# Transchromatic phenomena in the equivariant slice spectral sequence

Lennart Meier  Mathematical Institute, Utrecht University, Utrecht, 3584 CD,
the Netherlands<EMAIL_ADDRESS>,  XiaoLin Danny Shi  Department of
Mathematics, University of Washington, 4110 E Stevens Way NE, Seattle, WA
98195<EMAIL_ADDRESS>and  Mingcong Zeng  Department of Mathematics, Ny
Munkegade 118, 8000 Aarhus C, Denmark<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove a transchromatic phenomenon for Hill–Hopkins–Ravenel
and Lubin–Tate theories. This establishes a direct relationship between the
equivariant slice spectral sequences of height-$h$ and height-$(h/2)$
theories. As applications of this transchromatic phenomenon, we prove
periodicity and vanishing line results for these theories.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation and main theorem
    2. 1.2 Periodicities and vanishing lines
    3. 1.3 The stratification tower of Hill–Hopkins–Ravenel theories
    4. 1.4 Open questions and future directions
    5. 1.5 Organization of the paper
  2. 2 Stratification of the slice spectral sequence
  3. 3 Shearing isomorphisms of spectral sequences
    1. 3.1 The pullback functor
    2. 3.2 The dual slice tower
    3. 3.3 Shearing isomorphism
  4. 4 Isomorphism of localized slice towers
  5. 5 Correspondence formulas
  6. 6 The Transchromatic Isomorphism Theorem
  7. 7 The transchromatic tower
  8. 8 $RO(G)$-graded periodicities
  9. 9 Strong vanishing lines

##  1\. Introduction

###  1.1. Motivation and main theorem

The equivariant slice filtration was developed by Hill, Hopkins, and Ravenel
in their celebrated solution of the Kervaire invariant one problem [HHR16].
This filtration gives rise to the equivariant slice spectral sequence, which
is a valuable computational tool in equivariant homotopy theory for analyzing
the norms of the Real bordism spectrum
$MU^{(\\!(G)\\!)}:=N_{C_{2}}^{G}(MU_{\mathbb{R}})$.

Let $E_{h}=E(k,\Gamma_{h})$ be the Lubin–Tate spectrum associated to a formal
group law $\Gamma_{h}$ of height $h$ over a finite field $k$ of characteristic
2. Goerss, Hopkins, Miller, and Lurie have shown that $E_{h}$ is a commutative
ring spectrum, and there is an action of $\textup{Aut}(\Gamma_{h})$ on $E_{h}$
by commutative ring maps [Rez98, GH04, Lur18]. For any finite group $G$ of
$\textup{Aut}(\Gamma_{h})$, we can view $E_{h}$ as a $G$-equivariant
commutative ring spectrum using the cofree functor $F(EG_{+},-)$.

The theories $E_{h}^{hG}$ hold significant importance in chromatic homotopy
theory, as they are periodic theories capable of detecting periodic patterns
and nontrivial families of elements in the stable homotopy groups of spheres
[Ada66, Rav78, HHR16, LSWX19, BMQ23]. They are also used in the study of the
$K(h)$-local sphere [GHM04, GHMR05, Beh06, Hen07, HKM13, Lad13, GHM14, BG18,
BGH22]. At height 1, there is a well-established understanding of these
theories due to their relationship with complex and real $K$-theory [Ada74,
Bou79, Rav84, Bou85]. At height 2, they are closely related to topological
modular forms with level structures [HM98, Bau08, MR09, DFHH14, BO16, HM17,
HHR17].

In this paper, we focus our attention at the prime 2. Prior to [HS20, HSWX23],
computations of $E_{h}^{hG}$ at the prime 2 were limited to heights $h\leq 2$.
Explicitly computing these theories posed challenges, partly due to the lack
of geometric insight into the $G$-action on $E_{h}$.

In [HS20], Hahn and the second author produced a $G$-equivariant orientation

$MU^{(\\!(G)\\!)}\longrightarrow E_{h}.$

This orientation establishes a connection between the obstruction theoretic
$G$-action on $E_{h}$, induced from $\textup{Aut}(\Gamma)$, to the geometric
$G$-action on $MU^{(\\!(G)\\!)}$, which arises from complex conjugation. This
connection enables us to construct $E_{h}$ as a localization of a quotient of
$MU^{(\\!(G)\\!)}$.

For $G=C_{2^{n+1}}$ and $h=2^{n}\cdot m$, Beaudry–Hill–Shi–Zeng [BHSZ21]
constructed $G$-equivariant models of $E_{h}$ using quotients of the norm
$BP^{(\\!(G)\\!)}:=N_{C_{2}}^{G}(BP_{\mathbb{R}})$, where $BP_{\mathbb{R}}$ is
the Real Brown–Peterson spectrum. These quotients, denoted as
$BP^{(\\!(G)\\!)}\langle m\rangle$, are the equivariant analogues of the
classical truncated Brown–Peterson spectrum $BP\langle m\rangle$.
Consequently, computing the fixed points $E_{h}^{hG}$ is directly tied to
computing the $G$-fixed points of $BP^{(\\!(G)\\!)}\langle m\rangle$.

Using the equivariant slice spectral sequence, the following computations have
been made:

  1. (1)

The $C_{2}$-fixed points of $BP_{\mathbb{R}}\langle m\rangle$ for all $m\geq
1$. This yields $E_{m}^{hC_{2}}$ for all $m\geq 1$ ([HK01], [HS20]);

  2. (2)

The $C_{4}$-fixed points of $BP^{(\\!(C_{4})\\!)}\langle 1\rangle$. This
yields $E_{2}^{hC_{4}}$ ([BO16], [HHR17]);

  3. (3)

The $C_{4}$-fixed points of $BP^{(\\!(C_{4})\\!)}\langle 2\rangle$. This
yields $E_{4}^{hC_{4}}$ ([HSWX23]).

