twoside=false

#  $\mathbb{R}$-motivic $v_{1}$-periodic homotopy

Eva Belmont  Department of Mathematics, University of California San Diego, La
Jolla, CA 92093, USA<EMAIL_ADDRESS>,  Daniel C. Isaksen  Department of
Mathematics, Wayne State University, Detroit, MI 48202, USA<EMAIL_ADDRESS>and  Hana Jia Kong  School of Mathematics, Institute for Advanced Study,
Princeton, NJ 08540, USA<EMAIL_ADDRESS>
###### Abstract.

We compute the $v_{1}$-periodic $\mathbb{R}$-motivic stable homotopy groups.
The main tool is the effective slice spectral sequence. Along the way, we also
analyze $\mathbb{C}$-motivic and $\eta$-periodic $v_{1}$-periodic homotopy
from the same perspective.

###### Key words and phrases:

motivic stable homotopy group, $v_{1}$-periodicity, effective spectral
sequence

###### 2010 Mathematics Subject Classification:

Primary 55Q10; Secondary 14F42, 55Q51, 55T99

The second author was supported by NSF grant DMS-1904241. The third author was
supported by the National Science Foundation under Grant No. DMS-1926686. This
manuscript answers a question posed by Mark Behrens to the second author in
2010 at the Conference on Homotopy Theory and Derived Algebraic Geometry,
Fields Institute, Toronto, Ontario, Canada. The authors thank William
Balderrama, Robert Bruner, and John Rognes for helpful discussions.

##  1\. Introduction

The computation of the stable homotopy groups of spheres is a difficult but
central problem of stable homotopy theory. There is much that we do not know
about stable homotopy. However, the $v_{1}$-periodic stable homotopy groups
(also known for historical reasons as the image of $j$) are completely
understood, and they have interesting number-theoretic properties.

The goal of this article is to explore $v_{1}$-periodic stable homotopy in the
$\mathbb{R}$-motivic context. This choice of ground field represents a middle
ground between the well-understood $\mathbb{C}$-motivic situation and the much
more difficult situation of an arbitrary field, in which arithmetic
necessarily enters into the picture.

From our perspective, the field $\mathbb{R}$ introduces just one piece of
arithmetic: the failure of $-1$ to have a square root. This leads to
complications in $\mathbb{R}$-motivic homotopical computations, but they can
be managed with care and attention to detail.

Classically, $v_{1}$-periodic homotopy is detected by the connective spectrum
$j^{\mathrm{top}}$, which is defined to be the fiber of a map
$\mathrm{ko}^{\mathrm{top}}\overset{\psi^{3}-1}{\longrightarrow}\Sigma^{4}\mathrm{ksp}^{\mathrm{top}}$,
where $\mathrm{ko}^{\mathrm{top}}$ is the connective real $K$-theory spectrum,
$\mathrm{ksp}^{\mathrm{top}}$ is the connective symplectic $K$-theory
spectrum, and $\psi^{3}$ is an Adams operation. (The “$\mathrm{top}$”
superscripts indicate that we are discussing the classical context here,
rather than the motivic context.)

In fact, $\mathrm{ko}^{\mathrm{top}}$ itself is the more natural target for
the map $\psi^{3}-1$. However, the fiber of
$\mathrm{ko}^{\mathrm{top}}\overset{\psi^{3}-1}{\longrightarrow}\mathrm{ko}^{\mathrm{top}}$
has a minor defect. It has some additional homotopy classes in stems $-1$,
$0$, and $1$ that do not correspond to homotopy classes for the sphere
spectrum. In other words, the map from $S^{0}$ to this fiber is not surjective
in homotopy. If we change the target of $\psi^{3}-1$ from
$\mathrm{ko}^{\mathrm{top}}$ to its connective cover
$\Sigma^{4}\mathrm{ksp}^{\mathrm{top}}$, then this problem disappears, and the
map from $S^{0}$ to the fiber is onto in homotopy.

It is possible to mimic these constructions in motivic stable homotopy theory
[BH20]. At the prime $2$, one can define the motivic connective spectrum $j$
to be the fiber of a map
$\mathrm{ko}\overset{\psi^{3}-1}{\longrightarrow}\Sigma^{4,2}\mathrm{ksp}$,
where $\mathrm{ko}$ is the very effective connective Hermitian $K$-theory
spectrum, $\mathrm{ksp}$ is defined in terms of very effective covers of
$\mathrm{ko}$, and $\psi^{3}$ is a motivic lift of an Adams operation.

However, from a computational perspective, this definition of $j$ introduces
apparently unnecessary complications. It is possible to compute the homotopy
of $\mathbb{R}$-motivic $j$ using the techniques that appear later in this
manuscript. However, the computation is slightly messy, involving some
exceptional differentials and exceptional hidden extensions in low dimensions.
In any case, the homotopy of the $\mathbb{R}$-motivic sphere does not surject
onto the homotopy of $\mathbb{R}$-motivic $j$. In other words, the main
rationale for using $\mathrm{ksp}$ in the first place does not apply in the
motivic situation.

On the other hand, the computation of the homotopy of the $\mathbb{R}$-motivic
fiber of $\mathrm{ko}\overset{\psi^{3}-1}{\longrightarrow}\mathrm{ko}$ is much
cleaner. Moreover, it tells us just as much about $v_{1}$-periodic
$\mathbb{R}$-motivic homotopy as $j$. In other words, it has all of the
computational advantages of $j$, while avoiding some unfortunate
complications.

Consequently, in this manuscript, we will be solely concerned with the fiber
of $\mathrm{ko}\overset{\psi^{3}-1}{\longrightarrow}\mathrm{ko}$. We use the
notation $L$ for this fiber in order to avoid confusion with the traditional
meaning of $j$. The symbol $L$ is meant to draw a connection to the classical
$K(1)$-local sphere $L_{K(1)}S^{0}$, which is the fiber of
$\mathrm{KO}^{\mathrm{top}}\overset{\psi^{3}-1}{\longrightarrow}\mathrm{KO}^{\mathrm{top}}$.

However, the homotopy of the $\mathbb{R}$-motivic spheres does not surject
onto the homotopy of $\mathbb{R}$-motivic $L$. It is possible that we may have
not yet constructed the “correct” motivic version of the classical connective
spectrum $j^{\mathrm{top}}$. These considerations raise questions about vector
bundles and the motivic Adams conjecture. We make no attempt to study these
more geometric issues.

We claim to compute the $v_{1}$-periodic $\mathbb{R}$-motivic stable homotopy
groups, but this claim deserves some clarification. We do not use an intrinsic
definition of $v_{1}$-periodic $\mathbb{R}$-motivic homotopy, although such a
definition could probably be formulated in terms of the motivic $K(1)$-local
sphere. However, the theory of motivic $K(1)$-localization has not yet been
suitably developed.

Rather, we merely compute the homotopy of $L$, and we observe that it detects
large-scale structure in the stable homotopy of the $\mathbb{R}$-motivic
sphere, which was described in a range in [BI20]. In other words, we have a
practical description of $\mathbb{R}$-motivic $v_{1}$-periodic homotopy, not a
theoretical one.

The careful reader may object that our approach with effective spectral
sequences is long-winded and unnecessarily complicated. In fact, the homotopy
of $L$ could be determined by direct analysis of the long exact sequence
associated to the defining fiber sequence for $L$. However, there is a
disadvantage in this direct approach. We find that the effective filtration is
useful additional information about the homotopy of $L$ that helps us
understand the computation. The effective filtration is part of the “higher
structure” of the homotopy of $L$. While we have no immediate uses for this
higher structure, we know from experience that it inevitably becomes important
in deeper homotopical analyses.

###  1.1. Charts

We provide a series of charts that display the effective spectral sequences
for $\mathrm{ko}$ and $L$, as well as their $\mathbb{C}$-motivic counterparts.
We consider these charts to be the central achievement of this manuscript. We
encourage the reader to rely heavily on the charts. In a sense, they provide
an illustrated guide to our computations.

Caution must be exercised in the comparison to [BI20] since the Adams
filtrations and effective filtrations are different. As in [BI20], our charts
consider each coweight separately; we have found that this is a practical way
of studying $\mathbb{R}$-motivic homotopy groups. Periodicity by $\tau^{4}$
(which is not a permanent cycle, but should be thought of as a periodicity
operator in coweight 4) allows us to give a fairly compact depiction of the
homotopy of $L$ in coweights congruent to 0, 1, and 2 modulo 4; see Figures
13, 14, and 15.

The homotopy of $L$ in coweights congruent to 3 modulo 4 is much more
interesting but harder to describe. See Figures 17 and 18.

###  1.2. Completions

We are computing exclusively in the $2$-complete context. This simplifies all
questions surrounding convergence of spectral sequences. Also, the final
computational $2$-complete answers are easier to state than their
$2$-localized or integral counterparts.

We generally omit completions from our notation for brevity. For example, we
write $\mathbb{Z}$ for the $2$-adic integers, and we write $\mathrm{KO}$ for
the $2$-completed $\mathbb{R}$-motivic Hermitian $K$-theory spectrum.

Section 2.3 discusses these topics in slightly more detail.

###  1.3. Regarding the element $2$

When passing from the effective $E_{\infty}$-page to stable homotopy groups,
one must choose homotopy elements that are represented by each element of the
$E_{\infty}$-page. For the element $2$ in the $E_{\infty}$-page, there is more
than one choice in $\pi_{0,0}$ because of the presence of elements in the
$E_{\infty}$-page in higher effective filtration.

From the perspective of abelian groups, the element $2=1+1$ is the obvious
choice of homotopy element. However, there is another element $\mathsf{h}$,
also detected by $2$ in the effective spectral sequence, that turns out to be
a much more convenient choice. The difference between $\mathsf{h}$ and $2$ in
homotopy is detected by the element $\rho h_{1}$ in higher filtration (to be
discussed later). Experience has shown that the motivic stable homotopy groups
are easier to describe in terms of $\mathsf{h}$ than in terms of $2$. For
example, we have the relations $\mathsf{h}\rho=0$ and $\mathsf{h}\eta=0$,
where $\rho$ and $\eta$ are the homotopy elements detected by $\rho$ and
$h_{1}$ respectively. However, neither $2\rho$ nor $2\eta$ are zero.

There are two additional reasons why the element $\mathsf{h}$ plays a central
role. First, it corresponds to the hyperbolic plane under the isomorphism
between motivic $\pi_{0,0}$ and the Grothendieck–Witt group of symmetric
bilinear forms [Mor04]. Second, it plays the role of the zeroth Hopf map, in
the sense that the Steenrod operations on its cofiber are simpler than the
Steenrod operations for the cofiber of $2$.

Consequently, instead of describing motivic stable homotopy groups as a module
over the $2$-adic integers $\mathbb{Z}_{2}$ (i.e., in terms of the action of
$2$), it is easier to describe the homotopy groups in terms of the action of
$\mathsf{h}$.

###  1.4. Future directions

Our work points toward several open problems.

######  Problem 1.1.

Compute motivic $v_{1}$-periodic homotopy over an arbitrary base field. See
Section 1.5 for further discussion.

######  Problem 1.2.

Recompute the homotopy of $L$ using the $\mathbb{R}$-motivic Adams spectral
sequence. This would be a useful comparison object for further computations
with the Adams spectral sequence for the $\mathbb{R}$-motivic sphere. The
classical Adams spectral sequence for $j^{\mathrm{top}}$ was studied by Davis
[Dav75], but it was only recently computed completely by Bruner and Rognes
[BR21]. We are proposing a motivic analogue of their results.

######  Problem 1.3.

Carry out the effective spectral sequence for the $\mathbb{R}$-motivic sphere
in a range. These computations would serve as a useful companion to
$\mathbb{R}$-motivic Adams spectral sequence computations [BI20]. The idea is
to build on the techniques that are developed in this manuscript.

######  Problem 1.4.

Compute the $v_{1}$-periodic $C_{2}$-equivariant stable homotopy groups. More
precisely, carry out the $C_{2}$-effective spectral sequence for a
$C_{2}$-equivariant version of $L$. The details will be similar to but more
complicated than the computations in this manuscript. See [Kon21] for the
effective approach to the $C_{2}$-equivariant version of $\mathrm{ko}$.
Alternatively, one might compute the $v_{1}$-periodic $C_{2}$-equivariant
stable homotopy groups by periodicizing the $v_{1}$-periodic
$\mathbb{R}$-motivic groups with respect to $\tau$, as considered by Behrens
and Shah [BS20].

Recall that the $\mathbb{R}$-motivic and $C_{2}$-equivariant stable homotopy
groups are isomorphic in a range [BGI21]. Consequently, we anticipate that
some version of the structure described in this manuscript appears in the
$C_{2}$-equivariant context as well.

In the equivariant context, we mention Balderrama’s [Bal21] computation of the
homotopy groups of the Borel $C_{2}$-equivariant $K(1)$-local sphere, using
techniques that are entirely different from ours. Roughly speaking, Balderrama
computes the $\tau^{4}v_{1}^{4}$-periodicization of our result.

######  Problem 1.5.

Study $K(1)$-localization in the motivic context, which ought to be something
like localization with respect to $KGL/2$. Compute $K(1)$-local motivic
homotopy. This would provide an intrinsic definition of $v_{1}$-periodic
homotopy that would improve upon the practical computational perspective of
this manuscript.

A guide to the motivic situation could lie in the work of Balderrama [Bal21]
and Carrick [Car19] on equivariant localizations.

###  1.5. Towards $v_{1}$-periodic homotopy over general base fields

Our explicit computations point the way towards a complete computation of the
$v_{1}$-periodic motivic stable homotopy groups over arbitrary fields. The
situation here is analogous to the $\eta$-periodic $\mathbb{R}$-motivic
computations of [GI16], which foreshadowed the more general $\eta$-periodic
computations of [Wil18], [OR20], and [BH20].

