^{4}$
${\sf y}_{85,17}$ | $0$ | (1) |  |  |  |
${\sf y}_{86,12}$ | $0$ | (1) |  |  |  |
${\sf y}_{87,7}$ | ${\sf m}_{85,13}$ | (2) | $0$ | ${\sf s}_{85,13}$ | (1) | $\eta\Delta\bar{\kappa}^{3}$
${\sf y}_{88,6}$ | ${\sf m}_{86,12}$ | (2) | ${\sf s}_{85,13}$ |  | (2) | $\eta\Delta\bar{\kappa}^{3}$
${\sf y}_{90,14}$ | $0$ | (1) |  |  |  |
${\sf y}_{91,13}$ | $0$ | (1) |  |  |  |
${\sf y}_{92,8}$ | ${\sf m}_{90,10}$ | (2) | $0$ | ${\sf s}_{90,10}$ | (1) | $\eta^{2}\Delta^{2}\bar{\kappa}^{2}$
${\sf y}_{93,3}$ | ${\sf m}_{91,9}$ | (2) | ${\sf s}_{90,10}$ |  | (2) | $\eta^{2}\Delta^{2}\bar{\kappa}^{2}$
${\sf y}_{96,14}$ | $0$ | (1) |  |  |  |
${\sf y}_{97,9}$ | $0$ | (1) |  |  |  |
${\sf y}_{98,4}$ | ${\sf m}_{96,6}$ | (2) | $0$ | ${\sf s}_{96,0}$ | (1) | $2\Delta^{4}$
${\sf y}_{101,15}$ | $0$ | (1) |  |  |  |
${\sf y}_{102,2}$ | $0$ | (3) |  |  |  |
${\sf y}_{102,10}$ | ${\sf m}_{100,20}$ | (3) | $0$ | ${\sf s}_{100,20}$ | (1) | $\overline{\kappa}^{5}$
${\sf y}_{103,7}$ | ${\sf m}_{101,7}$ | (4) | ${\sf s}_{100,20}$ |  | (2) | $\bar{\kappa}^{5}$
${\sf y}_{105,21}$ | $0$ | (1) |  |  |  |
${\sf y}_{106,16}$ | $0$ | (1) |  |  |  |
${\sf y}_{107,3}$ | ${\sf m}_{105,3}$ | (5) | $0$ | ${\sf s}_{105,3}$ | (1,5) | $\nu^{3}\Delta^{4}$
${\sf y}_{107,11}$ | ${\sf m}_{105,11}$ | (5) | $0$ | ${\sf s}_{105,17}$ | (1,5) | $\eta\Delta\overline{\kappa}^{4}$
${\sf y}_{108,10}$ | ${\sf m}_{106,16}$ | (2) | ${\sf s}_{105,17}$ |  | (2) | $\eta\Delta\overline{\kappa}^{4}$
${\sf y}_{111,17}$ | $0$ | (1) |  |  |  |
${\sf y}_{112,12}$ | $0$ | (1) |  |  |  |
${\sf y}_{113,7}$ | ${\sf m}_{111,13}$ | (2,5) | ${\sf s}_{110,14}$ |  | (3) | $\eta^{2}\Delta^{2}\bar{\kappa}^{3}$
${\sf y}_{117,3}$ | $0$ | (1) |  |  |  |
${\sf y}_{117,13}$ | $0$ | (1) |  |  |  |
${\sf y}_{118,8}$ | ${\sf m}_{116,10}$ | (3) | $0$ | ${\sf s}_{116,4}$ | (1) | $2\Delta^{4}\bar{\kappa}$
${\sf y}_{119,3}$ | ${\sf m}_{117,3}$ | (2) | ${\sf s}_{116,4}$ |  | (5) | $2\Delta^{4}\cdot 2\bar{\kappa}$
${\sf y}_{122,4}$ | ${\sf m}_{120,6}$ | (5) | $0$ | ${\sf s}_{120,0}$ | (1) | $8\Delta^{5}$
${\sf y}_{122,14}$ | $0$ | (5) |  |  |  |
${\sf y}_{123,11}$ | $0$ | (1) |  |  |  |
${\sf y}_{127,15}$ | ${\sf m}_{125,21}$ | (2) | $0$ | ${\sf s}_{125,21}$ | (5) | $\eta\Delta\bar{\kappa}^{5}$
${\sf y}_{128,14}$ | ${\sf m}_{126,20}$ | (6) | ${\sf s}_{125,21}$ |  | (2) | $\eta\Delta\bar{\kappa}^{5}$
${\sf y}_{132,16}$ | $0$ | (1) |  |  |  |
${\sf y}_{133,11}$ | ${\sf m}_{131,17}$ | (2,5) | ${\sf s}_{130,18}$ |  | (2) | $\eta^{2}\Delta^{2}\bar{\kappa}^{4}$
${\sf y}_{137,17}$ | $0$ | (1) |  |  |  |
${\sf y}_{138,12}$ | ${\sf m}_{136,14}$ | (3) | $0$ | ${\sf s}_{136,8}$ | (3) | $\eta^{2}\Delta^{5}\kappa$
$y_{142,18}$ | $0$ | (1) |  |  |  |
${\sf y}_{143,15}$ | $0$ | (1) |  |  |  |
${\sf y}_{148,18}$ | $0$ | (1) |  |  |  |
${\sf y}_{150,2}$ | $0$ | (4) |  |  |  |
${\sf y}_{153,11}$ | $0$ | (3) |  |  |  |
${\sf y}_{153,15}$ | ${\sf m}_{151,21}$ | (3) | ${\sf s}_{150,22}$ |  | (2) | $\eta^{2}\Delta^{2}\bar{\kappa}^{5}$
${\sf y}_{155,3}$ | ${\sf m}_{153,3}$ | (2) | $0$ | ${\sf s}_{153,3}$ | (1) | $\nu\Delta^{6}\nu^{2}$
${\sf y}_{158,16}$ | ${\sf m}_{156,18}$ | (2) | $0$ | ${\sf s}_{156,12}$ | (3) | $\nu\Delta^{6}\eta\epsilon$
${\sf y}_{161,7}$ | $0$ | (1) |  |  |  |
${\sf y}_{165,3}$ | $0$ | (1) |  |  |  |
${\sf y}_{167,3}$ | ${\sf m}_{165,3}$ | (2) | ${\sf s}_{164,4}$ |  | (2) | $4\Delta^{6}\bar{\kappa}$
${\sf y}_{168,22}$ | $0$ | (1) |  |  |  |
${\sf y}_{170,4}$ | ${\sf m}_{168,6}$ | (2) | $0$ | ${\sf s}_{168,0}$ | (1) | $8\Delta^{7}$