The goal of this paper is to achieve a systematic understanding of the
structural properties of $E_{h}^{hG}$ across different groups and heights.
This includes differentials, periodicities, and vanishing lines. A key
conjecture by Hill–Hopkins–Ravenel states that the slice spectral sequences of
$E_{h}^{hG}$ exhibit connections to each other as we vary $h$ and $G$:

######  Conjecture 1.1 (Transchromatic Phenomenon).

The slice spectral sequences of higher height theories contain isomorphism
regions to the slice spectral sequences of lower height theories.

1.1 originated from the work of Hill, Hopkins, and Ravenel during their study
of $BP^{(\\!(C_{4})\\!)}\langle 1\rangle$ [HHR17], where they observed
isomorphism regions between the $C_{4}$-slice spectral sequence of
$BP^{(\\!(C_{4})\\!)}\langle 1\rangle$ ($E_{2}^{hC_{4}}$) and the
$C_{2}$-slice spectral sequence of $BP_{\mathbb{R}}\langle 1\rangle$
($E_{1}^{hC_{2}}$). A similar phenomenon was later discovered between the
$C_{4}$-slice spectral sequence of $BP^{(\\!(C_{4})\\!)}\langle 2\rangle$
($E_{4}^{hC_{4}}$) and the $C_{2}$-slice spectral sequence of
$BP_{\mathbb{R}}\langle 2\rangle$ ($E_{2}^{hC_{2}}$) [HSWX23].

To this end, the main theorem of this paper is the following:

######  Theorem A (Transchromatic Isomorphism).

For $G=C_{2^{n+1}}$ and $m\geq 1$, there is a shearing isomorphism
$d_{2r-1}\leftrightsquigarrow d_{r}$ between the following regions of spectral
sequences:

  1. (1)

The $G$-slice spectral sequence of $BP^{(\\!(G)\\!)}\langle m\rangle$ on or
above the line of slope $1$; and

  2. (2)

The $(G/C_{2})$-slice spectral sequence of $BP^{(\\!(G/C_{2})\\!)}\langle
m\rangle$.

More precisely, A states that all the differentials in the $G$-slice spectral
sequence of $BP^{(\\!(G)\\!)}\langle m\rangle$ on or above the line of slope 1
have lengths of the form $(2r-1)$, where $r\geq 2$. Additionally, the
transchromatic isomorphism induces a one-to-one correspondence between these
differentials and the $d_{r}$-differentials in the $(G/C_{2})$-slice spectral
sequence of $BP^{(\\!(G/C_{2})\\!)}\langle m\rangle$.

Combined with the results in [BHSZ21], A establishes an explicit
correspondence between differentials in the $C_{2^{n+1}}$-slice spectral
sequence of $E_{h}$ on or above the line of slope 1 and the differentials in
the $C_{2^{n}}$-slice spectral sequence of $E_{h/2}$.

For any finite subgroup $G\subset\mathbb{G}_{h}$, the 2-Sylow subgroup $H$ of
$G\cap\mathbb{S}_{h}$ is isomorphic to either a cyclic group of order a power
of $2$ or the quaternion group $Q_{8}$ [Hew95, Hew99, Buj12]. In light of this
classification, A resolves 1.1 for the case when the Sylow subgroup $H$ is
cyclic.

Computationally, A significantly simplifies the complexity of the slice
spectral sequences of Hill–Hopkins–Ravenel ($BP^{(\\!(C_{2^{n+1}})\\!)}\langle
m\rangle$) and Lubin–Tate theories ($E_{h}^{hC_{2^{n+1}}}$). More
specifically, the Stratification Theorem [MSZ23a, Theorem A] shows that the
$C_{2^{n+1}}$-slice spectral sequence is stratified into $(n+1)$ regions,
separated by the lines $\mathcal{L}_{2^{i}-1}$ of slope $(2^{i}-1)$, $0\leq
i\leq n+1$ (see Figure 1). A shows that all the differentials in the $n$
regions above the line $\mathcal{L}_{1}$ are accounted for by the
$C_{2^{n}}$-slice spectral sequence of $BP^{(\\!(C_{2^{n}})\\!)}\langle
m\rangle$ ($E_{h/2}^{hC_{2^{n}}}$). As a consequence, the new differentials
needing computation are limited to the conical region between
$\mathcal{L}_{0}$ and $\mathcal{L}_{1}$. The vanishing lines results in
Section 1.2 will further narrow down the possibilities for differentials in
this region.

Figure 1. The Transchromatic Isomorphism Theorem.

###  1.2. Periodicities and vanishing lines

A establishes explicit correspondence formulas between differentials in the
corresponding isomorphism regions of the slice spectral sequences. In addition
to providing an inductive approach for computing differentials, it also offers
a new method for studying $RO(G)$-graded periodicities and vanishing lines for
the higher-height theories through the lower-height theories.

The classical 2-periodicity of $E_{h}$ can be refined to a $C_{2}$-equivariant
$\rho_{2}$-periodicity, and, $G$-equivariantly, a $\rho_{G}$-periodicity
[HS20]. On top of these periodicities, there are numerous other $RO(G)$-graded
periodicities that are of the form $(|V|-V)$, where $V\in RO(G)$. They
correspond to the survival of orientation classes $u_{V}$ to the
$\mathcal{E}_{\infty}$-page of the slice spectral sequence of $E_{h}$.