######  Problem 1.6.

Let $k$ be an arbitrary field of characteristic different from $2$. Let
$GW(k)$ be the Grothendieck–Witt ring of symmetric bilinear forms over $k$.
Describe the 2-primary homotopy groups of the $k$-motivic spectrum $L$ in
terms of the cokernels and kernels of multiplication by various powers of $2$
and of $\mathsf{h}$ on $GW(k)$.

Problem 1.6 is stated only in terms of 2-primary computations because that is
the most interesting part. We expect that the generalization to odd primes is
straightforward.

The exact powers of $2$ and $\mathsf{h}$ that are required in Problem 1.6
depend not only on the coweight but also on the stem. Figures 17 and 18 show
that $2^{v(j)+3}$ is the relevant power of $2$ in most stems in coweight
$4j-1$. Here $v(j)$ is the $2$-adic valuation of $j$, i.e., largest number $v$
such that $2^{v}$ divides $j$. In coweight $4j-1$ and stem $4i-1$, we see
larger powers of $2$, as well as powers of $\mathsf{h}$.

Similar observations apply to the kernels that contribute to coweight $4i$.

###  1.6. Outline

Section 2 contains some background information that we will need to get
started on our computations. We briefly discuss convergence of the effective
spectral sequences that we will use. We recall some results of
Bachmann–Hopkins [BH20] about motivic Adams operations and of
Ananyevskiy–Röndigs–Østvær [ARØ20] about the slices of $\mathrm{ko}$.

The final subsection of Section 2 makes a precise connection between
$\mathbb{R}$-motivic computations and $\mathbb{C}$-motivic computations.
Namely, taking the cofiber of the map $\rho$ changes $\mathbb{R}$-motivic
computations into the analogous $\mathbb{C}$-motivic computations. This idea
originated in [BI20]. This simple observation has surprisingly powerful
consequences, especially for the analysis of hidden extensions. It allows us
to leverage well-understood $\mathbb{C}$-motivic information into
$\mathbb{R}$-motivic information.

In Section 2, we have taken some care to eliminate details that we do not use.
In other words, Section 2 describes the minimal hypotheses necessary in order
to carry out our computations.

Section 3 considers $\mathbb{C}$-motivic computations, which play two roles in
our work. First, they serve as a warmup to the more intricate
$\mathbb{R}$-motivic computations. Second, the comparison between
$\mathbb{R}$-motivic and $\mathbb{C}$-motivic homotopy is a necessary
ingredient for our computations. In this section, we describe the effective
spectral sequence for $\mathrm{ko}^{\mathbb{C}}$. This material is well-known,
since it is the same (up to regrading) as the $\mathbb{C}$-motivic
Adams–Novikov spectral sequence for $\mathrm{ko}^{\mathbb{C}}$, which is
nearly the same as the classical Adams–Novikov spectral sequence for
$\mathrm{ko}^{\mathrm{top}}$. We then use the fiber sequence

$L^{\mathbb{C}}\longrightarrow\mathrm{ko}^{\mathbb{C}}\overset{\psi^{3}-1}{\longrightarrow}\mathrm{ko}^{\mathbb{C}}$

in order to determine the $E_{1}$-page of the effective spectral sequence for
$L^{\mathbb{C}}$.

We next completely analyze the effective spectral sequence for the
$\eta$-periodicization $L^{\mathbb{C}}[\eta^{-1}]$. The $\eta$-periodic
spectral sequence is significantly simpler than the unperiodicized spectral
sequence. We note the close similarity between the homotopy of
$L^{\mathbb{C}}[\eta^{-1}]$ and the computations of Andrew–Miller [AM17].

The $\eta$-periodic effective differentials completely determine the
unperiodicized effective differentials for $L^{\mathbb{C}}$. Finally, we
determine hidden extensions in the effective $E_{\infty}$-page for
$L^{\mathbb{C}}$.

Section 3 completely computes the homotopy of $L^{\mathbb{C}}$, but the
effective spectral sequence is not necessarily the simplest way of obtaining
the computation. Nevertheless, we have chosen this approach because of its
relationship to our later $\mathbb{R}$-motivic computations.

Section 4 analyzes the effective spectral sequence for $\mathbb{R}$-motivic
$\mathrm{ko}$, including all differentials and hidden extensions. The
$E_{1}$-page is readily determined from the work of Ananyevskiy–Röndigs–Østvær
[ARØ20] on the slices of $\mathrm{ko}$. We draw particular attention to the
formula

(1.1) $(\tau h_{1})^{2}=\tau^{2}\cdot h_{1}^{2}+\rho^{2}\cdot v_{1}^{2}.$

This formula has a major impact on the shape of the answers that we obtain. In
a sense, our work merely draws algebraic conclusions from Equation (1.1) and
$\eta$-periodic information. The hidden extensions in the effective
$E_{\infty}$-page for $\mathrm{ko}$ are easily determined by comparison to the
$\mathbb{C}$-motivic case, using the relationship between $\mathbb{C}$-motivic
and $\mathbb{R}$-motivic homotopy that is described in Section 2.4.

Our computation of the homotopy of $\mathbb{R}$-motivic $\mathrm{ko}$ is not
original. See [Kon21] for a $C_{2}$-equivariant analogue of the effective
spectral sequence for $\mathrm{ko}$. The $\mathbb{R}$-motivic computation can
be extracted from the $C_{2}$-equivariant computation by dropping the
“negative cone” elements. Also, Hill [Hil11] computed the Adams spectral
sequence for $\mathrm{ko}$, although the $\mathbb{R}$-motivic spectrum
$\mathrm{ko}$ had not yet been constructed at the time.

The next step, undertaken in Section 4.2, is to analyze the effect of
$\psi^{3}$ on the effective spectral sequence of $\mathrm{ko}$. This follows
from a straightforward comparison to the classical case, together with careful
bookkeeping. In turn, this leads to a complete understanding of the effective
$E_{1}$-page of $L$, which is described in Section 5.1. Again, this is mostly
a matter of careful bookkeeping.

Section 5.2 completely analyzes the effective spectral sequence for
$\eta$-periodic $L[\eta^{-1}]$. This information is essentially already well-
known, either from [GI16] or from Ormsby–Röndigs [OR20], although those
references do not specifically mention $L$.

As in the $\mathbb{C}$-motivic situation of Section 3, $\eta$-periodic
information yields everything that we need to know about the unperiodic
situation, including all multiplicative relations in the effective
$E_{1}$-page for $L$ (see Section 5.3) and all differentials (see Sections 5.4
and 5.5). We again emphasize the significance of Equation (1.1) in carrying
out the details. Finally, Section 5.6 studies hidden extensions in the
effective $E_{\infty}$-page for $L$. As for $\mathrm{ko}$, these hidden
extensions follow by comparison to the $\mathbb{C}$-motivic case.

###  1.7. Notation

We use the following notation conventions.

  * •

$v(n)$ is the $2$-adic valuation of $n$, i.e., the largest integer $v$ such
that $2^{v}$ divides $n$.

  * •

Except in Section 2, everything is implicitly $2$-completed. For example, $S$
is actually the $2$-complete $\mathbb{R}$-motivic sphere spectrum. Similarly,
$\mathbb{Z}$ is the $2$-adic integers.

  * •

$s_{*}(X)$ are the slices of a motivic spectrum $X$.

  * •

$E_{r}(X)$ is the $E_{r}$-page of the effective spectral sequence for a
motivic spectrum $X$.

  * •

We find the effective slice filtration to be slightly inconvenient for our
purposes. We prefer to use the “Adams–Novikov filtration”, which equals twice
the effective filtration minus the stem.

  * •

Coweight equals the stem minus the motivic weight.

  * •

Elements in $E_{r}(X)$ are tri-graded. We write $E^{s,f,w}_{r}(X)$ to denote
the part with topological dimension $s$, Adams–Novikov filtration $f$, and
motivic weight $w$.

  * •

We use unadorned symbols for $\mathbb{R}$-motivic spectra. For example,
$\mathrm{ko}$ is the very effective cover of the $\mathbb{R}$-motivic
Hermitian $K$-theory spectrum.

  * •

$X^{\mathbb{C}}$ is the $\mathbb{C}$-motivic extension-of-scalars spectrum of
an $\mathbb{R}$-motivic spectrum $X$.

  * •

$X^{\mathrm{top}}$ is the Betti realization of an $\mathbb{R}$-motivic
spectrum $X$.

  * •

$S$ is the $\mathbb{R}$-motivic sphere spectrum.

  * •

$\mathrm{KO}$ is the $\mathbb{R}$-motivic spectrum that represents Hermitian
$K$-theory (also known as $\mathrm{KQ}$).

  * •

$\mathrm{ko}$ is the very effective connective cover of $\mathrm{KO}$.

  * •

$HA$ is the $\mathbb{R}$-motivic Eilenberg–Mac Lane spectrum on the group $A$.

  * •

$\psi^{3}$ is an Adams operation. We use the same symbol in the
$\mathbb{R}$-motivic, $\mathbb{C}$-motivic, and classical situations.

  * •

$L$ is the fiber of
$\mathrm{ko}\overset{\psi^{3}-1}{\longrightarrow}\mathrm{ko}$.

  * •

$\Sigma^{s,w}X$ is a (bigraded) suspension of a motivic spectrum $X$.

  * •

$\pi_{*,*}(X)$ are the bigraded stable homotopy groups of an
$\mathbb{R}$-motivic or $\mathbb{C}$-motivic spectrum.

  * •

Recall that $\varepsilon$ is the motivic homotopy class that is represented by
the twist map $S\wedge S\to S\wedge S$, where $S$ is the motivic sphere
spectrum. Let $\mathsf{h}$ be the element $1-\varepsilon$, which corresponds
to the hyperbolic plane under the isomorphism between $\pi_{0,0}(S)$ and the
Grothendieck-Witt ring $GW(\mathbb{R})$ [Mor04].

  * •

The element $\rho$ belongs to the $\mathbb{R}$-motivic homology of a point. It
is the class represented by $-1$ in the Milnor $K$-theory of $\mathbb{R}$.
Since $\rho$ survives all of the spectral sequences under consideration, we
use the same symbol for the corresponding homotopy class. However, there is a
choice of homotopy class represented by $\rho$ because of the presence of
elements in higher filtration. There is an inconsistency in the literature
about this choice. Following [Bac18], we define $\rho$ such that
$\varepsilon=\rho\eta-1$, or equivalently $2=\rho\eta+\mathsf{h}$.

We frequently use names for indecomposables that consist of more than one
symbol. For example, Theorem 2.1 discusses the indecomposable element
$v_{1}^{2}$ of the effective $E_{1}$-page for $\mathrm{ko}^{\mathbb{C}}$.
These longer names are slightly more cumbersome. This is especially the case
when we consider products. We will use expressions of the form $x\cdot y$ for
clarity.

On the other hand, our names are particularly convenient because they reflect
the origins of the elements in terms of the spectral sequences that we use.
For example, consider the indecomposable element $2v_{1}^{2}$ of the effective
$E_{\infty}$-page for $\mathrm{ko}^{\mathbb{C}}$, as discussed in Theorem 3.3
(see also Figure 2). This name reflects the element’s origin in the effective
$E_{1}$-page. It also illuminates relations such as

$2v_{1}^{2}\cdot 2v_{1}^{2}=4\cdot v_{1}^{4}$

However, one must be careful about possible error terms in such formulas; see
especially Equation (1.1).

##  2\. Background

In this section only, we write $\mathrm{ko}$ for the integral version of the
very effective cover of the Hermitian $K$-theory spectrum, and we use the
usual decorations to indicate localizations and completions of $\mathrm{ko}$.
In the rest of the manuscript, $\mathrm{ko}$ is assumed to be $2$-completed.

###  2.1. The effective slices of $\mathrm{ko}$

We recall the structure of the effective slices of $\mathrm{ko}$.

######  Theorem 2.1 ([ARØ20, Theorem 17]).

The slices of $\mathrm{ko}$ are

$s_{*}(\mathrm{ko})=H\mathbb{Z}[h_{1},v_{1}^{2}]/(2h_{1}),$

where $v_{1}^{2}$ and $h_{1}$ have degrees $(4,0,2)$ and $(1,1,1)$
respectively.

We explain the expression in Theorem 2.1. Each monomial of degree $(s,f,w)$
contributes a summand of $\Sigma^{s,w}HA$ in the
$\left(\frac{s+f}{2}\right)$th slice. Here $HA$ is the motivic Eilenberg–Mac
Lane spectrum associated to $A$. The abelian group $A$ is $\mathbb{F}_{2}$
when the monomial is $2$-torsion, and is $\mathbb{Z}$ when the monomial is
torsion free. We list the first three slices as examples:

$\displaystyle s_{0}(\mathrm{ko})$ $\displaystyle=H\mathbb{Z}\\{1\\},$
$\displaystyle s_{1}(\mathrm{ko})$
$\displaystyle=\Sigma^{1,1}H\mathbb{F}_{2}\\{h_{1}\\},$ $\displaystyle
s_{2}(\mathrm{ko})$
$\displaystyle=\Sigma^{2,2}H\mathbb{F}_{2}\\{h_{1}^{2}\\}\vee\Sigma^{4,2}H\mathbb{Z}\\{v_{1}^{2}\\}.$

Beware that the multiplicative structure of $s_{*}(\mathrm{ko})$ is not
completely captured by the notation in Theorem 2.1. The essential
multiplicative relation is Equation (1.1), which follows immediately from the
general formulas in [ARØ20].