##  3\. New infinite families

We begin by studying the commutative diagram of long exact sequences

(14)
${\cdots}$${\pi_{k}\mathrm{Y}}$${\pi_{k}\mathrm{A}_{1}}$${\pi_{k-3}\mathrm{Y}}$${\cdots}$${\cdots}$${{\sf
tmf}_{k}\mathrm{Y}}$${{\sf tmf}_{k}\mathrm{A}_{1}}$${{\sf
tmf}_{k-3}\mathrm{Y}}$${\cdots}$$\scriptstyle{i_{3}}$$\scriptstyle{{\sf
h}_{\sf tmf}}$$\scriptstyle{p_{3}}$$\scriptstyle{{\sf h}_{\sf
tmf}}$$\scriptstyle{v_{*}}$$\scriptstyle{{\sf h}_{\sf
tmf}}$$\scriptstyle{i_{3}}$$\scriptstyle{p_{3}}$$\scriptstyle{v_{*}}$

associated to the cofiber sequence (4).

######  Lemma 3.1.

Any nonzero element of the form $p_{3}(a)\in{\sf tmf}_{*}\mathrm{Y}$ admits a
nonzero lift in $\pi_{*}\mathrm{Y}$ along the ${\sf tmf}$-Hurewicz
homomorphism.

###### Proof.

This is a straightforward consequence of the fact that the ${\sf
tmf}$-Hurewicz map for $\mathrm{A}_{1}$ in (5) is a surjection [Pha23], along
with the commutativity of (14). ∎

Next, we study the commutative diagram of long exact sequences

(15)
${\cdots}$${\pi_{k-3}\mathrm{M}}$${\pi_{k-3}\mathrm{Y}}$${\pi_{k-5}\mathrm{M}}$${\cdots}$${\cdots}$${{\sf
tmf}_{k-3}\mathrm{M}}$${{\sf tmf}_{k-3}\mathrm{Y}}$${{\sf
tmf}_{k-5}\mathrm{M}}$${\cdots}$$\scriptstyle{i_{2}}$$\scriptstyle{{\sf
h}_{\sf tmf}}$$\scriptstyle{p_{2}}$$\scriptstyle{{\sf h}_{\sf
tmf}}$$\scriptstyle{\eta_{*}}$$\scriptstyle{{\sf h}_{\sf
tmf}}$$\scriptstyle{i_{2}}$$\scriptstyle{p_{2}}$$\scriptstyle{\eta_{*}}$

associated to the cofiber sequence (3).