The $RO(G)$-graded periodicities of $E_{h}$ play a crucial role in the proof
of the Kervaire invariant one problem, producing the 256-periodicity of the
detecting spectrum $\Omega$ [HHR16, Theorem 1.7]. They are also valuable for
understanding periodic behaviors for differentials in the slice spectral
sequence [HHR17, HSWX23, BHL+22], as well as for computing Picard groups
[MS16, HMS17, HM17, BBHS20, HLS21].

######  Theorem B (Periodicity, 8.3).

Let $G=C_{2^{n+1}}$, $h=2^{n}\cdot m$, and suppose $V\in RO(G/C_{2})$. Then
$(|V|-V)$ is an $RO(G)$-graded periodicity for $E_{h}$ if and only if
$(|V|-V)$ is an $RO(G/C_{2})$-graded periodicity for $E_{h/2}$. Here, $V$ is
treated as an element in $RO(G)$ through the pullback along the map $G\to
G/C_{2}$.

B provides an inductive method for establishing $RO(G)$-graded periodicities.
It shows that the periodicities obtained for the smaller height theories
immediately extend to the higher height theories. As we move up the height and
group, the $RO(G)$-graded periodicities build up and accumulate, rather than
being random.

More specifically, when $G=C_{2^{n+1}}$ is the cyclic group of order $2^{n+1}$
and we are working 2-locally, the $RO(G)$-graded homotopy groups are graded by
the irreducible real representations
$\\{1,\sigma,\lambda_{1},\ldots,\lambda_{n}\\}$, where 1 is the trivial
representation, $\sigma$ is the sign representation, and $\lambda_{i}$ is the
2-dimensional real representation corresponding to rotation by
$\left(\frac{\pi}{2^{i}}\right)$. Similarly, when considering the quotient
group $G/C_{2}$, the $RO(G/C_{2})$-graded homotopy groups are graded by the
representations $\\{1,\sigma,\lambda_{1},\ldots,\lambda_{n-1}\\}$. By applying
B to the $RO(G/C_{2})$-graded periodicities of $E_{h/2}$, we can immediately
obtain all the $RO(G)$-graded periodicities of $E_{h}$ that are generated by
the representations $\\{1,\sigma,\lambda_{1},\ldots,\lambda_{n-1}\\}$.

Our second application of A is in establishing vanishing lines. Classically,
it is a consequence of the Nilpotence Theorem of Devinatz, Hopkins, and Smith
that the homotopy fixed point spectral sequences of $E_{h}$ all admit strong
horizontal vanishing lines at some finite filtration (see [DH04, Section 5]
and [BGH22, Section 2.3]). This means that there exists a finite number $R>0$
such that the spectral sequence collapses after the $\mathcal{E}_{R}$-page,
with no elements of filtration greater than or equal to $R$ surviving to the
$\mathcal{E}_{\infty}$-page.

While theoretically useful, the existence of the horizontal vanishing line
cannot be used to compute any differentials, because there is no knowledge of
exactly where the vanishing occurs. Knowing a bound (or even better, the exact
location) for the vanishing line would significantly aid in computing
differentials and in studying their structural properties.

In [DLS22], Duan, Li, and the second author established a bound for the strong
horizontal vanishing line in the $C_{2^{n+1}}$-homotopy fixed point spectral
sequence and the $C_{2^{n+1}}$-slice spectral sequence of $E_{h}$.

Combining A with the results in [DLS22], we prove the following result:

######  Theorem C (Vanishing Lines, 9.3).

Let $G=C_{2^{n+1}}$ and $h=2^{n}\cdot m$. For all $V\in RO(G)$ and $0\leq
k\leq n$, the line $\mathcal{L}_{2^{k}-1,N_{k}}$ defined by the equation

$s=(2^{k}-1)(t-s)+N_{k}$

is a strong vanishing line in the slice spectral sequence of $E_{h}$ within
the region $t-s\geq 0$. Here, $N_{k}$ is the constant
$(2^{h/2^{k}+n+1}-2^{n+1}+2^{k})$. This strong vanishing line satisfies the
following properties:

  1. (1)

There are no classes on or above $\mathcal{L}_{2^{k}-1,N_{k}}$ that survive to
the $\mathcal{E}_{\infty}$-page;

  2. (2)

Differentials originating on or above the line $\mathcal{L}_{2^{k}-1}$ have
lengths at most ${N_{k}-(2^{k}-1)=2^{h/2^{k}+n+1}-2^{n+1}+1}$;

  3. (3)

All differentials originating below $\mathcal{L}_{2^{k}-1}$ have targets
strictly below $\mathcal{L}_{2^{k}-1,N_{k}}$.

Figure 2. Vanishing Lines in the slice spectral sequence.

The collection of strong vanishing lines established in C is unique to the
equivariant slice spectral sequence. With the exception of the horizontal
vanishing line $\mathcal{L}_{0,N_{0}}$, the remaining vanishing lines do not
appear in the homotopy fixed point spectral sequence of $E_{h}$. This absence
is due to the absence of stratification regions in the homotopy fixed point
spectral sequence, which, in turn, prevents the observation of transchromatic
phenomena.

In contrast, the slice spectral sequence has $(n+1)$ stratification regions
[MSZ23a]. A grants each stratification region with a vanishing line of its
own. Consequently, in addition to the original horizontal vanishing line
established in [DLS22], there are $n$ more vanishing lines of slopes $1$, $3$,
$\ldots$, $(2^{n}-1)$, respectively.