######  Remark 2.2.

The calculation of the slices of the motivic sphere spectrum, due to Röndigs,
Spitzweck, and Østvær [RSØ19], is commonly expressed at the prime $2$ as

$s_{*}(S)=H\mathbb{Z}\otimes\operatorname{Ext}_{BP_{*}BP}^{*,*}(BP_{*},BP_{*}).$

Analogously, Theorem 2.1 says that

$s_{*}(\mathrm{ko})=H\mathbb{Z}\otimes\operatorname{Ext}_{BP_{*}BP}^{*,*}(BP_{*},BP_{*}(\mathrm{ko}^{\mathrm{top}})).$

However, we do not know of a general theorem relating the slices of a motivic
spectrum with the Adams–Novikov $E_{2}$-page for its topological counterpart.

###  2.2. The Adams operation $\psi^{3}$ and the spectrum $L$

Bachmann and Hopkins [BH20] constructed a motivic analogue of the classical
Adams operation $\psi^{3}$. We summarize the results that we need.

######  Theorem 2.3 ([BH20]).

There is a unital ring map
$\psi^{3}:\mathrm{ko}\left[\frac{1}{3}\right]\to\mathrm{ko}\left[\frac{1}{3}\right]$
whose Betti realization is the classical Adams operation $\psi^{3}$.

###### Proof.

There is a unital ring map
$\psi^{3}:\mathrm{KO}\left[\frac{1}{3}\right]\to\mathrm{KO}\left[\frac{1}{3}\right]$
[BH20, Theorem 3.1], which is an $E_{\infty}$-map. Its Betti realization is
also an $E_{\infty}$-map whose action on the classical Bott element is
multiplication by $81$. These properties uniquely characterize the classical
Adams operation.

Now apply very effective covers, and the result about $\mathrm{ko}$ follows
formally. ∎

The original result is more general in more than one sense. First, it works
over general base schemes in which $2$ is invertible, while we only use the
construction over $\mathbb{R}$. Second, its values are computed more precisely
than just compatibility with the classical values.

######  Corollary 2.4.

  1. (1)

$\psi^{3}:\pi_{*,*}(\mathrm{ko}^{\wedge}_{2})\to\pi_{*,*}(\mathrm{ko}^{\wedge}_{2})$
is a ring map.

  2. (2)

If $x$ is in the image of the unit map
$\pi_{{*,*}}(S^{\wedge}_{2})\to\pi_{{*,*}}(\mathrm{ko}^{\wedge}_{2})$, then
$\psi^{3}(x)=x$.

  3. (3)

There is a commutative diagram

$\textstyle{\pi_{*,*}(\mathrm{ko}^{\wedge}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{3}}$$\textstyle{\pi_{*,*}(\mathrm{ko}^{\wedge}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{*}(\mathrm{ko}^{\mathrm{top}})^{\wedge}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{3}}$$\textstyle{\pi_{*}(\mathrm{ko}^{\mathrm{top}})^{\wedge}_{2},}$

where the vertical maps are Betti realization homomorphisms.

###### Proof.

These are computational consequences of Theorem 2.3. Part (1) follows from the
fact that $\psi^{3}$ is a ring map. Part (2) follows from the fact that
$\psi^{3}$ is unital. Part (3) follows from the fact that the Betti
realization of the motivic Adams operation is the classical Adams operation. ∎

######  Remark 2.5.

Corollary 2.4 can also be stated in a localized sense rather than completed
sense, but we will not need that.

######  Definition 2.6.

Let $L$ be the fiber of the map
$\mathrm{ko}\left[\frac{1}{3}\right]\overset{\psi^{3}-1}{\longrightarrow}\mathrm{ko}\left[\frac{1}{3}\right]$.

Note that our definition of $L$ is already localized; we do not consider an
integral version. Except for this section, $L$ is assumed to be $2$-completed.

The most important point for us is that there is a fiber sequence

$L^{\wedge}_{2}\longrightarrow\mathrm{ko}^{\wedge}_{2}\overset{\psi^{3}-1}{\longrightarrow}\mathrm{ko}^{\wedge}_{2}$

of completed spectra since completion preserves fiber sequences.

###  2.3. Convergence of the effective spectral sequence

The _effective spectral sequence_ for a motivic spectrum $X$ denotes the
spectral sequence associated to the effective slice filtration of $X$. We
refer to [Lev13, RSØ19] for details on the construction and properties of this
spectral sequence.

The effective slice filtration [Voe02] has truncations $f^{q}(X)$ and
quotients (i.e., slices) $s_{q}(X)$. The $E_{1}$-page of the effective
spectral sequence is $\pi_{*,*}(s_{*}(X))$. In good cases, it converges to the
homotopy groups of a completion of $X$. We also use the very effective slice
filtration [SØ12], but only to define $\mathrm{ko}$.

The slice functors do not necessarily commute with completions, i.e.,
$s_{*}(X)^{\wedge}_{2}$ and $s_{*}(X^{\wedge}_{2})$ are not always equivalent.
Consequently, we must carefully define the spectral sequences that we use to
study completed spectra. On the other hand, the effective slices do interact
nicely with localizations [Spi08, Corollary 4.6].

######  Theorem 2.7.

There are strongly convergent spectral sequences

$E_{1}^{s,f,w}(\mathrm{ko})=\pi_{s,w}\left(s_{\frac{s+f}{2}}(\mathrm{ko})^{\wedge}_{2}\right)\implies\pi_{s,w}(\mathrm{ko}^{\wedge}_{2})$

and

$E_{1}^{s,f,w}(L)=\pi_{s,w}\left(s_{\frac{s+f}{2}}(L)^{\wedge}_{2}\right)\implies\pi_{s,w}(L^{\wedge}_{2})$

with differentials $d_{r}:E_{r}^{s,f,w}\to E_{r}^{s-1,f+2r-1,w}$.

We remind the reader that our grading of the effective spectral sequence is
different than the standard grading in the literature. Briefly, $s$ represents
the topological stem, $f$ represents the Adams–Novikov filtration (not the
effective filtration), and $w$ represents the motivic weight. See Section 1.7
for more discussion.

###### Proof.

We discuss the spectral sequence for $\mathrm{ko}$ in detail; most of the
argument for $L$ is the same.

Consider the effective slice tower

$f^{0}(\mathrm{ko})\leftarrow f^{1}(\mathrm{ko})\leftarrow
f^{2}(\mathrm{ko})\leftarrow\cdots.$

Now take the $2$-completion of this tower to obtain

$f^{0}(\mathrm{ko})^{\wedge}_{2}\leftarrow
f^{1}(\mathrm{ko})^{\wedge}_{2}\leftarrow
f^{2}(\mathrm{ko})^{\wedge}_{2}\leftarrow\cdots.$

The resulting layers are the same as $s_{*}(\mathrm{ko})^{\wedge}_{2}$ since
completion respects cofiber sequences. Beware that this is not necessarily the
same as the slice tower of the completion $\mathrm{ko}^{\wedge}_{2}$, since
slices do not interact nicely with completions. The associated spectral
sequence of this tower is the one described in the statement of the theorem.

It remains to determine the target of the completed spectral sequence. The
limit of the uncompleted slice tower of $\mathrm{ko}$ is equivalent to its
$\eta$-completion [RSØ19], [ARØ20], i.e.,

$\operatorname*{holim}f^{n}(\mathrm{ko})\simeq\mathrm{ko}^{\wedge}_{\eta}.$

Completion respects limits, so the limit
$\operatorname*{holim}(f^{n}(\mathrm{ko})^{\wedge}_{2})$ of the completed
slice tower is equivalent to $(\mathrm{ko}^{\wedge}_{\eta})^{\wedge}_{2}$,
which is equivalent to $\mathrm{ko}^{\wedge}_{2}$ by [HKO11, Theorem 1].
Consequently, the completed effective spectral sequence of $\mathrm{ko}$
converges to the homotopy of $\mathrm{ko}^{\wedge}_{2}$, as desired.

Strong convergence follows from [Boa99, Theorem 7.1], which has a technical
hypothesis involving derived $E_{\infty}$-pages. For $\mathrm{ko}$, this
technical hypothesis follows directly from the computations of Section 4. For
$L$, the technical hypothesis follows directly from the computations in
Sections 5.4 and 5.5. ∎

######  Remark 2.8.

By construction, we have a fiber sequence

$s_{*}(L)^{\wedge}_{2}\longrightarrow
s_{*}(\mathrm{ko})^{\wedge}_{2}\overset{\psi^{3}-1}{\longrightarrow}s_{*}(\mathrm{ko})^{\wedge}_{2},$

which yields a long exact sequence

$\cdots\longrightarrow E_{1}^{s,f,w}(L)\longrightarrow
E_{1}^{s,f,w}(\mathrm{ko})\overset{\psi^{3}-1}{\longrightarrow}E_{1}^{s,f,w}(\mathrm{ko})\longrightarrow\cdots.$

This long exact sequence will be our main tool for computing $E_{1}(L)$ in
Section 5.1.

###  2.4. Comparison between $\mathbb{R}$-motivic and $\mathbb{C}$-motivic
homotopy

Extension of scalars induces a functor $(-)^{\mathbb{C}}$ from
$\mathbb{R}$-motivic stable homotopy theory to $\mathbb{C}$-motivic stable
homotopy theory. Following [BI20, Section 3], we recall its computational
interpretation. These results will play a central role later in Section 5.6
when we consider hidden extensions.

The homotopy element $\rho$ maps to zero under extension of scalars.
Therefore, extension of scalars induces a natural homomorphism

$\pi_{*,*}(X/\rho)\to\pi_{*,*}(X^{\mathbb{C}})$

for any $\mathbb{R}$-motivic spectrum $X$.

######  Proposition 2.9.

The maps

$\pi_{*,*}(\mathrm{ko}/\rho)\to\pi_{*,*}(\mathrm{ko}^{\mathbb{C}})$

and

$\pi_{*,*}(L/\rho)\to\pi_{*,*}(L^{\mathbb{C}})$

are isomorphisms.

###### Proof.

The argument for the first isomorphism is essentially identical to the
argument in [BI20, Section 3]. We need that the Adams $E_{2}$-page for
$\mathrm{ko}$ is computed by the cobar complex associated to the subalgebra
$A(1)$ of the motivic Steenrod algebra, and multiplication by $\rho$ is
injective on this cobar complex.

For the second isomorphism, consider the diagram

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{*,*}(L/\rho)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{*,*}(\mathrm{ko}/\rho)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{*,*}(\mathrm{ko}/\rho)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{*,*}(L^{\mathbb{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{*,*}(\mathrm{ko}^{\mathbb{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{*,*}(\mathrm{ko}^{\mathbb{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots.}$

The vertical maps are induced by extension of scalars. The horizontal maps
arise from the fiber sequence

$L\longrightarrow\mathrm{ko}\overset{\psi^{3}-1}{\longrightarrow}\mathrm{ko},$

so the rows are exact. We already know that the second and third vertical
arrows are isomorphisms, so the first vertical arrow is also an isomorphism by
the Five Lemma. ∎

##  3\. $\mathbb{C}$-motivic computations

In this section, we carry out a preliminary computation of the effective
spectral sequences for $\mathrm{ko}^{\mathbb{C}}$ and $L^{\mathbb{C}}$. We
also consider the $\eta$-periodic spectral sequences. We are primarily
interested in $\mathbb{R}$-motivic computations, but we will need to compare
our $\mathbb{R}$-motivic computations to their $\mathbb{C}$-motivic
counterparts.

###  3.1. The effective spectral sequence for $\mathrm{ko}^{\mathbb{C}}$

We review the effective spectral sequence for $\mathrm{ko}^{\mathbb{C}}$.

######  Proposition 3.1.

The effective spectral sequence for $\mathrm{ko}^{\mathbb{C}}$ takes the form

$E_{1}(\mathrm{ko}^{\mathbb{C}})=\mathbb{Z}[\tau,h_{1},v_{1}^{2}]/2h_{1}.$

###### Proof.

This follows from Theorem 2.1 by taking stable homotopy groups. There are no
possible error terms to complicate the multiplicative structure. ∎

Table LABEL:tab:ko-C-E1-generators lists the generators of
$E_{1}(\mathrm{ko}^{\mathbb{C}})$. Figure 1 depicts
$E_{1}(\mathrm{ko}^{\mathbb{C}})$ in graphical form.

Table 1. Multiplicative generators for $E_{1}(\mathrm{ko}^{\mathbb{C}})$ coweight | $(s,f,w)$ | $x$ | $d_{1}(x)$ | $\psi^{3}(x)$
---|---|---|---|---
$0$ | $(1,1,1)$ | $h_{1}$ |  | $h_{1}$
$1$ | $(0,0,-1)$ | $\tau$ |  | $\tau$
$2$ | $(4,0,2)$ | $v_{1}^{2}$ | $\tau h_{1}^{3}$ | $9v_{1}^{2}$
Table 1. Multiplicative generators for $E_{1}(\mathrm{ko}^{\mathbb{C}})$

######  Proposition 3.2.

Table LABEL:tab:ko-C-E1-generators gives the values of the effective $d_{1}$
differential on the multiplicative generators of
$E_{1}(\mathrm{ko}^{\mathbb{C}})$.

###### Proof.

The $\mathbb{C}$-motivic effective spectral sequence is identical to the
$\mathbb{C}$-motivic Adams–Novikov spectral sequence up to reindexing. This
claim does not appear to be cleanly stated in the literature, but it is a
computational consequence of the weight $0$ result of [Lev15, Theorem 1].
Alternatively, there is only one pattern of effective differentials that
computes the motivic stable homotopy groups of $\mathrm{ko}^{\mathbb{C}}$,
which were previously described using the $\mathbb{C}$-motivic Adams spectral
sequence [IS11]. ∎

######  Theorem 3.3.

The $E_{\infty}$-page of the effective spectral sequence for
$\mathrm{ko}^{\mathbb{C}}$ takes the form

$E_{\infty}(\mathrm{ko}^{\mathbb{C}})=\frac{\mathbb{Z}[\tau,h_{1},2v_{1}^{2},v_{1}^{4}]}{2h_{1},\tau
h_{1}^{3},(2v_{1}^{2})^{2}=4\cdot v_{1}^{4}}.$

###### Proof.

For degree reasons, there can be no higher differentials in the effective
spectral sequence for $\mathrm{ko}^{\mathbb{C}}$. ∎

Table LABEL:tab:ko-C-Einfty-generators lists the multiplicative generators of
$E_{\infty}(\mathrm{ko}^{\mathbb{C}})$. Figure 2 depicts
$E_{\infty}(\mathrm{ko}^{\mathbb{C}})$ in graphical form.