######  Lemma 3.2.

Any nonzero element of the form $p_{2}(p_{3}(a))\in{\sf tmf}_{*}\mathrm{M}$
admits a nonzero lift in $\pi_{*}\mathrm{M}$ along the ${\sf tmf}$-Hurewicz
homomorphism.

###### Proof.

If $p_{2}(p_{3}(a))\neq 0$ then, in particular, $p_{3}(a)\neq 0$. Thus, by
Lemma 3.1, there exists

$\tilde{y}\neq 0\in\pi_{*}\mathrm{Y}$

such that ${\sf h}_{{\sf tmf}}(\tilde{y})=p_{3}(a)$. The result then follows
from commutativity of (15). ∎

######  Remark 3.3.

The action of $\Delta^{8}\in{\sf tmf}_{192}$ is faithful on ${\sf
tmf}_{*}\mathrm{A}_{1}$ [Pha23], ${\sf tmf}_{*}\mathrm{Y}$ [BBPX22], ${\sf
tmf}_{*}\mathrm{M}$ [BBPX22], ${\sf tmf}_{*}$ [Bau08], the Hurewicz image of
${\sf tmf}_{*}$ [BMQ23], and the cokernel of the ${\sf tmf}$-Hurewicz map
[BMQ23].

###  3.1. Infinite families in $2$-local stable stems

Our final step studies the commutative diagram of long exact sequences

(16)
${\cdots}$${\pi_{k-5}\SS}$${\pi_{k-5}\mathrm{M}}$${\pi_{k-6}\SS}$${\cdots}$${\cdots}$${{\sf
tmf}_{k-5}}$${{\sf tmf}_{k-5}\mathrm{M}}$${{\sf
tmf}_{k-6}}$${\cdots}$$\scriptstyle{i_{1}}$$\scriptstyle{{\sf h}_{\sf
tmf}}$$\scriptstyle{p_{1}}$$\scriptstyle{{\sf h}_{\sf
tmf}}$$\scriptstyle{\cdot 2}$$\scriptstyle{{\sf h}_{\sf
tmf}}$$\scriptstyle{i_{1}}$$\scriptstyle{p_{1}}$$\scriptstyle{\cdot 2}$

associated to the cofiber sequence (2).

Suppose first the case when $p_{1}(p_{2}(p_{3}(a)))\neq 0$ for some $a\in{\sf
tmf}_{k}\mathrm{A}_{1}$. Then, using Lemma 3.2, Remark 3.3, and (14), it is
easy to see that there is a $192$-periodic infinite family

$\\{\tilde{\sf s}_{k-6+192i}\in\pi_{k-6+192i}(\SS):i\in\mathbb{N}\\}$

such that

  1. (1)

${\sf h}_{\sf tmf}(\tilde{\sf s}_{k-6})=p_{1}(p_{2}(p_{3}(a)))$,

  2. (2)

${\sf h}_{\sf tmf}(\tilde{\sf s}_{k-6+192i})\neq 0$ for all $i\in\mathbb{N}$.

######  Theorem 3.4.

Suppose $a\in{\sf tmf}_{k}\mathrm{A}_{1}$ such that $p_{2}(p_{3}(a))\in{\sf
tmf}_{k-5}\mathrm{M}$ is nonzero but $p_{1}(p_{2}(p_{3}(a)))=0$ $\in{\sf
tmf}_{k-6}$.

  1. (I)

If $i_{1}^{-1}(p_{2}(p_{3}(a)))\cap{\sf img}({\sf h}_{{\sf
tmf}})\neq\emptyset,$ then there exists a $192$-periodic infinite family of
elements in the stable stems

$\\{\tilde{\sf s}_{k-5+192i}\in\pi_{k-5+192i}(\SS):i\in\mathbb{N}\\}$

such that $i_{1}({\sf h}_{\sf tmf}(\tilde{\sf s}_{k-5}))=p_{2}(p_{3}(a)))$ and
${\sf h}_{\sf tmf}(\tilde{\sf s}_{k-5+192i})\neq 0$ for all $i\in\mathbb{N}$.