Together, these vanishing lines offer more precise bounds for both the lengths
and locations of differentials, as illustrated in Figure 2. Rather than
computing the entire spectral sequence all at once, one can focus on proving
differentials in each of the $(n+1)$ conical region individually. This
significantly reduces the number of potential differential possibilities. This
feature makes the slice spectral sequence an effective computational tool with
the potential to greatly expand our ability to analyze and compute
$E_{h}^{hG}$ at higher heights.

###  1.3. The stratification tower of Hill–Hopkins–Ravenel theories

We will now outline our strategy for proving A. It is worth noting that,
although 1.1 originally emerged from explicit spectral sequence computations,
our proof of A does not require the computation of explicit differentials.
Instead, we will prove a more general statement on the level of filtered
objects, which establishes $RO(G)$-graded shearing isomorphisms for all
generalized Hill–Hopkins–Ravenel theories $BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle$.

By the work of Hill–Hopkins–Ravenel [HHR16, Section 5], we have the following
isomorphism:

$\pi_{*\rho_{2}}^{C_{2}}BP^{(\\!(C_{2^{n}})\\!)}\cong\mathbb{Z}_{(2)}[C_{2^{n}}\cdot\bar{t}_{1}^{C_{2^{n}}},C_{2^{n}}\cdot\bar{t}_{2}^{C_{2^{n}}},\ldots].$

Here,
$\bar{t}_{i}^{C_{2^{n}}}\in\pi_{(2^{i}-1)\rho_{2}}^{C_{2}}BP^{(\\!(C_{2^{n}})\\!)}$,
and $C_{2^{n}}\cdot\bar{t}_{i}^{C_{2^{n}}}$ denotes the set

$\left\\{\bar{t}_{i}^{C_{2^{n}}},\gamma\bar{t}_{i}^{C_{2^{n}}},\ldots,\gamma^{2^{n-1}-1}\bar{t}_{i}^{C_{2^{n}}}\right\\}$

containing $2^{n-1}$ elements, where $\gamma$ represents the generator of
$C_{2^{n}}$. The method of twisted monoid rings [HHR16, Section 2] allows us
to form quotients of $BP^{(\\!(C_{2^{n}})\\!)}$ by collections of permutation
summands of the form $C_{2^{n}}\cdot\bar{t}_{i}^{C_{2^{n}}}$.

######  Definition 1.2.

For $I\subseteq\mathbb{N}$, the generalized Hill–Hopkins–Ravenel theory
$BP^{(\\!(C_{2^{n}})\\!)}\langle I\rangle$ is defined as

$BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle:=BP^{(\\!(G)\\!)}/(C_{2^{n}}\cdot\bar{t}_{j}^{C_{2^{n}}}\,|\,j\notin
I).$

The theories $BP^{(\\!(C_{2^{n}})\\!)}\langle I\rangle$ are studied in
[BHL+22]. In particular, [BHL+22, Theorem A] shows that the slice associated
graded of $BP^{(\\!(C_{2^{n}})\\!)}\langle I\rangle$ is the generalized
Eilenberg–Mac Lane spectrum

$H\underline{\mathbb{Z}}\left[C_{2^{n}}\cdot\bar{t}_{i}^{C_{2^{n}}}\,|\,i\in
I\right].$

When $I=\\{1,2,\ldots,m\\}$, the spectrum $BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle$ is the classical Hill–Hopkins–Ravenel theory

$BP^{(\\!(C_{2^{n}})\\!)}\langle
m\rangle=BP^{(\\!(C_{2^{n}})\\!)}/(C_{2^{n}}\cdot\bar{t}_{m+1}^{C_{2^{n}}},C_{2^{n}}\cdot\bar{t}_{m+2}^{C_{2^{n}}},\ldots).$

For a fixed $I\subseteq\mathbb{N}$, the key to proving A lies in isolating and
comparing regions in the slice spectral sequences associated with the theories
$BP^{(\\!(C_{2^{n}})\\!)}\langle I\rangle$ as we vary $n$. To achieve this, we
will utilize the localized slice spectral sequence and the stratification
tower, developed by the authors in [MSZ23b] and [MSZ23a].

######  Theorem D (Localized Slice Tower Isomorphism, 4.1).

Suppose $n$ is a positive integer. For all $1\leq k\leq n$, we have the
following equivalence of towers:

$\widetilde{E}\mathcal{F}[C_{2^{k+1}}]\wedge
P_{\bullet}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle\right)\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2^{k}}}\mathscr{D}^{k}\left(\widetilde{E}\mathcal{F}[C_{2}]\wedge
P_{\bullet}(BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle I\rangle)\right).$

Here, $P_{\bullet}(-)$ denotes the dual slice tower,
$\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2^{k}}}(-)$ is the pullback functor as
defined in 3.1, and $\mathscr{D}(-)$ is the doubling operation as defined in
3.5.

D establishes equivalences between various localized dual slice towers,
leading to corresponding equivalences in the localized slice spectral
sequences across different groups. This equivalence results in a shearing
isomorphism of the corresponding localized slice spectral sequences. This
shearing isomorphism, combined with the Slice Recovery Theorem [MSZ23a,
Theorem 3.3], produces shearing isomorphisms for the $RO(G)$-graded slice
spectral sequences of $BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle$, $n\geq 0$.

######  Theorem E (Shearing Isomorphism, 3.7, 6.2).

For all $V\in RO(C_{2^{n+1}})$ and $1\leq k\leq n$, there is a shearing
isomorphism $d_{2^{k}r-(2^{k}-1)}\leftrightsquigarrow d_{r}$ between the
following regions of spectral sequences:

  1. (1)

The $(V+t-s,s)$-graded page of
${\textup{SliceSS}(BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle)}$ on or above
the line $\mathcal{L}^{V}_{2^{k}-1}$ within the region $t-s\geq 0$; and

  2. (2)

The $(V^{C_{2^{k}}}+t-s,s)$-graded page of
$\textup{SliceSS}(BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle I\rangle)$ on or above
the horizontal line $s=C_{V,k}$ within the region $t-s\geq 0$.