Table 2. Multiplicative generators for $E_{\infty}(\mathrm{ko}^{\mathbb{C}})$ coweight | $(s,f,w)$ | $x$ | $\psi^{3}(x)$
---|---|---|---
$0$ | $(1,1,1)$ | $h_{1}$ | $h_{1}$
$1$ | $(0,0,-1)$ | $\tau$ | $\tau$
$2$ | $(4,0,2)$ | $2v_{1}^{2}$ | $9\cdot 2v_{1}^{2}$
$4$ | $(8,0,4)$ | $v_{1}^{4}$ | $81v_{1}^{4}$
Table 2. Multiplicative generators for $E_{\infty}(\mathrm{ko}^{\mathbb{C}})$

######  Remark 3.4.

There are no possible hidden extensions in
$E_{\infty}(\mathrm{ko}^{\mathbb{C}})$ for degree reasons. Therefore, Theorem
3.3 describes $\pi_{*,*}(\mathrm{ko}^{\mathbb{C}})$ as a ring.

###  3.2. The effective $E_{1}$-page for $L^{\mathbb{C}}$

Our next goal is to describe the effective $E_{1}$-page
$E_{1}(L^{\mathbb{C}})$. First we must study the values of $\psi^{3}$ on
$\mathrm{ko}^{\mathbb{C}}$.

######  Lemma 3.5.

The map $E_{\infty}(\mathrm{ko}^{\mathbb{C}})\to
E_{\infty}(\mathrm{ko}^{\mathbb{C}})$ induced by $\psi^{3}$ on effective
$E_{\infty}$-pages takes the values shown in Table LABEL:tab:ko-C-Einfty-
generators.

###### Proof.

All values follow immediately by comparison along Betti realization to the
values of classical $\psi^{3}$. ∎

######  Lemma 3.6.

The map $E_{1}(\mathrm{ko}^{\mathbb{C}})\to E_{1}(\mathrm{ko}^{\mathbb{C}})$
induced by $\psi^{3}$ on effective $E_{1}$-pages takes the values shown in
Table LABEL:tab:ko-C-E1-generators.

###### Proof.

The values of $\psi^{3}$ on $E_{1}(\mathrm{ko}^{\mathbb{C}})$ are compatible
with the values of $\psi^{3}$ on $E_{\infty}(\mathrm{ko}^{\mathbb{C}})$, as
shown in Table LABEL:tab:ko-C-Einfty-generators (see also Lemma 3.5). This
immediately yields all values. ∎

In order to describe $E_{1}(L^{\mathbb{C}})$, we need some elementary number
theory.

######  Definition 3.7.

Let $v(n)$ be the $2$-adic valuation of $n$, i.e., the exponent of the largest
power of $2$ that divides $n$.

######  Lemma 3.8.

$v(3^{n}-1)=\left\\{\begin{array}[]{l@{\quad\mathrm{if} \quad}l}1&v(n)=0\\\
2+v(n)&v(n)>0\end{array}\right.$

###### Proof.

Let $n=2^{a}\cdot b$, where $b$ is an odd number, so $v(n)=a$. Then

$3^{n}-1=\left(1+3^{2^{a}}+(3^{2^{a}})^{2}+\cdots+(3^{2^{a}})^{b-1}\right)(3-1)\prod_{i=0}^{a-1}(1+3^{2^{i}}).$

The first factor is odd, so it does not contribute to the $2$-adic valuation.
The factor $(1+3^{2^{i}})$ has valuation 1 if $i>0$, and it has valuation 2 if
$i=0$. ∎

######  Proposition 3.9.

The chart in Figure 3 depicts the effective $E_{1}$-page of $L^{\mathbb{C}}$.

###### Proof.

The long exact sequence

$\cdots\longrightarrow E_{1}(L^{\mathbb{C}})\longrightarrow
E_{1}(\mathrm{ko}^{\mathbb{C}})\overset{\psi^{3}-1}{\longrightarrow}E_{1}(\mathrm{ko}^{\mathbb{C}})\longrightarrow\cdots$

induces a short exact sequence

$0\longrightarrow\Sigma^{-1}C\longrightarrow
E_{1}(L^{\mathbb{C}})\longrightarrow K\longrightarrow 0,$

where $C$ and $K$ are the cokernel and kernel of
$E_{1}(\mathrm{ko}^{\mathbb{C}})\overset{\psi^{3}-1}{\longrightarrow}E_{1}(\mathrm{ko}^{\mathbb{C}})$
respectively. The cokernel and kernel can be computed directly from the
information given in Table LABEL:tab:ko-C-E1-generators (see also Lemma 3.6).

The kernel is additively generated by all multiples of $h_{1}$ in
$E_{1}(\mathrm{ko}^{\mathbb{C}})$, together with the elements $\tau^{k}$ for
$k\geq 0$.

The cokernel $C$ is nearly the same as $E_{1}(\mathrm{ko}^{\mathbb{C}})$
itself. We must impose the relations $(3^{2k}-1)v_{1}^{2k}=0$ for all $k>0$.
Lemma 3.8 says that $3^{2k}-1$ equals $2^{v(2k)+2}\cdot u$, where $u$ is an
odd number, i.e., a unit in our $2$-adic context. Therefore, the relation
$(3^{2k}-1)v_{1}^{2k}=0$ is equivalent to the relation
$2^{v(2k)+2}v_{1}^{2k}=0$. ∎

Table LABEL:tab:L-C-E1-generators lists some elements of the effective
$E_{1}$-page of $L^{\mathbb{C}}$. In fact, these elements are multiplicative
generators for $E_{1}(L^{\mathbb{C}})$. By inspection, all elements of
$E_{1}(L^{\mathbb{C}})$ are of the form $\tau^{a}h_{1}^{b}x$, for some $x$ in
the table.

We use the same notation for elements of $E_{1}(L^{\mathbb{C}})$ and their
images in $E_{1}(\mathrm{ko}^{\mathbb{C}})$. On the other hand, we define the
elements $\iota x$ of $E_{1}(L^{\mathbb{C}})$ by the property that they are
the image of $x$ under the map $\iota:\Sigma^{-1}E_{1}(\mathrm{ko})\to
E_{1}(L)$. For example, the element $1$ of $E_{1}(\mathrm{ko})$ maps to
$\iota$.

Table 3. Multiplicative generators for $E_{1}(L^{\mathbb{C}})$: $k\geq 0$ coweight | $(s,f,w)$ | generator
---|---|---
$1$ | $(0,0,-1)$ | $\tau$
$2k$ | $(4k+1,1,2k+1)$ | $h_{1}v_{1}^{2k}$
$2k-1$ | $(4k-1,1,2k)$ | $\iota v_{1}^{2k}$
Table 3. Multiplicative generators for $E_{1}(L^{\mathbb{C}})$: $k\geq 0$

######  Remark 3.10.

Our choice of notation for elements of $E_{1}(L^{\mathbb{C}})$ is helpful for
the particular analysis at hand. The generators of $E_{1}(L^{\mathbb{C}})$
also have traditional names from the perspective of the Adams–Novikov spectral
sequence. Namely, $h_{1}v_{1}^{2k}$ and $\iota v_{1}^{2k}$ correspond to
$\alpha_{2k+1}$ and $\alpha_{2k/v(8k)}$ respectively. However, the
$\alpha$-family perspective is not so helpful for us.

###  3.3. The effective spectral sequence of $L^{\mathbb{C}}[\eta^{-1}]$

Next, we describe the effective spectral sequence of
$L^{\mathbb{C}}[\eta^{-1}]$.

In the $\eta$-periodic context, the element $h_{1}$ is a unit. Therefore,
powers of $h_{1}$ are inconsequential for computational purposes.
Consequently, we have removed these powers from all $\eta$-periodic formulas.
The appropriate powers of $h_{1}$ can be easily reconstructed from the degrees
of elements (although this reconstruction is typically not necessary).

######  Proposition 3.11.

The effective $E_{1}$-page for $L^{\mathbb{C}}[\eta^{-1}]$ is given by

$E_{1}(L^{\mathbb{C}}[\eta^{-1}])=\mathbb{F}_{2}[h_{1}^{\pm
1},\tau,v_{1}^{2},\iota]/\iota^{2}.$

###### Proof.

The functors $s_{*}$ commute with homotopy colimits [Spi08, Corollary 4.6].
Therefore, we can just invert $h_{1}$ in $E_{1}(\mathrm{ko}^{\mathbb{C}})$ to
obtain

$E_{1}(\mathrm{ko}^{\mathbb{C}}[\eta^{-1}])=\mathbb{F}_{2}[h_{1}^{\pm
1},\tau,v_{1}^{2}].$

See Proposition 3.1 (and Figure 1) for the description of
$E_{1}(\mathrm{ko}^{\mathbb{C}})$.

The map
$E_{1}(\mathrm{ko}^{\mathbb{C}}[\eta^{-1}])\overset{\psi^{3}-1}{\longrightarrow}E_{1}(\mathrm{ko}^{\mathbb{C}}[\eta^{-1})$
is trivial because $(\psi^{3}-1)(h_{1})=0$, as shown in Table LABEL:tab:ko-
C-E1-generators (see also Lemma 3.6). Therefore, the long exact sequence

$\cdots\longrightarrow E_{1}(L^{\mathbb{C}}[\eta^{-1})\longrightarrow
E_{1}(\mathrm{ko}^{\mathbb{C}}[\eta^{-1}])\overset{\psi^{3}-1}{\longrightarrow}E_{1}(\mathrm{ko}^{\mathbb{C}}[\eta^{-1}])\longrightarrow\cdots$

implies that $E_{1}(L^{\mathbb{C}}[\eta^{-1}])$ splits as

$E_{1}(\mathrm{ko}^{\mathbb{C}}[\eta^{-1}])\oplus\Sigma^{-1}E_{1}(\mathrm{ko}^{\mathbb{C}}[\eta^{-1}]).$

This establishes the additive structure of $E_{1}(L[\eta^{-1}])$, as well as
most of the multiplicative structure.

The relation $\iota^{2}=0$ is immediate because there are no possible non-zero
values for $\iota^{2}$. ∎

######  Proposition 3.12.

In the effective spectral sequence for $L^{\mathbb{C}}[\eta^{-1}]$, we have
$d_{1}(v_{1}^{2})=\tau$. The effective differentials are zero on all other
multiplicative generators on all pages.

###### Proof.

The value of $d_{1}(v_{1}^{2})$ in $E_{1}(L^{\mathbb{C}}[\eta^{-1}])$ follows
by comparison of effective spectral sequences along the maps
$L^{\mathbb{C}}\to L^{\mathbb{C}}[\eta^{-1}]$ and
$L^{\mathbb{C}}\to\mathrm{ko}^{\mathbb{C}}$. Table LABEL:tab:ko-
C-E1-generators (see also Proposition 3.2) gives the value of
$d_{1}(v_{1}^{2})$ in $E_{1}(\mathrm{ko}^{\mathbb{C}})$ . ∎

######  Remark 3.13.

The effective spectral sequence for $L^{\mathbb{C}}[\eta^{-1}]$ is very close
to the effective spectral sequence for the $\eta$-periodic sphere
$S^{\mathbb{C}}[\eta^{-1}]$. The effective spectral sequence for
$S^{\mathbb{C}}[\eta^{-1}]$ is the same (up to reindexing) as the motivic
Adams–Novikov spectral sequence for $S^{\mathbb{C}}[\eta^{-1}]$. This motivic
Adams–Novikov spectral sequence is analyzed in [AM17]. The element $\iota$ is
not present in $E_{1}(S^{\mathbb{C}}[\eta^{-1}])$, but its multiples
$\iota(v_{1}^{2})^{k}$ are present.

###  3.4. Effective differentials for $L^{\mathbb{C}}$

######  Proposition 3.14.

Table LABEL:tab:L-C-d1 gives the values of the effective $d_{1}$ differentials
on the multiplicative generators of $E_{1}(L^{\mathbb{C}})$. There are no
higher differentials in the effective spectral sequence for $L^{\mathbb{C}}$.

###### Proof.

All of these differentials follow immediately from the effective $d_{1}$
differentials for $L^{\mathbb{C}}[\eta^{-1}]$, which are determined by
Proposition 3.12.

For degree reasons, there are no possible higher differentials. ∎

Table 4. Effective $d_{1}$ differentials for $L^{\mathbb{C}}$: $k\geq 0$ coweight | $(s,f,w)$ | $x$ | $d_{1}(x)$
---|---|---|---
$1$ | $(0,0,-1)$ | $\tau$ |
$4k$ | $(8k+1,1,4k+1)$ | $h_{1}v_{1}^{4k}$ |
$4k+2$ | $(8k+5,1,4k+3)$ | $h_{1}v_{1}^{4k+2}$ | $\tau h_{1}^{3}\cdot h_{1}v_{1}^{4k}$
$4k-1$ | $(8k-1,1,4k)$ | $\iota v_{1}^{4k}$ |
$4k+1$ | $(8k+3,1,4k+2)$ | $\iota v_{1}^{4k+2}$ | $\tau h_{1}^{3}\cdot\iota v_{1}^{4k}$
Table 4. Effective $d_{1}$ differentials for $L^{\mathbb{C}}$: $k\geq 0$

######  Theorem 3.15.

The $E_{\infty}$-page of the effective spectral sequence for $L^{\mathbb{C}}$
is depicted in Figure 4.