  2. (II)

If $i_{1}^{-1}(p_{2}(p_{3}(a)))\cap{\sf img}({\sf h}_{{\sf tmf}})=\emptyset,$
then there exists a $192$-periodic infinite family of elements in the stable
stems

(17) $\\{\bar{\sf s}_{k-6+192i}\in\pi_{k-6+192i}(\SS):i\in\mathbb{N}\\}$

such that $\bar{\sf s}_{k-6+192i}\neq 0$ and ${\sf h}_{\sf tmf}(\bar{\sf
s}_{k-6+192i})=0$ for all $i\in\mathbb{N}$.

###### Proof.

If $p_{2}(p_{3}(a))\in{\sf tmf}_{k-5}\mathrm{M}$ is nonzero, then by Remark
3.3,

$p_{2}(p_{3}(\Delta^{8i}\cdot a))=\Delta^{8i}\cdot p_{2}(p_{3}(a))\neq 0.$

Thus, by Lemma 3.2, there exist nonzero elements

(18) $\tilde{m}_{k-5+192i}\in\pi_{k-5+192i}(\mathrm{M})$

such that ${\sf h}_{\sf tmf}(\tilde{m}_{k-5+192i})=\Delta^{8i}\cdot
p_{2}(p_{3}(a))$ for all $i\in\mathbb{N}$.

Suppose $p_{2}(p_{3}(a))$ admits a lift $s_{k-5}\in{\sf tmf}_{k-5}$ along
$i_{1}$ which is in the Hurewicz image. Then, by Remark 3.3, $\Delta^{8i}\cdot
s_{k-5}$ is also in the Hurewicz image, and a collection

$\\{\tilde{\sf s}_{k-5+192i}\in\pi_{k-5+192i}(\SS):i\in\mathbb{N}\\}$

such that ${\sf h}_{{\sf tmf}}(\tilde{\sf s}_{k-5+192i})=\Delta^{8i}\cdot
s_{k-5}$ forms an infinite family with the desired properties.

On the other hand, if none of the lifts of $p_{2}(p_{3}(a))$ along $i_{1}$ is
in the Hurewicz image, then the same holds for $\Delta^{8i}\cdot
p_{2}(p_{3}(a))$ for all $i\in\mathbb{N}$ by Remark 3.3. Thus,
$p_{1}(\tilde{m}_{k-5+192i})\neq 0$ for all $i\in\mathbb{N}$, and

$\\{\overline{\sf s}_{k-6+192i}=p_{2}(\tilde{m}_{k-5+192i}):i\in\mathbb{N}\\}$

is the desired infinite family. ∎

###### Proof of 1.

From LABEL:Table:A1lifts we notice that there exists an element $a\in{\sf
tmf}_{k}\mathrm{A}_{1}$ such that

  1. (i)

$p_{2}(p_{3}(a))\neq 0$,

  2. (ii)

$p_{1}(p_{2}(p_{3}(a)))=0$,

  3. (iii)

there exists $s\in{\sf tmf}_{k-5}^{\sf tor}$ such that
$i_{1}(s)=p_{2}(p_{3}(a))$,

for each $k\in\\{29,53,77,80,101,119,173\\}$. However, the Hurewicz image is
trivial in degrees $24$, $48$, $72$, $75$, $96$, $114$, and $168$. Thus, the
result follows from case (II) of 3.4. ∎

######  Remark 3.5.

To summarize, the seven infinite families in 1 are a consequence of the fact
that the elements

(19)
$8\Delta,4\Delta^{2},8\Delta^{3},(\eta\Delta)^{3},2\Delta^{4},8\Delta^{5},8\Delta^{7},$

which are not in the Hurewicz image of ${\sf tmf}_{*}$ are the lifts of
nonzero elements in the image of $p_{2}\circ p_{1}:{\sf
tmf}_{*}\mathrm{A}_{1}\longrightarrow{\sf tmf}_{*-5}\mathrm{M}$ along $i_{1}$.

###  3.2. Infinite families in $\mathrm{K}(2)$-local stable stems

######  Theorem 3.6.

_All elements listed in 1 have nonzero images in the $\mathrm{K}(2)$-local
stable stems. _

######  Notation 3.7.

For any spectrum $\mathrm{X}$, let $\widehat{\mathrm{X}}$ denote
$\mathrm{X}\wedge\SS_{\mathrm{K}(2)}$.