Here, the line $\mathcal{L}_{2^{k}-1}^{V}$ is a line of slope $(2^{k}-1)$ as
defined in 2.3, and $C_{V,k}$ is a constant as defined in 6.1.

A more detailed description of the shearing isomorphism in E is as follows:
let $\underline{\mathcal{E}}_{C_{2^{n+1}}}$ and
$\underline{\mathcal{E}}_{C_{2^{n-k+1}}}$ represent the slice spectral
sequences of $BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle$ and
$BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle I\rangle$, respectively. Then the
following statements hold:

  1. (1)

In the specified regions of their respective $\mathcal{E}_{2}$-pages, there is
an isomorphism

$\underline{\mathcal{E}}_{C_{2^{n+1}},2}^{s^{\prime},V+t^{\prime}}\cong\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2^{k}}}\left(\underline{\mathcal{E}}_{C_{2^{n-k+1}},2}^{s,V^{C_{2^{k}}}+t}\right),$
(1.1)

where

$\displaystyle t^{\prime}$ $\displaystyle=\left(|V^{C_{2^{k}}}|\cdot
2^{k}-|V|\right)+2^{k}t,$ $\displaystyle s^{\prime}$
$\displaystyle=\left(|V^{C_{2^{k}}}|\cdot
2^{k}-|V|\right)+(2^{k}-1)(t-s)+2^{k}s.$

  2. (2)

In the specified region for $\underline{\mathcal{E}}_{C_{2^{n+1}}}$, all the
differentials have lengths of the form

$2^{k}r-(2^{k}-1),\,\,\,r\geq 2.$

  3. (3)

The isomorphism (1.1) induces a one-to-one correspondence between the
$d_{2^{k}r-(2^{k}-1)}$-differentials in
$\underline{\mathcal{E}}_{C_{2^{n+1}}}$ originating on or above the line
$\mathcal{L}^{V}_{2^{k}-1}$ and the $d_{r}$-differentials in
$\underline{\mathcal{E}}_{C_{2^{n-k+1}}}$ originating on or above the
horizontal line $s=C_{V,k}$.

In particular, when we set $V=0$ in E, a shearing isomorphism
${d_{2^{k}r-(2^{k}-1)}\leftrightsquigarrow d_{r}}$ is established between the
following regions of spectral sequences:

  1. (1)

The slice spectral sequence of $BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle$ on
or above the line $\mathcal{L}_{2^{k}-1}$ (the line of slope $(2^{k}-1)$
through the origin); and

  2. (2)

The entire slice spectral sequence of $BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle
I\rangle$.

By setting $I=\\{1,2,\ldots,m\\}$, A immediately follows from E. As an
important application, it is an immediate consequence of A that in the slice
spectral sequence of $BP^{(\\!(C_{8})\\!)}\langle 1\rangle$
($E_{4}^{hC_{8}}$), which detects all the Kervaire invariant elements, the
differentials on or above the line of slope 1 are completely determined by the
differentials in the slice spectral sequence of $BP^{(\\!(C_{4})\\!)}\langle
1\rangle$ ($E_{2}^{hC_{4}}$).

Another notable case is when we set $I=\\{m\\}$. In this case, the spectrum
$BP^{(\\!(C_{2^{n}})\\!)}\langle I\rangle$ becomes

$BP^{(\\!(C_{2^{n}})\\!)}\langle m,m\rangle:=BP^{(\\!(C_{2^{n}})\\!)}\langle
m\rangle/(C_{2^{n}}\cdot\bar{t}_{1}^{C_{2^{n}}},\ldots,C_{2^{n}}\cdot\bar{t}_{m-1}^{C_{2^{n}}}).$

These theories have been explored in [BHL+22]. By setting $I=\\{m\\}$, E also
establishes a transchromatic isomorphism theorem for the theories
$BP^{(\\!(C_{2^{n+1}})\\!)}\langle m,m\rangle$.

We would like to emphasize the crucial role played by the pullback functor
$\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2^{k}}}(-)$, which is the right-adjoint to
the geometric fixed point functor $\Phi^{C_{2^{k}}}(-)$. This functor is
crucial in establishing the equivalence in D and the shearing isomorphism in
E. While comparing the slice spectral sequence of a $C_{2^{n+1}}$-spectrum
with a $C_{2^{n-k+1}}$-spectrum, it may be tempting to consider
$C_{2^{n-k+1}}$ as a subgroup of $C_{2^{n+1}}$. However, a key observation is
that, to establish the desired equivalence, one should instead treat
$C_{2^{n-k+1}}$ as the quotient group $C_{2^{n+1}}/C_{2^{k}}$.

The shearing isomorphism of E induces an explicit correspondence between
classes located in the isomorphism regions.