###### Proof.

Because there are no higher effective differentials for $L^{\mathbb{C}}$, we
obtain the effective $E_{\infty}$-page immediately from the effective $d_{1}$
differentials in Table LABEL:tab:L-C-d1 (see also Proposition 3.14). ∎

###  3.5. Hidden extensions in $E_{\infty}(L^{\mathbb{C}})$

######  Proposition 3.16.

In the effective spectral sequence for $L^{\mathbb{C}}$, there are hidden
$\mathsf{h}$ extensions from $\iota 4v_{1}^{4k+2}$ to $\tau h_{1}^{2}\cdot
h_{1}v_{1}^{4k}$ for all $k\geq 0$.

###### Proof.

Recall that $\tau\eta^{2}=\langle\mathsf{h},\eta,\mathsf{h}\rangle$ in the
homotopy of the $\mathbb{C}$-motivic sphere [Isa19, Table 7.23]. If $\alpha$
is a homotopy element of $L^{\mathbb{C}}$ such that $\mathsf{h}\alpha$ is
zero, then

$\alpha\cdot\tau\eta^{2}=\alpha\langle\mathsf{h},\eta,\mathsf{h}\rangle=\langle\alpha,\mathsf{h},\eta\rangle\mathsf{h}.$

In particular, let $\alpha$ be detected by $h_{1}v_{1}^{4k}$. Then $\tau
h_{1}^{2}\cdot h_{1}v_{1}^{4k}$ detects a homotopy element that is divisible
by $\mathsf{h}$, so $\tau h_{1}^{2}\cdot h_{1}v_{1}^{4k}$ must be the target
of a hidden $\mathsf{h}$ extension. There is only one possible source for this
extension. ∎

##  4\. The effective spectral sequence for $\mathrm{ko}$

We now study the effective spectral sequence for $\mathbb{R}$-motivic
$\mathrm{ko}$.

######  Proposition 4.1.

The effective spectral sequence for $\mathrm{ko}$ takes the form

$E_{1}(\mathrm{ko})=\frac{\mathbb{Z}[\rho,\tau^{2},h_{1},\tau
h_{1},v_{1}^{2}]}{2\rho,2h_{1},2\cdot\tau h_{1},(\tau h_{1})^{2}=\tau^{2}\cdot
h_{1}^{2}+\rho^{2}\cdot v_{1}^{2}}$

###### Proof.

The additive structure follows from Theorem 2.1 by taking stable homotopy
groups. We need that the homotopy groups of $\mathbb{R}$-motivic $H\mathbb{Z}$
are

$H\mathbb{Z}_{*,*}=\mathbb{Z}[\tau^{2},\rho]/2\rho,$

and the homotopy groups of $\mathbb{R}$-motivic $H\mathbb{F}_{2}$ are

$(H\mathbb{F}_{2})_{*,*}=\mathbb{F}_{2}[\tau,\rho].$

The multiplicative structure is mostly also immediate from Theorem 2.1. As
explained in [Kon21], our formula for $(\tau h_{1})^{2}$ is equivalent to the
formula $\eta^{2}\overset{\delta}{\to}\sqrt{\alpha}$ given in [ARØ20, p.
1029]. ∎

Table LABEL:tab:ko-E1-generators lists the generators of $E_{1}(\mathrm{ko})$.
Figure 5 depicts $E_{1}(\mathrm{ko})$ in graphical form.

Table 5. Multiplicative generators for $E_{1}(\mathrm{ko})$ coweight | $(s,f,w)$ | $x$ | $d_{1}(x)$ | $\psi^{3}(x)$ | image in $E_{1}(\mathrm{ko}[\eta^{-1}])$
---|---|---|---|---|---
$0$ | $(-1,1,-1)$ | $\rho$ |  | $\rho$ | $\rho$
$0$ | $(1,1,1)$ | $h_{1}$ |  | $h_{1}$ | $1$
$1$ | $(1,1,0)$ | $\tau h_{1}$ |  | $\tau h_{1}$ | $\tau$
$2$ | $(0,0,-2)$ | $\tau^{2}$ | $\rho^{2}\cdot\tau h_{1}$ | $\tau^{2}$ | $\tau^{2}+\rho^{2}\cdot v_{1}^{2}$
$2$ | $(4,0,2)$ | $v_{1}^{2}$ | $\tau h_{1}\cdot h_{1}^{2}$ | $9v_{1}^{2}$ | $v_{1}^{2}$
Table 5. Multiplicative generators for $E_{1}(\mathrm{ko})$

######  Proposition 4.2.

Table LABEL:tab:ko-E1-generators gives the values of the effective $d_{1}$
differential on the multiplicative generators of $E_{1}(\mathrm{ko})$.

###### Proof.

The value of $d_{1}(\tau^{2})$ follows from [ARØ20, Theorem 20] and
$\mathbb{R}$-motivic Steenrod algebra actions. Then the value of
$d_{1}(v_{1}^{2})$ follows from Equation (1.1).

Alternatively, there is only one pattern of effective differentials that
computes the motivic stable homotopy groups of $\mathrm{ko}$, which were
previously computed with the $\mathbb{R}$-motivic Adams spectral sequence
[Hil11]. ∎

The entire $d_{1}$ differential in the effective spectral sequence for
$\mathrm{ko}$ can easily be deduced from Proposition 4.2 and the Leibniz rule.

######  Theorem 4.3.

The $E_{\infty}$-page of the effective spectral sequence for $\mathrm{ko}$ is
depicted in Figures 6, 7, and 8.

###### Proof.

The Leibniz rule, together with the values in Table LABEL:tab:ko-E1-generators
(see also Proposition 4.2), completely determines the effective $d_{1}$
differential on $E_{1}(\mathrm{ko})$. The $E_{2}$-page can then be determined
directly. However, the computation is not entirely straightforward. Of
particular note is the differential

$d_{1}(\tau^{2}\cdot\tau h_{1}\cdot v_{1}^{2})=\tau^{4}\cdot
h_{1}^{4}+\rho^{4}\cdot v_{1}^{4},$

which yields the relation

(4.1) $\tau^{4}\cdot h_{1}^{4}=\rho^{4}\cdot v_{1}^{4}$

in $E_{2}(\mathrm{ko})$.

For degree reasons, there can be no higher differentials in the effective
spectral sequence for $\mathrm{ko}$. ∎

For legibility, Figures 6, 7, and 8 display $E_{\infty}(\mathrm{ko})$ in three
different charts separated by coweight modulo 4. There is no chart for
coweights 3 mod 4 because $E_{\infty}(\mathrm{ko})$ is zero in those
coweights.

Figure 9 illustrates part of the analysis of the $d_{1}$ differentials and the
determination of $E_{2}(\mathrm{ko})$; it is meant to be representative, not
thorough. The chart shows some of the elements in coweights $1$ and $2$ mod 4,
together with the $d_{1}$ differentials that relate these elements. In this
chart, one can see that $\tau^{2}\cdot h_{1}^{2}+\rho^{2}\cdot v_{1}^{2}$
survives to $E_{2}(\mathrm{ko})$. This element survives to
$E_{\infty}(\mathrm{ko})$. It is labelled $(\tau h_{1})^{2}$ in Figure 8, in
accordance with Equation (1.1).

######  Remark 4.4.

There is an alternative, slightly more structured, method for obtaining
$E_{\infty}(\mathrm{ko})$. One can filter $E_{1}(\mathrm{ko})$ by powers of
$\tau h_{1}$ and obtain a spectral sequence that converges to
$E_{2}(\mathrm{ko})$. In this spectral sequence, we have the relation
$\tau^{2}\cdot h_{1}^{2}=\rho^{2}\cdot v_{1}^{2}$. There are differentials
$d_{1}(\tau^{2})=\rho^{2}\cdot\tau h_{1}$ and
$d_{1}(v_{1}^{2})=h_{1}^{2}\cdot\tau h_{1}$. Then there is a higher
differential $d_{3}(\tau^{2}\cdot v_{1}^{2})=(\tau h_{1})^{3}$. None of this
is essential to our study, but the interested reader may wish to carry out the
details.

Table LABEL:tab:ko-Einfty-generators lists the multiplicative generators of
$E_{\infty}(\mathrm{ko})$. It is possible to give a complete list of
relations. However, the long list is not so helpful for understanding the
structure of $E_{\infty}(\mathrm{ko})$. The charts in Figures 6, 7, and 8 are
more useful for this purpose.

Table 6. Multiplicative generators for $E_{\infty}(\mathrm{ko})$ coweight | $(s,f,w)$ | $x$ | $\psi^{3}(x)$
---|---|---|---
$0$ | $(-1,1,-1)$ | $\rho$ | $\rho$
$0$ | $(1,1,1)$ | $h_{1}$ | $h_{1}$
$1$ | $(1,1,0)$ | $\tau h_{1}$ | $\tau h_{1}$
$2$ | $(0,0,-2)$ | $2\tau^{2}$ | $2\tau^{2}$
$2$ | $(4,0,2)$ | $2v_{1}^{2}$ | $9\cdot 2v_{1}^{2}$
$4$ | $(0,0,-4)$ | $\tau^{4}$ | $\tau^{4}$
$4$ | $(4,0,0)$ | $2\tau^{2}v_{1}^{2}$ | $9\cdot 2\tau^{2}v_{1}^{2}$
$4$ | $(8,0,4)$ | $v_{1}^{4}$ | $81v_{1}^{4}$
Table 6. Multiplicative generators for $E_{\infty}(\mathrm{ko})$

######  Proposition 4.5.

Table LABEL:tab:ko-hidden lists some hidden extensions by $\rho$,
$\mathsf{h}$, and $\eta$ in the effective spectral sequence for $\mathrm{ko}$.
All other hidden extensions by $\rho$, $\mathsf{h}$, and $\eta$ are
$v_{1}^{4}$-multiples and $\tau^{4}$-multiples of these.

###### Proof.

Recall from Proposition 2.9 that the homotopy of $\mathrm{ko}/\rho$ is
isomorphic to the homotopy of $\mathrm{ko}^{\mathbb{C}}$. Therefore, we
completely understand the homotopy of $\mathrm{ko}/\rho$ from Theorem 3.3 and
Figure 2.

The hidden $\rho$ extensions follow from inspection of the long exact sequence
associated to the cofiber sequence

$\Sigma^{-1,-1}\mathrm{ko}\overset{\rho}{\longrightarrow}\mathrm{ko}\to\mathrm{ko}/\rho.$

The map $\mathrm{ko}\to\mathrm{ko}/\rho$ takes the elements $\tau^{4}\cdot
h_{1}^{3}$ and $(\tau h_{1})^{2}h_{1}$ to zero because there are no possible
targets in the homotopy of $\mathrm{ko}/\rho$. Therefore, those two elements
must receive hidden $\rho$ extensions, and there is only one possibility in
both cases.

The relation $\tau^{4}\cdot h_{1}^{4}=\rho^{4}\cdot v_{1}^{4}$ (see Equation
(4.1)) then implies that $2\tau^{2}v_{1}^{2}$ also supports an $h_{1}$
extension.

The map $\mathrm{ko}/\rho\to\Sigma^{0,-1}\mathrm{ko}$ takes $\tau^{3}$ and
$\tau^{3}h_{1}$ to $2\tau^{2}$ and $\rho(\tau h_{1})^{2}$ respectively. There
is an $h_{1}$ extension connecting $\tau^{3}$ and $\tau^{3}h_{1}$ in
$\mathrm{ko}/\rho$, so there must be a hidden $\eta$ extension from
$2\tau^{2}$ to $\rho(\tau h_{1})^{2}$.

The hidden $\mathsf{h}$ extension on $\tau h_{1}$ follows from the analogous
hidden extension in the homotopy groups of the $\mathbb{R}$-motivic sphere
[DI17] [BI20], using the unit map $S\to\mathrm{ko}$. Alternatively, this
hidden extension is computed in [Hil11, Proposition 4.3] in the context of the
$\mathbb{R}$-motivic Adams spectral sequence for $\mathrm{ko}$.

Finally, multiply by $\tau h_{1}$ to obtain the hidden $\mathsf{h}$ extension
on $(\tau h_{1})^{2}$.

For degree reasons, there are no other possible hidden extensions to consider.
∎

Table 7. Hidden extensions in $E_{\infty}(\mathrm{ko})$ coweight | source | type | target | $(s,f,w)$
---|---|---|---|---
$2$ | $2v_{1}^{2}$ | $\rho$ | $(\tau h_{1})^{2}h_{1}$ | $(3,3,1)$
$4$ | $2\tau^{2}v_{1}^{2}$ | $\rho$ | $\tau^{4}\cdot h_{1}^{3}$ | $(3,3,-1)$
$4$ | $2\tau^{2}v_{1}^{2}$ | $\eta$ | $\rho^{3}\cdot v_{1}^{4}$ | $(5,3,1)$
$2$ | $2\tau^{2}$ | $\eta$ | $\rho(\tau h_{1})^{2}$ | $(1,3,-1)$
$1$ | $\tau h_{1}$ | $\mathsf{h}$ | $\rho\cdot\tau h_{1}\cdot h_{1}$ | $(1,3,0)$
$2$ | $(\tau h_{1})^{2}$ | $\mathsf{h}$ | $\rho(\tau h_{1})^{2}h_{1}$ | $(2,4,0)$
Table 7. Hidden extensions in $E_{\infty}(\mathrm{ko})$

######  Remark 4.6.

We have completely analyzed the $E_{\infty}$-page of the effective spectral
sequence for $\mathrm{ko}$, but this is not quite the same as completely
describing the homotopy of $\mathrm{ko}$. In particular, one must choose an
element of $\pi_{*,*}\mathrm{ko}$ that is represented by each multiplicative
generator of $E_{\infty}(\mathrm{ko})$ (see Table LABEL:tab:ko-Einfty-
generators). In some cases, there is more than one choice because of the
presence of elements in higher filtration in the $E_{\infty}$-page. The
choices of $\rho$, $h_{1}$, $\tau h_{1}$, and $\tau^{4}$ can be made
arbitrarily; the ring structure is unaffected by these choices. The elements
$2\tau^{2}$ and $2v_{1}^{2}$ are already well-defined because there are no
elements in higher filtration. Finally, the choices of $2\tau^{2}v_{1}^{2}$
and $v_{1}^{4}$ can then be uniquely specified by the relations $\rho\cdot
2\tau^{2}v_{1}^{2}=\tau^{4}\cdot h_{1}^{3}$ and $\rho^{4}\cdot
v_{1}^{4}=\tau^{4}\cdot h_{1}^{4}$.