The work in [Pha23] shows that the $\mathrm{K}(2)$-local Hurewicz map of
$\mathrm{A}_{1}$

${{\sf
h}_{\mathrm{TMF}}:\pi_{*}\widehat{\mathrm{A}}_{1}}$${\mathrm{TMF}_{*}\mathrm{A}_{1}}$

is a surjection.

Since $\mathrm{TMF}_{*}\mathrm{A}_{1}\cong(\Delta^{8})^{-1}{\sf
tmf}_{*}\mathrm{A}_{1}$ and the action of $\Delta^{8}$ on ${\sf
tmf}_{*}\mathrm{A}_{1}$ is faithful (see Remark 3.3), the natural map

${\ell:{\sf tmf}_{*}\mathrm{A}_{1}}$${\mathrm{TMF}_{*}\mathrm{A}_{1}}$

is an injection. Thus the image of
$\ell(a)\in\mathrm{TMF}_{*}\widehat{\mathrm{A}}_{1}$ under the map

${p_{2}\circ
p_{3}:\mathrm{TMF}_{*}\mathrm{A}_{1}}$${\mathrm{TMF}_{*}\mathrm{M}}$

is nonzero if and only if $p_{2}(p_{3}(a))\in{\sf tmf}_{*}\mathrm{M}$ is
nonzero. Similarly, the image of
$\ell(a)\in\mathrm{TMF}_{*}\widehat{\mathrm{A}}_{1}$ under the map

${p_{1}\circ p_{2}\circ
p_{3}:\mathrm{TMF}_{*}\widehat{\mathrm{A}}_{1}}$${\mathrm{TMF}_{*-6}}$

is zero if and only if $p_{1}(p_{2}(p_{3}(a)))\in\mathrm{TMF}_{*-6}$.
Therefore, the proof of 3.6 can follow the exact same arguments to that of 1
provided

$8\Delta,4\Delta^{2},8\Delta^{3},(\eta\Delta)^{3},2\Delta^{4},8\Delta^{5},8\Delta^{7}$

are not in the $\mathrm{K}(2)$-local Hurewicz image of $\mathrm{TMF}_{*}$
(also see Remark 3.5).

###### Proof of 3.6.

Since the elements

$8\Delta,4\Delta^{2},8\Delta^{3},2\Delta^{4},8\Delta^{5},8\Delta^{7}$

are of infinite order, and $\pi_{*}\widehat{\SS}$ is a finite group in degrees
$24,48,72,96,120$, and $168$ (see [BSSW24, Theorem A]), they cannot be in the
$\mathrm{K}(2)$-local Hurewicz image. Further, $(\eta\Delta)^{3}$ is also not
in the Hurewicz image by Lemma 3.8. Then the proof of 1 goes through _mutatis
mutandis_ to yield the result. ∎

######  Lemma 3.8.

The element $(\eta\Delta)^{3}$ is not in the image of the
$\mathrm{K}(2)$-local Hurewicz map of $\mathrm{TMF}$

(20) ${{\sf h}_{\mathrm{TMF}}:\pi_{*}\widehat{\SS}}$${\mathrm{TMF}_{*}.}$

###### Proof.

By [Lau04, Corollary 3] there is a $K(1)$-local equivalence

$v_{1}^{-1}\mathrm{TMF}\simeq\mathrm{KO}\llbracket j^{\pm 1}\rrbracket,$

which implies that the composition of the $\mathrm{K}(2)$-local Hurewicz map
with the localization map to $v_{1}^{-1}\mathrm{TMF}$

${\mathrm{KO}_{*}}$${\pi_{*}\widehat{\SS}}$${\pi_{*}\mathrm{TMF}}$${v_{1}^{-1}\mathrm{TMF}_{*}\cong\mathrm{KO}_{*}\llbracket
j^{\pm 1}\rrbracket}$

factors through $\mathrm{KO}_{*}$. Then we simply implement the arguments of
[BMQ23, Theorem 6.1].