######  Theorem F (Correspondence formulas, 5.5).

We have the following correspondences between classes in
$\textup{SliceSS}(BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle)$ and
$\textup{SliceSS}(BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle I\rangle):$

  1. (1)

Suppose $V\in RO(C_{2^{n+1}}/C_{2^{k}})=RO(C_{2^{n-k+1}})$ is an actual
representation, then we have the correspondence

$a_{V}\leftrightsquigarrow a_{V}.$

  2. (2)

Suppose $V\in RO(C_{2^{n+1}}/C_{2^{k}})=RO(C_{2^{n-k+1}})$ is an orientable
representation, then we have the correspondence

$u_{V}\leftrightsquigarrow u_{V}.$

  3. (3)

For all $i\in I$ and $1\leq j\leq n-k+1$, we have the correspondence

$N_{C_{2}}^{C_{2^{k+j}}}(\bar{t}_{i}^{C_{2^{n+1}}})\cdot\left(\frac{a_{\bar{\rho}_{2^{k+j}}}}{a_{\bar{\rho}_{2^{k+j}/2^{k}}}}\right)^{2^{i}-1}\leftrightsquigarrow
N_{C_{2}}^{C_{2^{j}}}(\bar{t}_{i}^{C_{2^{n-k+1}}}).$

In (1) and (2), $V$ is also viewed as an element of $RO(C_{2^{n+1}})$ through
pullback along the map $C_{2^{n+1}}\to C_{2^{n+1}}/C_{2^{k}}$. In (3), the
representation $\bar{\rho}_{2^{k+j}/2^{k}}\in RO(C_{2^{k+j}})$ is defined as
in 5.4.

The correspondence formulas in F allows us to directly translate differentials
and permanent cycles from the slice spectral sequence of
$BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle I\rangle$ to the slice spectral sequence
of $BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle$. They play a key role in
establishing the Periodicity Theorem (B) and the Vanishing Lines Theorem (C).

###  1.4. Open questions and future directions

#### Other types of transchromatic phenomena

In addition to the transchromatic phenomenon proved in A, the following
conjecture offers another transchromatic approach to studying the
$C_{2^{n}}$-fixed points of $E_{h}$ across different heights:

######  Conjecture 1.3.

The differentials in the slice spectral sequence of $BP^{(\\!(C_{2^{n}})\\!)}$
can be classified by “height”. With this classification, the $C_{2^{n}}$-slice
spectral sequence of $BP^{(\\!(C_{2^{n}})\\!)}\langle m\rangle$ contains all
the differentials in the slice spectral sequence of $BP^{(\\!(C_{2^{n}})\\!)}$
that are of height at most $h=2^{n-1}m$.

To prove 1.3, it is worth noting that the map

$\textup{SliceSS}(BP^{(\\!(C_{2^{n}})\\!)}\langle
m+1\rangle)\longrightarrow\textup{SliceSS}(BP^{(\\!(C_{2^{n}})\\!)}\langle
m\rangle)$ (1.2)

is a surjection on the $\mathcal{E}_{2}$-page for all $m,n\geq 1$.

######  Conjecture 1.4.

For all $m\geq 1$ and $r\geq 2$, the map (1.2) induces a surjection on the
$\mathcal{E}_{r}$-page and of the $d_{r}$-differentials.

Proving 1.4 would imply that the higher-height theories $E_{(m+1)\cdot
2^{n-1}}^{hC_{2^{n}}}$ inherit all the differentials from the lower-height
theories $E_{m\cdot 2^{n-1}}^{hC_{2^{n}}}$. This constitutes a significant
step toward fully resolving 1.3. Currently, 1.4 has been verified for all the
known computations listed in Section 1.1.

Utilizing A, there is a possible inductive approach to proving 1.4. The base
case of the induction is when the group is $C_{2}$. In this scenario, the
surjectivity of (1.2) has already been established for all $m\geq 1$. Now,
suppose that we have proved the claim for the group $C_{2^{n-1}}$. For
$C_{2^{n}}$, we have the following diagram:

${\textup{SliceSS}(BP^{(\\!(C_{2^{n}})\\!)}\langle
m+1\rangle)}$${\textup{SliceSS}(BP^{(\\!(C_{2^{n}})\\!)}\langle
m\rangle)}$${\textup{SliceSS}(BP^{(\\!(C_{2^{n-1}})\\!)}\langle
m+1\rangle)}$${\textup{SliceSS}(BP^{(\\!(C_{2^{n-1}})\\!)}\langle
m\rangle).}$$\scriptstyle{\mathbf{1}}$shearingshearing$\scriptstyle{\mathbf{2}}$

The vertical arrows in the diagram correspond to the shearing isomorphisms
established in A. Given that we have already demonstrated that map
$\mathbf{2}$ satisfies the claim outlined in 1.4, it suffices to prove
surjectivity of map $\mathbf{1}$ for the region below the line of slope 1.

#### Transchromatic isomorphism for the negative cone

For any $G$-spectrum $X$, its equivariant slice spectral sequence is
concentrated in two conical regions. On the integer-graded page, one of these
regions is in the first quadrant, and is known as the “positive cone”, while
the other, located in the third quadrant, is referred to as the “negative
cone”. Both regions are bounded by lines of slopes $0$ and $(|G|-1)$.

For connective theories such as $BP^{(\\!(G)\\!)}\langle m\rangle$, the slice
spectral sequence will only have a positive cone. However, for periodic
theories such as $D^{-1}BP^{(\\!(G)\\!)}\langle m\rangle$ and Lubin–Tate
theories, a negative cone also appears.

On the integer-graded page, the behavior of the positive cone can be
completely understood by utilizing the localized slice spectral sequence, as
demonstrated in our use of the stratification tower and our proof of A. Given
this result, it is reasonable to expect that the negative cone should also
exhibit similar transchromatic phenomena. Investigating the negative cone is
particularly important in the context of duality. Explicit computations have
unveiled duality phenomena in the $C_{2}$-slice spectral sequences of
$E_{\mathbb{R}}(n):=\bar{v}_{n}^{-1}BP_{\mathbb{R}}\langle n\rangle$ [GM17]
and in the $C_{4}$-slice spectral sequence of
$D^{-1}BP^{(\\!(C_{4})\\!)}\langle 1\rangle$ [HHR17].