###  4.1. $\eta$-periodic $\mathrm{ko}$

Later we will need some information about the $\eta$-periodic spectrum
$\mathrm{ko}[\eta^{-1}]$. As in Section 3.3, powers of $h_{1}$ are
inconsequential for computational purposes in the $\eta$-periodic context.
Consequently, we have removed these powers from all $\eta$-periodic formulas.

######  Proposition 4.7.

The effective $E_{1}$-page for $\mathrm{ko}$ is given by

$E_{1}(\mathrm{ko}[\eta^{-1}])=\mathbb{F}_{2}[h_{1}^{\pm
1},\tau,\rho,v_{1}^{2}].$

Moreover, the periodicization map $\mathrm{ko}\to\mathrm{ko}[\eta^{-1}]$
induces the map on effective $E_{1}$-pages whose values are given in Table
LABEL:tab:ko-E1-generators.

The first part of Proposition 4.7 was first proved in [ARØ20, Theorem 19],
although the notation is different.

###### Proof.

The functors $s_{*}$ commute with homotopy colimits [Spi08, Corollary 4.6].
Therefore, we can just invert $h_{1}$ in the description of
$E_{1}(\mathrm{ko})$ given in Proposition 4.1 (see also Figure 5). ∎

######  Remark 4.8.

Table LABEL:tab:ko-E1-generators gives an unexpected value for $\tau^{2}$.
Recall that $\tau^{2}$ is indecomposable in $E_{1}(\mathrm{ko})$, so there is
no inconsistency. The unexpected value arises from Equation (1.1).

###  4.2. The Adams operation $\psi^{3}$ in effective spectral sequences

Our goal in this section is to study $\psi^{3}$ as a map of effective spectral
sequences. This will allow us to compute the $E_{1}$-page of the effective
spectral sequence for $L$.

######  Lemma 4.9.

The map $E_{\infty}(\mathrm{ko})\to E_{\infty}(\mathrm{ko})$ induced by
$\psi^{3}$ on effective $E_{\infty}$-pages takes the values shown in Table
LABEL:tab:ko-Einfty-generators.

###### Proof.

Corollary 2.4(2) gives the values of $\psi^{3}$ on $\rho$, $h_{1}$, and $\tau
h_{1}$.

The value of $\psi^{3}$ on $\tau^{4}$ is determined immediately by comparison
along Betti realization to the classical value $\psi^{3}(1)=1$. The
computation is greatly simplified by ignoring terms in higher effective
filtration. Similarly, the value of $\psi^{3}$ on $2\tau^{2}$ is determined by
the classical value $\psi^{3}(2)=2$.

The remaining values in Table LABEL:tab:ko-Einfty-generators are also
determined by comparison along Betti realization to the classical values
$\psi^{3}(2v_{1}^{2})=9\cdot 2v_{1}^{2}$ and
$\psi^{3}(v_{1}^{4})=81v_{1}^{4}$. ∎

######  Lemma 4.10.

The map $E_{1}(\mathrm{ko})\to E_{1}(\mathrm{ko})$ induced by $\psi^{3}$ on
effective $E_{1}$-pages takes the values shown in Table
LABEL:tab:ko-E1-generators.

###### Proof.

The values of $\psi^{3}$ on $E_{1}(\mathrm{ko})$ are compatible with the
values of $\psi^{3}$ on $E_{\infty}(\mathrm{ko})$, as shown in Table
LABEL:tab:ko-Einfty-generators. This immediately yields the value of
$\psi^{3}$ on $\rho$, $h_{1}$, and $\tau h_{1}$.

The value of $\psi^{3}((\tau^{2})^{2})$ must be $(\tau^{2})^{2}$ by
compatibility with the value of $\psi^{3}(\tau^{4})$ in
$E_{\infty}(\mathrm{ko})$. Then the relation
$\psi^{3}((\tau^{2})^{2})=(\psi^{3}(\tau^{2}))^{2}$ implies that
$\psi^{3}(\tau^{2})=\tau^{2}$.

Similarly, the value of $\psi^{3}((v_{1}^{2})^{2})$ must be
$81(v_{1}^{2})^{2}$ by compatibility with the value of $\psi^{3}(v_{1}^{4})$
in $E_{\infty}(\mathrm{ko})$. Then the relation
$\psi^{3}((v_{1}^{2})^{2})=(\psi^{3}(v_{1}^{2}))^{2}$ implies that
$\psi^{3}(v_{1}^{2})=9v_{1}^{2}$. ∎

######  Remark 4.11.

Since $\psi^{3}$ is a ring homomorphism, all values of $\psi^{3}$ on
$E_{1}(\mathrm{ko})$ are readily determined by the values on multiplicative
generators given in Table LABEL:tab:ko-E1-generators. In particular, for all
$k\geq 0$,

$\psi^{3}(v_{1}^{2k})=9^{k}v_{1}^{2k}.$

######  Remark 4.12.

Table LABEL:tab:ko-E1-generators implies that
$\psi^{3}(v_{1}^{4})=81v_{1}^{4}$. The careful reader will notice that this
expression appears to be simpler than the analogous formula in [BH20, Theorem
3.1(2)]. The difference is explained by the fact that we are working only up
to higher filtration. In particular, our formulas do not reflect the
difference between $2$ and $\mathsf{h}$. This also means that our formulas are
less precise, but that has no consequence for our computational results.

##  5\. The effective spectral sequence for $L$

###  5.1. The effective $E_{1}$-page of $L$

In this section we compute the $E_{1}$-page of the effective spectral sequence
for $L$.

The fiber sequence
$L\to\mathrm{ko}\overset{\psi^{3}-1}{\longrightarrow}\mathrm{ko}$ induces a
fiber sequence

$s_{*}L\longrightarrow
s_{*}\mathrm{ko}\overset{\psi^{3}-1}{\longrightarrow}s_{*}\mathrm{ko}$

on slices. Upon taking homotopy groups, we obtain a long exact sequence

$\cdots\longrightarrow E_{1}(L)\longrightarrow
E_{1}(\mathrm{ko})\overset{\psi^{3}-1}{\longrightarrow}E_{1}(\mathrm{ko})\longrightarrow\cdots.$

Table LABEL:tab:ko-E1-generators (see also Lemma 4.10) gives us complete
computational knowledge of the map $E_{1}(\mathrm{ko})\to E_{1}(\mathrm{ko})$.
This allows us to compute $E_{1}(L)$.

######  Proposition 5.1.

The chart in Figure 10 depicts the effective $E_{1}$-page of $L$.

###### Proof.

The long exact sequence

$\cdots\longrightarrow E_{1}(L)\longrightarrow
E_{1}(\mathrm{ko})\overset{\psi^{3}-1}{\longrightarrow}E_{1}(\mathrm{ko})\longrightarrow\cdots$

induces a short exact sequence

$0\longrightarrow\Sigma^{-1}C\longrightarrow E_{1}(L)\longrightarrow
K\longrightarrow 0,$

where $C$ and $K$ are the cokernel and kernel of
$E_{1}(\mathrm{ko})\overset{\psi^{3}-1}{\longrightarrow}E_{1}(\mathrm{ko})$.
The cokernel and kernel can be computed directly from the information given in
Lemma 4.10. See also Remark 4.11.

The kernel consists of all elements in $E_{1}(\mathrm{ko})$ with the exception
of the integer multiples of $\tau^{2j}\cdot v_{1}^{2k}$ for $j\geq 0$ and
$k>0$.

The cokernel $C$ is nearly the same as $E_{1}(\mathrm{ko})$ itself. We must
impose the relations $(3^{2k}-1)v_{1}^{2k}=0$ for all $k>0$. Lemma 3.8 says
that $3^{2k}-1$ equals $2^{v(2k)+2}\cdot u$, where $u$ is an odd number, i.e.,
a unit in our $2$-adic context. Therefore, the relation
$(3^{2k}-1)v_{1}^{2k}=0$ is equivalent to the relation
$2^{v(2k)+2}v_{1}^{2k}=0$. ∎

Table LABEL:tab:L-E1-generators lists some elements of the effective
$E_{1}$-page of $L$. In fact, by inspection these elements are multiplicative
generators for $E_{1}(L)$.

We use the same notation for elements of $E_{1}(L)$ and their images in
$E_{1}(\mathrm{ko})$. On the other hand, we define the element $\iota x$ of
$E_{1}(L)$ to be the image of $x$ under the map
$\iota:\Sigma^{-1}E_{1}(\mathrm{ko})\to E_{1}(L)$. For example, the element
$1$ of $E_{1}(\mathrm{ko})$ maps to $\iota$ in $E_{1}(L)$.

Table 8. Multiplicative generators for $E_{1}(L)$: $k\geq 0$ coweight | $(s,f,w)$ | generator | image in $E_{1}(L[\eta^{-1}])$
---|---|---|---
$2$ | $(0,0,-2)$ | $\tau^{2}$ | $\tau^{2}+\rho^{2}\cdot v_{1}^{2}$
$2k+1$ | $(4k+1,1,2k)$ | $\tau h_{1}v_{1}^{2k}$ | $\tau(v_{1}^{2})^{k}$
$2k$ | $(4k-1,1,2k-1)$ | $\rho v_{1}^{2k}$ | $\rho(v_{1}^{2})^{k}$
$2k$ | $(4k+1,1,2k+1)$ | $h_{1}v_{1}^{2k}$ | $(v_{1}^{2})^{k}$
$2k-1$ | $(4k-1,1,2k)$ | $\iota v_{1}^{2k}$ | $\iota(v_{1}^{2})^{k}$
Table 8. Multiplicative generators for $E_{1}(L)$: $k\geq 0$

###  5.2. The effective spectral sequence for $L[\eta^{-1}]$

In Section 5.1, we determined the effective $E_{1}$-page of $L$. The next
steps in the analysis of the effective spectral sequence for $L$ are to
determine the multiplicative structure of $E_{1}(L)$ (see Section 5.3) and to
determine the effective differentials (see Sections 5.4 and 5.5).

Before doing so, we collect some information on the $\eta$-periodicization
$L[\eta^{-1}]$. We will study $L[\eta^{-1}]$ by comparing to the more easily
understood $\mathrm{ko}[\eta^{-1}]$.

As in Sections 3.3 and 4.1, powers of $h_{1}$ are inconsequential for
computational purposes in the $\eta$-periodic context. Consequently, we have
removed these powers from all $\eta$-periodic formulas.

######  Proposition 5.2.

The effective $E_{1}$-page for $L[\eta^{-1}]$ is given by

$E_{1}(L[\eta^{-1}])=\mathbb{F}_{2}[\tau,\rho,v_{1}^{2},\iota]/\iota^{2}.$

Moreover, the periodicization map $L\to L[\eta^{-1}]$ induces the map
$E_{1}(L)\to E_{1}(L[\eta^{-1}])$ whose values are given in Table
LABEL:tab:L-E1-generators.

###### Proof.

As in Proposition 4.7, we can just invert $h_{1}$ in the additive description
of $E_{1}(L)$ given in Proposition 5.1.

The map
$E_{1}(\mathrm{ko}[\eta^{-1}])\overset{\psi^{3}-1}{\longrightarrow}E_{1}(\mathrm{ko}[\eta^{-1})$
is trivial because $(\psi^{3}-1)(h_{1})=0$, as shown in Table
LABEL:tab:ko-E1-generators (see also Lemma 4.10). Therefore, the long exact
sequence

$\cdots\longrightarrow E_{1}(L[\eta^{-1})\longrightarrow
E_{1}(\mathrm{ko}[\eta^{-1}])\overset{\psi^{3}-1}{\longrightarrow}E_{1}(\mathrm{ko}[\eta^{-1}])\longrightarrow\cdots$

splits as

$E_{1}(L[\eta^{-1}])\cong
E_{1}(\mathrm{ko}[\eta^{-1}])\oplus\Sigma^{-1}E_{1}(\mathrm{ko}[\eta^{-1}]).$

With Proposition 4.7, this establishes the additive structure of
$E_{1}(L[\eta^{-1}])$, as well as most of the multiplicative structure.

The relation $\iota^{2}=0$ is immediate because there are no possible non-zero
values for $\iota^{2}$. ∎

######  Remark 5.3.

As in Remark 4.8, Table LABEL:tab:L-E1-generators gives an unexpected value
for $\tau^{2}$, which arises from Equation (1.1). Also, the last column of
Table LABEL:tab:L-E1-generators leaves out of $h_{1}$ for readability.

######  Remark 5.4.

Note that $E_{1}(L[\eta^{-1}])$ is very close to the effective $E_{1}$-page
for the $\eta$-periodic sphere $S[\eta^{-1}]$ [RSØ19, Theorem 2.32] [OR20,
Theorem 2.3]. The element $\iota$ is not present in $E_{1}(S[\eta^{-1}])$, but
the elements $\iota v_{1}^{2k}$ are present.