More precisely, we observe that $(\eta\Delta)^{3}$ lifts to an element in
$\mathrm{TMF}_{\ast}\mathrm{M}({\infty}),$ where
$\mathrm{M}({\infty}):=\underset{i\to\infty}{\mathrm{colim}}\ \mathrm{M}(i)$
(see 3.11), whose image after inverting $c_{4}$ is

$\overline{v_{1}^{38}}j^{-3}\in
v_{1}^{-1}\mathrm{TMF}_{\ast}\mathrm{M}({\infty})$

in the notation of [BMQ23, $\mathsection$6]. If $(\eta\Delta)^{3}$ is in the
image of the Hurewicz map (20), then $\overline{v_{1}^{38}}j^{-3}$ must also
be in the image of $\ell\circ{\sf h}_{\mathrm{TMF}}$ in the diagram

${\mathrm{KO}_{*}\mathrm{M}({\infty})}$${\pi_{*}\widehat{\mathrm{M}({\infty})}}$${\mathrm{TMF}_{*}\mathrm{M}({\infty})}$${v_{1}^{-1}\mathrm{TMF}_{*}\mathrm{M}({\infty})}$$\scriptstyle{{\sf
h}_{\mathrm{TMF}}}$$\scriptstyle{\ell}$

which contradicts the fact that $\ell\circ{\sf h}_{\mathrm{TMF}}$ factors
through $\mathrm{KO}_{*}\mathrm{M}(\infty)$. ∎

###  3.3.  Infinite families in $\mathrm{T}(2)$-local stable stems

Notice that the unit map of the $\mathrm{K}(2)$-local sphere spectrum factors
through $\SS_{\mathrm{T}(2)}$ (see (1)). Therefore, the proof of 3.6 also
shows that the elements listed in 1 are nontrivial after
$\mathrm{T}(2)$-localization completing the proof of 2. Nevertheless, we take
this opportunity to give another proof of this fact independent of
$\mathrm{K}(2)$-local Hurewicz map calculations of Section 3.2. Instead, we
make use of a $v_{2}$-self-map of $\mathrm{A}_{1}$.

######  Notation 3.9.

For a spectrum $\mathrm{E}$ and a finite spectrum $\mathrm{X}$ with a
$v_{n}$–self-map $v:\Sigma^{|v|}\mathrm{X}\to\mathrm{X}$, let

$\Phi_{\mathrm{X}}(\mathrm{E}):=\underset{\to}{\operatorname{colim}}\
\\{\mathrm{E}^{\mathrm{X}}\overset{{\sf v}^{*}}{\longrightarrow}\Sigma^{-|{\sf
v}|}\mathrm{E}^{\mathrm{X}}\overset{{\sf
v}^{*}}{\longrightarrow}\Sigma^{-2|{\sf
v}|}\mathrm{E}^{\mathrm{X}}\longrightarrow\dots\\}.$

There is a natural map from $\mathrm{E}$ to $\Phi_{\mathrm{X}}(\mathrm{E})$
which we will denote by $\alpha$ (sometimes with subscripts).

It is well-known that $\mathrm{A}_{1}$ admits a $v_{2}^{32}$-self-map

${v:\Sigma^{192}\mathrm{A}_{1}}$${\mathrm{A}_{1}}$

detected by $\Delta^{8}\in{\sf tmf}_{*}$ [BEM17]. Therefore, for any lift
$\tilde{a}\in\pi_{k}\mathrm{A}_{1}$ of $a\in{\sf tmf}_{k}(\mathrm{A}_{1})$ we
have

${\sf h}_{{\sf tmf}}(v^{i}\cdot\tilde{a})=\Delta^{8i}\cdot a$

for all $i\in\mathbb{N}$.

Suppose $\tilde{a}\in\pi_{k}\mathrm{A}_{1}$ pinches to a nonzero element ${\sf
p}_{*}(\tilde{a})$ in $\pi_{k-6}(\SS)$ which is listed in 1. Then, we may
choose $\tilde{m}_{k-5+192i}$ of (18) as

$\tilde{m}_{k-5+192i}:=p_{2}(p_{3}(v^{i}\cdot\tilde{a}))$

for all $i\in\mathbb{N}$ in the proof of 3.4 (II). In that case, the element
$\tilde{s}_{k-6+192i}$ in the infinite family of (17) will be equal to ${\sf
p}_{*}(v^{i}\cdot\tilde{a})$ for all $i\in\mathbb{N}$. Consequently:

######  Lemma 3.10.

The image of an element $\tilde{\sf s}\in\pi_{*}(\SS)$ listed in 1 under the
map

${\alpha_{1}:\pi_{*}\SS}$${\pi_{*}(\Phi_{\Sigma^{-6}\mathrm{A}_{1}}(\SS))}$

is nonzero.