In collaboration with Yutao Liu and Guoqi Yan, we will analyze the isomorphism
regions in the generalized Tate spectral sequence with respect to different
families $\mathcal{F}$. This analysis will provide a stratification for the
negative cone of the slice spectral sequence for non-connective theories. We
will also leverage the shearing isomorphisms already established in this paper
to deduce further shearing isomorphisms and vanishing line results for the
negative cone of the slice spectral sequence.

####  $Q_{8}$-equivariant transchromatic isomorphisms

According to the classification of maximal finite 2-groups in the stabilizer
group $\mathbb{S}_{h}$ [Hew95, Hew99, Buj12], there are two possible types:
cyclic groups of order a power of 2 and $Q_{8}$. In this paper, our focus has
been on cyclic groups.

######  Question 1.5.

Are there $Q_{8}$-equivariant analogues of the Transchromatic Isomorphism
Theorem (A)?

This question is particularly relevant when the height $h$ is congruent to $2$
modulo $4$. A crucial step in the proof of A relies on analyzing the
stratification tower of $BP^{(\\!(C_{2^{n+1}})\\!)}\langle m\rangle$. In the
context of $Q_{8}$, the work of [HS20] also implies the existence of a
$Q_{8}$-equivariant orientation $BP^{(\\!(Q_{8})\\!)}\to E_{h}$.

It would be interesting to attempt to create chromatically meaningful
quotients from $BP^{(\\!(Q_{8})\\!)}$ and study their stratification towers.
Drawing a parallel to the comparison made with cyclic groups, where we
compared the stratification towers of quotients of $BP^{(\\!(G)\\!)}$ and
$BP^{(\\!(G/C_{2})\\!)}$, one might expect similar isomorphism results between
the stratification towers of $BP^{(\\!(Q_{8})\\!)}$ and $BP^{(\\!(C_{2}\times
C_{2})\\!)}$, given that $Q_{8}/C_{2}=C_{2}\times C_{2}$. Interestingly,
$C_{2}\times C_{2}$ is not a finite subgroup of $\mathbb{S}_{h/2}$, making it
intriguing to explore where the transchromatic phenomenon may manifest itself.

#### Transchromatic phenomena at odd primes, other equivariant filtrations

When $p$ is an odd prime and the height $h$ is of the form
$m\cdot(p-1)p^{n-1}$, the Lubin–Tate spectrum $E_{h}$ admits a
$C_{p^{n}}$-action, as shown in [Hew95, Hew99, Buj12]. Consequently, at height
$h/p=m\cdot(p-1)p^{n-2}$, the spectrum $E_{h/p}$ admits a
$C_{p^{n-1}}$-action. Given this, the authors believe that there should be
analogues of the Transchromatic Isomorphism Theorem (A) for odd primes. In the
following conjecture, we will leave certain terms vague in our statement, as
discussed below.

######  Conjecture 1.6.

There is a shearing isomorphism $d_{p\cdot r-1}\leftrightsquigarrow d_{r}$
between the following regions of spectral sequences:

  1. (1)

The $C_{p^{n}}$-spectral sequence of $E_{h}$ on or above the line of slope
$(p-1)$ within the region $t-s\geq 0$; and

  2. (2)

The $(C_{p^{n}}/C_{p})$-spectral sequence of $E_{h/p}$ within the region
$t-s\geq 0$.

Should 1.6 hold true, an important calculation is to determine the
$C_{9}$-fixed points of $E_{6}$ at the prime 3. The theory $E_{6}^{hC_{9}}$
could play a crucial role in resolving the 3-primary Kervaire invariant
problem. Utilizing the shearing isomorphism outlined in 1.6, the computation
of $E_{2}^{hC_{3}}$ by Hopkins and Miller ([Nav10]) can be inductively used to
compute $E_{6}^{hC_{9}}$. This approach may help simplify its computation.

An essential aspect of 1.6 is the need to specify which equivariant spectral
sequence we are using to compute $E_{h}$. In contrast to the situation at the
prime 2, odd primes may offer alternative equivariant spectral sequences that
are more efficient for computations. It will be interesting to investigate how
to construct a $C_{p^{n}}$-filtration for $E_{h}$ whose associated spectral
sequence satisfies the shearing isomorphism proposed in 1.6.

In a more general context, for any equivariant filtration $\\{P^{\bullet}\\}$,
we can create a new filtration $\\{\widetilde{E}\mathcal{F}\wedge
P^{\bullet}\\}$ by smashing it with $\widetilde{E}\mathcal{F}$. The resulting
map of towers $\\{P^{\bullet}\\}\to\\{\widetilde{E}\mathcal{F}\wedge
P^{\bullet}\\}$ induces a map of the corresponding spectral sequences.
However, it is important to note that we are not always guaranteed that this
map will produce isomorphism regions. One crucial fact that establishes the
existence of the stratification tower and the Transchromatic Isomorphism
Theorem is the observation that this map indeed yields isomorphism regions
when ${P^{\bullet}}$ is the equivariant slice filtration, as demonstrated by
the Slice Recovery Theorem ([MSZ23a, Theorem 3.3]).

######  Question 1.7.

For what other equivariant filtrations are there stratification results and
shearing isomorphisms?

###  1.5. Organization of the paper

In Section 2, we revisit the construction of the stratification tower and its
relevant properties from [MSZ23a]. In Section 3, we discuss the pullback
functor, the dual slice tower, and the shearing isomorphism. We prove a result
(3.7) on shearing isomorphism between two interrelated spectral sequences.
This relationship emerges through the interplay of a pullback operation and a
modification of filtrations.