######  Proposition 5.5.

Some values of the differentials in the effective spectral sequence of
$L[\eta^{-1}]$ are:

  1. (1)

$d_{1}(v_{1}^{2})=\tau$.

  2. (2)

$d_{n+1}(v_{1}^{2^{n}})=\rho^{n+1}\cdot\iota v_{1}^{2^{n}}$ for $n\geq 2$.

The effective differentials are zero on all other multiplicative generators on
all pages.

Following our convention throughout this section, we have omitted the powers
of $h_{1}$ from the formulas in Proposition 5.5.

###### Proof.

The $d_{1}$ differential follows from [RSØ19, Lemma 4.2] or [OR20, Theorem
2.6].

To study the higher differentials, consider the map $S[\eta^{-1}]\to
L[\eta^{-1}]$. This map induces an isomorphism on stable homotopy groups,
except in coweight $-1$. This follows from a minor adjustment to [BH20,
Theorem 1.1]. The adjustment arises from the fact that our $L[\eta^{-1}]$ is
the fiber of
$\mathrm{ko}[\eta^{-1}]\overset{\psi^{3}-1}{\longrightarrow}\mathrm{ko}[\eta^{-1}]$,
while [BH20, Theorem 1.1] refers to the fiber of
$\mathrm{ko}[\eta^{-1}]\overset{\psi^{3}-1}{\longrightarrow}\Sigma^{8,4}\mathrm{ko}[\eta^{-1}]$.

The homotopy of $S[\eta^{-1}]$ is completely computed in [GI16], so the
homotopy of $L[\eta^{-1}]$ is known (except in coweight $-1$). There is only
one pattern of differentials that is compatible with the known values for
$L[\eta^{-1}]$. ∎

######  Remark 5.6.

In the language of [OR20, Section 4], Proposition 5.5 establishes the profile
of the $\eta$-periodic effective spectral sequence over $\mathbb{R}$.

###  5.3. Multiplicative relations for $E_{1}(L)$

In this section, we will completely describe the product structure on
$E_{1}(L)$. We do not need all of this structure for our later computations,
but we include it for completeness.

######  Proposition 5.7.

Table LABEL:tab:E1-products lists some products in $E_{1}(L)$.

Table 9. Products in $E_{1}(L)$: $j\geq 0$ and $k\geq 0$ | $\rho v_{1}^{2j}$ | $h_{1}v_{1}^{2j}$ | $\tau h_{1}v_{1}^{2j}$ | $\iota v_{1}^{2j}$
---|---|---|---|---
$\rho v_{1}^{2k}$ | $\rho\cdot\rho v_{1}^{2j+2k}$ |  |  |
$h_{1}v_{1}^{2k}$ | $\rho\cdot h_{1}v_{1}^{2j+2k}$ | $h_{1}\cdot h_{1}v_{1}^{2j+2k}$ |  |
$\tau h_{1}v_{1}^{2k}$ | $\rho\cdot\tau h_{1}v_{1}^{2j+2k}$ | $h_{1}\cdot\tau h_{1}v_{1}^{2j+2k}$ | $\tau^{2}\cdot h_{1}\cdot h_{1}v_{1}^{2j+2k}+$ |
|  |  | $+\rho\cdot\rho v_{1}^{2j+2k+2}$ |
$\iota v_{1}^{2k}$ | $\rho\cdot\iota v_{1}^{2j+2k}$ | $h_{1}\cdot\iota v_{1}^{2j+2k}$ | $\tau h_{1}\cdot\iota v_{1}^{2j+2k}$ | $0$
Table 9. Products in $E_{1}(L)$: $j\geq 0$ and $k\geq 0$

###### Proof.

All of these products are detected by $E_{1}(L[\eta^{-1}])$, which is
described in Proposition 5.2. We need the values of the periodicization map
$E_{1}(L)\to E_{1}(L[\eta^{-1}])$ given in Table LABEL:tab:L-E1-generators. ∎

###  5.4. The effective $d_{1}$ differential for $L$

Our next task is to compute the differentials in the effective spectral
sequence for $L$.

######  Proposition 5.8.

Table LABEL:tab:L-d1 gives the values of the effective $d_{1}$ differential on
the multiplicative generators of $E_{1}(L)$.

Table 10. Effective $d_{1}$ differentials for $L$: $k\geq 0$ coweight | $(s,f,w)$ | $x$ | $d_{1}(x)$
---|---|---|---
$2$ | $(0,0,-2)$ | $\tau^{2}$ | $\rho^{2}\cdot\tau h_{1}$
$4k$ | $(8k-1,1,4k-1)$ | $\rho v_{1}^{4k}$ |
$4k+2$ | $(8k+3,1,4k+1)$ | $\rho v_{1}^{4k+2}$ | $\rho h_{1}^{2}\cdot\tau h_{1}v_{1}^{4k}$
$4k$ | $(8k+1,1,4k+1)$ | $h_{1}v_{1}^{4k}$ |
$4k+2$ | $(8k+5,1,4k+3)$ | $h_{1}v_{1}^{4k+2}$ | $h_{1}^{3}\cdot\tau h_{1}v_{1}^{4k}$
$4k+3$ | $(8k+5,1,4k+2)$ | $\tau h_{1}v_{1}^{4k+2}$ | $\tau^{2}\cdot h_{1}^{3}\cdot h_{1}v_{1}^{4k}+\rho^{2}h_{1}\cdot h_{1}v_{1}^{4k+2}$
$4k+1$ | $(8k+1,1,4k)$ | $\tau h_{1}v_{1}^{4k}$ |
$4k+1$ | $(8k+3,1,4k+2)$ | $\iota v_{1}^{4k+2}$ | $\tau h_{1}\cdot h_{1}^{2}\cdot\iota v_{1}^{4k}$
$4k-1$ | $(8k-1,1,4k)$ | $\iota v_{1}^{4k}$ |
Table 10. Effective $d_{1}$ differentials for $L$: $k\geq 1$

###### Proof.

All of these differentials follow immediately from the effective $d_{1}$
differentials for $L[\eta^{-1}]$, which are all determined by Proposition
5.5(1) Beware that the exact values of the map $E_{1}(L)\to
E_{1}(L[\eta^{-1}])$, as shown in Table LABEL:tab:L-E1-generators, are
important.

For example, consider the differential on the element $\tau
h_{1}v_{1}^{4k+2}$. It maps to $\tau(v_{1}^{2})^{2k+1}$ in
$E_{1}(L[\eta^{-1}])$ (up to $h_{1}$ multiples, which as usual we ignore in
the $\eta$-periodic situation). The $\eta$-periodic differential on this
latter element is $\tau^{2}(v_{1}^{2})^{2k}$. Finally, we need to find an
element of $E_{1}(L)$ in the correct degree whose $\eta$-periodicization is
$\tau^{2}(v_{1}^{2})^{2k}$, The only possibility is $\tau^{2}\cdot
h_{1}^{3}\cdot h_{1}v_{1}^{4k}+\rho^{2}h_{1}\cdot h_{1}v_{1}^{4k+2}$. ∎

######  Remark 5.9.

All $d_{1}$ differentials in $E_{1}(L)$ can be deduced from the information in
Table LABEL:tab:L-d1 and the Leibniz rule, but the computations can be
complicated by the multiplicative relations of Table LABEL:tab:E1-products.
For example,

$d_{1}(\tau^{2}\cdot\tau h_{1}v_{1}^{2})=\rho^{2}\cdot\tau h_{1}\cdot\tau
h_{1}v_{1}^{2}+\tau^{2}(\tau^{2}\cdot h_{1}^{4}+\rho^{2}h_{1}\cdot
h_{1}v_{1}^{2})=\tau^{4}\cdot h_{1}^{4}+\rho^{4}\cdot v_{1}^{4}.$

Having completely analyzed the slice $d_{1}$ differentials for $E_{1}(L)$, it
is now possible to compute the $E_{2}$-page of the slice spectral sequence for
$L$.

######  Proposition 5.10.

The $E_{2}$-page of the effective spectral sequence for $L$ is depicted in
Figures 11, 12, 14, and 15.

For legibility, Figures 11, 12, 14, and 15 display $E_{2}(L)$ in four
different charts separated by coweight modulo $4$. Note that Figures 14 and 15
also serve as $E_{\infty}$-page charts in coweights $1$ and $2$ modulo $4$
because there are no higher differentials that affect these coweights.

###### Proof.

The Leibniz rule, together with the values in Table LABEL:tab:L-d1, completely
determines the effective $d_{1}$ differential on $E_{1}(L)$. The $E_{2}$-page
can then be determined directly. However, as in the proof of Theorem 4.3, the
computation is not entirely straightforward.

It turns out that the $d_{1}$ differential preserves the image of the map
$\Sigma^{-1}E_{1}(\mathrm{ko})\to E_{1}(L)$. Moreover, it turns out that all
$d_{1}$ differentials with values in the image of
$\Sigma^{-1}E_{1}(\mathrm{ko})\to E_{1}(L)$ also have source in this image.
(This is not for formal reasons; in fact, the higher effective differentials
do not have this property.) Consequently, the determination of the
$E_{2}$-page splits into two separate computations: one for the image of
$\Sigma^{-1}E_{1}(\mathrm{ko})\to E_{1}(L)$, and one for the cokernel of the
same map.

In more concrete terms, we can determine $E_{2}(L)$ by first considering only
elements of the form $\iota x$, and then separately considering only elements
that are not of this form.

The $d_{1}$ differential on the image of $\Sigma^{-1}E_{1}(\mathrm{ko})\to
E_{1}(L)$ is identical to the $d_{1}$ differential for $\mathrm{ko}$ discussed
in Section 4. The $d_{1}$ differential on the cokernel of
$\Sigma^{-1}E_{1}(\mathrm{ko})\to E_{1}(L)$ is similar to the $d_{1}$
differential on $E_{1}(\mathrm{ko})$, but slightly different. The difference
is created by the absence of the elements $v_{1}^{2k}$ in $E_{1}(L)$. ∎

###  5.5. Higher differentials

We now consider the higher differentials in the effective spectral sequence
for $L$.

By inspection of the charts for $E_{2}(L)$, the only possible higher
differentials have source in coweight congruent to $0$ modulo $4$ and value in
coweight congruent to $3$ modulo $4$. In other words, in coweights congruent
to $1$ and $2$ modulo 4, we have that $E_{2}(L)$ equals $E_{\infty}(L)$.

It turns out that there are many higher differentials. In fact, nearly all of
the elements in $E_{2}(L)$ in coweight congruent to 0 modulo 4 support
differentials. While it is possible to write down explicit formulas for all of
these differentials, the formulas would be cumbersome and not so helpful.
Rather, we give a more qualitative description of the differentials because it
is more useful for computation.

######  Proposition 5.11.

Consider the elements of $E_{2}(L)$ in coweights congruent to $0$ modulo $4$
that belong to the cokernel of the map $\Sigma^{-1}E_{2}(\mathrm{ko})\to
E_{2}(L)$.

  1. (1)

The only permanent cycles are the multiples of $1$, the multiples of
$2\tau^{4k}$ for $k\geq 0$, and $\rho^{a}h_{1}^{b}$ for all $a\geq 0$ and
$b\geq 0$.

  2. (2)

Excluding the elements listed in (1), if an element has coweight congruent to
$2^{r-1}$ modulo $2^{r}$, then it supports a $d_{r}$ differential.

Proposition 5.11 may seem imprecise because it does not give the values of the
differentials. However, there is only one non-zero possible value in every
case, so there is no ambiguity.

###### Proof.

These differentials follow immediately from the $\eta$-periodic differentials
of Proposition 5.5, together with multiplicative relations in $E_{2}(L)$.

For example, consider the element $\tau^{8}\cdot\rho v_{1}^{12}$ in coweight
$20$, which is congruent to $2^{2}$ modulo $2^{3}$. Using Table
LABEL:tab:L-E1-generators, we find that this element maps to
$\rho^{9}(v_{1}^{2})^{10}$ in $E_{2}(L[\eta^{-1}])$. Here we are using that
$\tau^{2}$ is zero in $E_{2}(L[\eta^{-1}])$ since it is hit by an
$\eta$-periodic $d_{1}$ differential. Proposition 5.5 says that this element
supports an $\eta$-periodic $d_{3}$ differential. It follows that
$\tau^{8}\cdot\rho v_{1}^{12}$ also supports a $d_{3}$ differential. ∎

######  Theorem 5.12.

The $E_{\infty}$-page of the effective spectral sequence for $L$ is depicted
in Figures 13, 14, 15, 16, 17, 18, and 19.

###### Proof.

The $E_{\infty}$-page can be deduced directly from the higher differentials
described in Proposition 5.11. ∎

The $E_{\infty}$-page in coweights congruent to $3$ modulo $4$ is by far the
most complicated case. Figures 17, 18, and 19 display $E_{\infty}(L)$ in
coweights congruent to 3 modulo 8, 7 modulo 16, and 15 modulo 32 respectively.

In each case (and more generally in coweights congruent to $2^{n-1}-1$ modulo
$2^{n}$, we see similar patterns with minor variations. The lower boundary of
each chart takes the same shape. The upper boundary of the $\tau$-periodic
portion of each chart also takes the same shape. However, the filtration jump
between the lower and upper boundaries increases linearly with $n$.

In addition to the $\tau$-periodic portion of each chart, there are also
$\tau$-torsion, $\eta$-periodic regions. These consist of bands of infinite
$h_{1}$-towers of width $n$ that repeat every $2^{n+1}$ stems. The first such
band starts at $\iota v_{1}^{2^{n-1}}$.