######  Notation 3.11.

Let $\mathrm{M}({\sf i},{\sf j})$ denote the cofiber of a $v_{1}^{\sf
j}$-self-map on $\mathrm{M}({\sf i})$, the cofiber of multiplication by
$2^{\sf i}$ on $\SS$.

######  Proposition 3.12.

The map ${\sf p}:\Sigma^{-6}\mathrm{A}_{1}\longrightarrow\SS$ factors through
$\Sigma^{-10}\mathrm{M}(1,4)$.

###### Proof.

Let ${\sf ko}$ denote the connective real $\mathrm{K}$-theory. Since ${\sf
ko}_{6}\mathrm{M}=0$, it follows that the composite

${\Sigma^{6}\mathbb{S}}$${\Sigma^{6}\mathrm{Y}}$${\mathrm{Y}}$${\Sigma^{2}\mathrm{M}}$$\scriptstyle{v_{1}^{3}}$$\scriptstyle{{\sf
p}_{2}}$

is nonzero in ${\sf ko}$-homology, and hence, in stable homotopy. Using the
fact that ${\sf ko}\wedge\mathrm{Y}\simeq{\sf k}(1)$, a ${\sf ko}$-homology
analysis reveals that the composite

${\Sigma^{8}\mathrm{Y}}$${\mathrm{Y}}$${\Sigma^{2}\mathrm{M}}$${\Sigma^{3}\SS}$$\scriptstyle{v_{1}^{4}}$$\scriptstyle{{\sf
p}_{2}}$$\scriptstyle{{\sf p}_{1}}$

is nonzero. Consequently, we have a commutative diagram (also see [DM81, p.
616])

${\Sigma^{6}\mathrm{Y}}$${\Sigma^{4}\mathrm{Y}}$${\Sigma^{8}{\mathrm{M}}}$${{\mathrm{M}}}$$\scriptstyle{v_{1}}$$\scriptstyle{{\sf
p}_{2}}$$\scriptstyle{{\sf p}_{2}\circ v_{1}^{3}}$$\scriptstyle{v_{1}^{4}}$

which implies there is a map
$\Sigma^{4}\mathrm{A}_{1}\longrightarrow\mathrm{M}(1,4)$ which factors the
pinch map of $\Sigma^{4}\mathrm{A}_{1}$ to its top cell. ∎

A consequence of Proposition 3.12 is that we have a directed system

(21)
${\Sigma^{-6}\mathrm{A}_{1}}$${\Sigma^{-10}\mathrm{M}(1,4)}$${\Sigma^{-18}\mathrm{M}(2,8)}$${\dots}$${\dots}$${\SS}$$\scriptstyle{\sf
p}$

of type $2$ spectra which is cofinal among all type $2$ spectra with a ‘pinch’
map to $\SS$.

######  Notation 3.13.

Let $\Phi_{k}(-)$ denote $\Phi_{\mathrm{V}_{k}}(-)$, where $\mathrm{V}_{k}$ is
the $k$-th entry of the sequence (21).

######  Theorem 3.14.

All elements listed in 1 have nonzero images in the $\mathrm{T}(2)$-local
stable stems.

###### Proof.

We will make use of the standard theory of Bousfield-Kuhn functors (see
[Kuh08]) which implies

$\SS_{\mathrm{T}(2)}\simeq\underset{\leftarrow}{\lim}\ \Phi_{k}(\SS).$

Since any element ${\sf s}\in\pi_{*}(\SS)$ that is listed in 1 has a nonzero
image under $\alpha_{1}$ (see Lemma 3.10), it follows from the diagram

${\pi_{*}(\SS)}$${\dots}$${\pi_{*}(\Phi_{1}(\SS))}$${\pi_{*}(\Phi_{2}(\SS))}$${\pi_{*}(\Phi_{3}(\SS))}$${\text{
}\cdots}$$\scriptstyle{\alpha_{1}}$$\scriptstyle{\alpha_{2}}$

that its image in $\underset{\leftarrow}{\lim}\ \pi_{*}(\Phi_{k}(\SS))$ is
also nonzero. Then the result follows from the fact that the natural map

${\pi_{*}\left(\SS_{\mathrm{T}(2)}\right)\cong\pi_{*}\left(\underset{\leftarrow}{\lim}\
{\Phi_{k}(\SS)}\right)}$${\underset{\leftarrow}{\lim}\
\pi_{*}(\Phi_{k}(\SS))}$

is a surjection (with Milnor $\lim^{1}$ term as the kernel). ∎

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