###  5.6. Hidden extensions

Our last goal is to compute hidden extensions by $\rho$, $\mathsf{h}$, and
$\eta$. See [Isa19, Section 4.1] for a precise definition of a hidden
extension. Fortunately, none of the complications associated with crossing
extensions occur in this manuscript.

######  Proposition 5.13.

Table LABEL:tab:L-hidden lists some hidden extensions by $\rho$, $\mathsf{h}$,
and $\eta$ in the effective spectral sequence for $L$.

###### Proof.

The last column of Table LABEL:tab:L-hidden indicates the reason for each
hidden extension. Some of the hidden extensions follow from the analogous
extensions for $\mathrm{ko}$ given in Table LABEL:tab:ko-hidden, using the
maps $\Sigma^{-1}\mathrm{ko}\to L$ and $L\to\mathrm{ko}$.

Other extensions follow from the long exact sequence associated to the cofiber
sequence

$\Sigma^{-1,-1}L\overset{\rho}{\longrightarrow}L\overset{}{\longrightarrow}L/\rho.$

Here we need that the homotopy of $L/\rho$ is isomorphic to the homotopy of
$L^{\mathbb{C}}$, as shown in Proposition 2.9. For example, the hidden
$\mathsf{h}$ extensions of Proposition 3.16 give hidden $\mathsf{h}$
extensions in $L/\rho$, which then imply the hidden extension from $\iota
4v_{1}^{2}$ to $h_{1}^{2}\cdot\tau h_{1}$.

∎

Table 11. Hidden extensions in $E_{\infty}(L)$ coweight | source | type | target | $(s,f,w)$ | proof
---|---|---|---|---|---
$0$ | $\iota\cdot\tau h_{1}$ | $\mathsf{h}$ | $\iota\cdot\rho h_{1}\cdot\tau h_{1}$ | $(0,2,0)$ | $\Sigma^{-1}\mathrm{ko}\to L$
$1$ | $\tau h_{1}$ | $\mathsf{h}$ | $\rho h_{1}\cdot\tau h_{1}$ | $(1,1,0)$ | $L\to\mathrm{ko}$
$1$ | $\iota(\tau h_{1})^{2}$ | $\mathsf{h}$ | $\iota\cdot\rho h_{1}(\tau h_{1})^{2}$ | $(1,3,0)$ | $\Sigma^{-1}\mathrm{ko}\to L$
$1$ | $\iota\cdot 2\tau^{2}$ | $\eta$ | $\iota\cdot\rho(\tau h_{1})^{2}$ | $(-1,1,-2)$ | $\Sigma^{-1}\mathrm{ko}\to L$
$1$ | $\iota 2v_{1}^{2}$ | $\rho$ | $\iota\cdot h_{1}(\tau h_{1})^{2}$ | $(3,1,2)$ | $\Sigma^{-1}\mathrm{ko}\to L$
$1$ | $\iota 4v_{1}^{2}$ | $\mathsf{h}$ | $h_{1}^{2}\cdot\tau h_{1}$ | $(3,1,2)$ | $L/\rho$
$2$ | $(\tau h_{1})^{2}$ | $\mathsf{h}$ | $\rho h_{1}(\tau h_{1})^{2}$ | $(2,2,0)$ | $L\to\mathrm{ko}$
$3$ | $\iota 4\tau^{2}v_{1}^{2}$ | $\mathsf{h}$ | $(\tau h_{1})^{3}$ | $(3,1,0)$ | $L/\rho$
$3$ | $\iota 2\tau^{2}v_{1}^{2}$ | $\rho$ | $\iota\tau^{4}\cdot h_{1}^{3}$ | $(3,1,0)$ | $\Sigma^{-1}\mathrm{ko}\to L$
$3$ | $\iota 2\tau^{2}v_{1}^{2}$ | $\eta$ | $\rho^{3}\cdot\iota v_{1}^{4}$ | $(3,1,0)$ | $\Sigma^{-1}\mathrm{ko}\to L$
$2$ | $2\tau^{2}$ | $\eta$ | $\rho(\tau h_{1})^{2}$ | $(0,0,-2)$ | $L\to\mathrm{ko}$
$3$ | $(\tau h_{1})^{3}$ | $\mathsf{h}$ | $\iota\tau^{4}\cdot\rho^{2}h_{1}^{6}$ | $(3,3,0)$ | $L/\rho$
$5$ | $\iota v_{1}^{4}\cdot 8\tau^{2}$ | $\mathsf{h}$ | $\rho^{2}\cdot\tau h_{1}v_{1}^{4}$ | $(7,1,2)$ | $L/\rho$
Table 11. Hidden extensions in $E_{\infty}(L)$

######  Remark 5.14.

The hidden extensions in Table LABEL:tab:L-hidden are $\tau^{4}$-periodic in
the following sense. If we take the source and target of each extension in
$E_{1}(L)$ and multiply by $\tau^{4}$, then we obtain permanent cycles that
are related by a hidden extension. For example, the hidden $\mathsf{h}$
extension from $\tau h_{1}$ to $\rho h_{1}\cdot\tau h_{1}$ generalizes to a
family of hidden extensions from $\tau^{4k+1}h_{1}$ to $\rho
h_{1}\cdot\tau^{4k+1}h_{1}$ for all $k\geq 0$.

######  Remark 5.15.

Similarly to the $\tau^{4}$-periodicity discussed in Remark 5.14, most of the
hidden extensions in Table LABEL:tab:L-hidden are $v_{1}^{4}$-periodic as
well. For example, the hidden $\mathsf{h}$ extension from $\tau h_{1}$ to
$\rho h_{1}\cdot\tau h_{1}$ generalizes to a family of hidden extensions from
$\tau h_{1}v_{1}^{4k}$ to $\rho h_{1}\cdot\tau h_{1}v_{1}^{4k}$ for all $k\geq
0$. There are three exceptions, which appear below the horizontal divider at
the bottom of the table. These exceptions are discussed in more detail in
Remarks 5.16, 5.17, and 5.18.

######  Remark 5.16.

The hidden $\eta$ extension from $2\tau^{2}$ to $\rho(\tau h_{1})^{2}$ is
$\tau^{4}$-periodic as in Remark 5.14, but it is not $v_{1}^{4}$-periodic. The
elements $2\tau^{2}v_{1}^{4k}$ are not permanent cycles for $k\geq 1$.

######  Remark 5.17.

The hidden $\mathsf{h}$ extension from $\iota v_{1}^{4}\cdot 8\tau^{2}$ to
$\rho^{2}\cdot\tau h_{1}v_{1}^{4}$ is $v_{1}^{4}$-periodic, but the situation
is slightly more complicated than in Remark 5.15. For all $k$,
$\rho^{2}\cdot\tau h_{1}v_{1}^{4k}$ receives a hidden $\mathsf{h}$ extension
from an appropriate multiple of $\iota v_{1}^{4k}\cdot 2\tau^{2}$. For
example, as shown in Figure 14, there is a hidden $\mathsf{h}$ extension from
$\iota v_{1}^{4k}\cdot 16\tau^{2}$ to $\rho^{2}\cdot\tau h_{1}v_{1}^{8}$.

######  Remark 5.18.

The hidden $\mathsf{h}$ extension from $(\tau h_{1})^{3}$ to
$\iota\tau^{4}\cdot\rho^{2}h_{1}^{6}$ is $v_{1}^{4}$-periodic, but the
situation is more complicated than in Remarks 5.15 and 5.17. For all $k\geq
0$, the element $(\tau h_{1})^{2}\tau h_{1}v_{1}^{4k}$ supports a hidden
$\mathsf{h}$ extension to the element of $E_{\infty}(L)$ of highest filtration
in the appropriate degree. For example, as shown in Figure 18, there is a
hidden $\mathsf{h}$ extension from $(\tau h_{1})^{2}\cdot\tau^{5}h_{1}$ to
$\iota\tau^{8}\cdot\rho^{3}h_{1}^{7}$. Figures 17, 18, and 19 show several
extensions of this type.

##  6\. Charts

We explain the notation used in the charts.

  * •

The horizontal coordinate is the stem $s$. The vertical coordinate is the
Adams-Novikov filtration $f$ (see Section 1.7 for further discussion).

  * •

Black or green circles represent copies of $\mathbb{F}_{2}$, periodicized by
some power of $\tau$. The relevant power of $\tau$ varies from chart to chart.

  * •

Black or green unfilled boxes represent copies of $\mathbb{Z}$ (the $2$-adic
integers), periodicized by some power of $\tau$. The relevant power of $\tau$
varies from chart to chart.

  * •

Black or green boxes containing a number $n$ represent copies of
$\mathbb{Z}/2^{n}$, periodicized by some power of $\tau$. The relevant power
of $\tau$ varies from chart to chart.

  * •

Red unfilled boxes represent copies of $\mathbb{Z}$ (the $2$-adic integers)
that are not $\tau^{k}$-periodic for any $k$.

  * •

Green objects represent elements in the image of the map
$\Sigma^{-1}\mathrm{ko}\to L$ (or $\Sigma^{-1}\mathrm{ko}^{\mathbb{C}}\to
L^{\mathbb{C}}$).

  * •

Black objects represent elements in the cokernel of the map
$\Sigma^{-1}\mathrm{ko}\to L$ (or $\Sigma^{-1}\mathrm{ko}^{\mathbb{C}}\to
L^{\mathbb{C}}$). In other words, they are detected by the map
$L\to\mathrm{ko}$ (or $L^{\mathbb{C}}\to\mathrm{ko}^{\mathbb{C}}$).

  * •

Lines of slope $1$ represent $h_{1}$-multiplications.

  * •

Black or green arrows of slope $1$ represent infinite sequences of elements
that are $\tau^{k}$-periodic for some $k>0$ and are connected by
$h_{1}$-multiplications.

  * •

Red arrows of slope $1$ represent infinite sequences of elements that are
connected by $h_{1}$-multiplications and are not $\tau^{k}$-periodic for any
$k$.

  * •

Lines of slope $-1$ represent $\rho$-multiplications.

  * •

Dashed lines of slope $-1$ represent $\rho$-multiplications whose values are
multiples of $\tau^{k}$ for some $k>0$. For example, in Figure 6, we have
$\rho\cdot\rho^{3}v_{1}^{4}$ equals $\tau^{4}\cdot h_{1}^{4}$.

  * •

Black or green arrows of slope $-1$ represent infinite sequences of elements
that are $\tau^{k}$-periodic for some $k>0$ and are connected by
$\rho$-multiplications.

  * •

Light blue lines of slope $-3$ represent effective $d_{1}$ differentials.

  * •

Dashed light blue lines of slope $-3$ represent effective $d_{1}$
differentials that hit multiples of $\tau^{k}$, for some $k>0$. For example,
the dashed line in Figure 1 indicates that $d_{1}(v_{1}^{2})$ equals $\tau
h_{1}^{3}$.

  * •

Dark blue lines indicate hidden extensions by $\mathsf{h}$, $\rho$, or
$h_{1}$.

  * •

Dashed dark blue lines indicate hidden extensions whose value is a multiple of
$\tau^{k}$ for some $k>0$. For example, in Figure 4, there is a hidden
$\mathsf{h}$ extension from $\iota 4v_{1}^{2}$ to $\tau h_{1}^{3}$.

Figure 1. The $E_{1}$-page of the effective spectral sequence for
$\mathrm{ko}^{\mathbb{C}}$

Figure 2. The $E_{\infty}$-page of the effective spectral sequence for
$\mathrm{ko}^{\mathbb{C}}$

Figure 3. The $E_{1}$-page of the effective spectral sequence for
$L^{\mathbb{C}}$

Figure 4. The $E_{\infty}$-page of the effective spectral sequence for
$L^{\mathbb{C}}$

Figure 5. The $E_{1}$-page of the effective spectral sequence for
$\mathrm{ko}$

Figure 6. The $E_{\infty}$-page of the effective spectral sequence for
$\mathrm{ko}$ in coweights $0$ mod $4$

paperlandscape,10.4in:11.0in DIVlast

Figure 7. The $E_{\infty}$-page of the effective spectral sequence for
$\mathrm{ko}$ in coweights $1$ mod $4$

Figure 8. The $E_{\infty}$-page of the effective spectral sequence for
$\mathrm{ko}$ in coweights $2$ mod $4$

Figure 9. Some differentials in the effective spectral sequence for
$\mathrm{ko}$ Figure 10. The $E_{1}$-page of the effective spectral sequence
for $L$

paperlandscape,10.4in:7.0in DIVlast

Figure 11. The $E_{2}$-page of the effective spectral sequence for $L$ in
coweights $0$ mod $4$

paperlandscape,9.6in:7.0in DIVlast

Figure 12. The $E_{2}$-page of the effective spectral sequence for $L$ in
coweights $3$ mod $4$

paperlandscape,11.3in:10.2in DIVlast

Figure 13. The $E_{\infty}$-page of the effective spectral sequence for $L$ in
coweights $0$ mod $4$ Figure 14. The $E_{\infty}$-page of the effective
spectral sequence for $L$ in coweights $1$ mod $4$

paperlandscape,10.4in:8.2in DIVlast

Figure 15. The $E_{\infty}$-page of the effective spectral sequence for $L$ in
coweights $2$ mod $4$ Figure 16. The $E_{\infty}$-page of the effective
spectral sequence for $L$ in coweight $-1$

paperlandscape,20in:7.8in DIVlast

Figure 17. The $E_{\infty}$-page of the effective spectral sequence for $L$ in
coweights $3$ mod $8$

paperlandscape,33.78in:8.66in DIVlast

Figure 18. The $E_{\infty}$-page of the effective spectral sequence for $L$ in
coweights $7$ mod $16$

paperlandscape,23.4in:10.4in DIVlast

Figure 19. The $E_{\infty}$-page of the effective spectral sequence for $L$ in
coweights $15$ mod $32$

paperportrait,8.5in:11in DIVlast

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