In Section 4, we compare the localized dual slice towers of
$BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle$ and
$BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle I\rangle$, proving D (4.1). Combined with
3.7, the equivalence of localized dual slice towers induces a shearing
isomorphism of the corresponding localized slice spectral sequences. In
Section 5, we establish correspondence formulas (5.5) for classes in these
localized slice spectral sequences.

In Section 6, we prove E (6.2) and state several corollaries that will be
useful for our discussions in the subsequent sections. F is a direct
consequence of 5.5 and 6.2. We also provide an alternative proof of the Slice
Differentials Theorem by Hill, Hopkins, and Ravenel (6.7).

Our main theorem, A, directly follows from E by setting
$I=\\{1,2,\ldots,m\\}$. This is discussed in more detail in Section 7.

In the final two sections, we turn our attention to applications of A. In
Section 8, we prove B (8.3), which establishes $RO(G)$-graded periodicities of
height-$h$ theories from $RO(G/{C_{2}})$-graded periodicities of
height-$(h/2)$-theories. In Section 9, we prove C (9.3).

### Acknowledgements

We would like to thank William Balderrama, Agnès Beaudry, Mark Behrens, Mike
Hopkins, Hana Jia Kong, Tyler Lawson, Guchuan Li, Wenao Li, Yutao Liu, Peter
May, Doug Ravenel, Guozhen Wang, Guoqi Yan, and Foling Zou for helpful
conversations. We would like to especially thank Mike Hill and Zhouli Xu for
comments on an earlier draft of our paper. The first author is supported by
the NWO grant VI.Vidi.193.111 and the second author is supported in part by
NSF Grant DMS-2313842.

### Notations and conventions

  1. (1)

The slice tower we consider will always be the _regular_ slice towers (see
[Ull13a, Ull13b]).

  2. (2)

For a $G$-spectrum $X$, we denote by $\pi_{*}^{G}X$ the integer-graded
homotopy group of $X$; by $\pi_{\star}^{G}X$ the $RO(G)$-graded homotopy group
of $X$; by ${\underline{\pi}}_{*}X$ the integer-graded Mackey functor valued
homotopy group of $X$; and by ${\underline{\pi}}_{\star}X$ the $RO(G)$-graded
Mackey functor valued homotopy groups of $X$.

  3. (3)

All of our spectral sequences are $RO(G)$-graded. For a fixed $V\in RO(G)$,
the “$(V+t-s,s)$-graded page” is the part of the spectral sequence consisting
of all the classes with degrees $(V+t-s,s)$, where $t,s\in\mathbb{Z}$. The
“integer-graded page” is when we set $V=0$.

  4. (4)

We will denote the regular representation of $C_{2^{n}}$ by $\rho_{2^{n}}$.

  5. (5)

For $0\leq i\leq n-1$, we will denote the real 2-dimensional
$C_{2^{n}}$-representation corresponding to rotation by
$\left(\frac{\pi}{2^{i}}\right)$ by $\lambda_{i}$. With this notation,
$\lambda_{0}=2\sigma$, where $\sigma$ is the 1-dimensional sign
representation.

  6. (6)

We will denote by $\textup{Sp}_{G}$ the homotopy category of genuine
$G$-spectra and by $\wedge$ the derived smash product on it.

##  2\. Stratification of the slice spectral sequence

In this section, we will recall the relevant definitions and results in
[MSZ23a] that will be needed in the following sections. The construction of
the stratification tower in [MSZ23a] relies on the utilization of the
localized slice spectral sequence, which is a modification of the slice
spectral sequence studied by the authors in [MSZ23b].

Let $G$ be a finite group, $X$ a $G$-spectrum, and $\mathcal{F}$ a family of
subgroups of $G$. The localized slice spectral sequence of $X$ with respect to
$\mathcal{F}$ is the spectral sequence associated with the tower
$\widetilde{E}\mathcal{F}\wedge P^{\bullet}X$, obtained by smashing the
universal space $\widetilde{E}\mathcal{F}$ with the slice tower of $X$. This
spectral sequence is denoted by
$\widetilde{E}\mathcal{F}\wedge\textup{SliceSS}(X)$.

Whenever there is an inclusion $\mathcal{F}\subset\mathcal{F}^{\prime}$ of
families, it induces a map
$\widetilde{E}\mathcal{F}\to\widetilde{E}\mathcal{F}^{\prime}$ of universal
spaces, consequently leading to a map

$\varphi:\widetilde{E}\mathcal{F}\wedge\textup{SliceSS}(X)\longrightarrow\widetilde{E}\mathcal{F}^{\prime}\wedge\textup{SliceSS}(X)$

of the corresponding localized slice spectral sequences.

######  Construction 2.1 (Stratification tower).

Let $\mathcal{F}_{\leq h}$ be the family consisting of all subgroups of $G$ of
order $\leq h$. Note from definition, $\mathcal{F}_{\leq 0}$ is the empty
family, and $\mathcal{F}_{\leq|G|}$ is the family consisting of all subgroups
of $G$. The chain of inclusions

$\mathcal{F}_{\leq 0}\subset\mathcal{F}_{\leq
1}\subset\cdots\subset\mathcal{F}_{\leq|G|}$

induces the maps

$\widetilde{E}\mathcal{F}_{\leq
0}\longrightarrow\widetilde{E}\mathcal{F}_{\leq
1}\longrightarrow\cdots\widetilde{E}\mathcal{F}_{\leq|G|-1}\longrightarrow\widetilde{E}\mathcal{F}_{\leq|G|}.$

Since $\widetilde{E}\mathcal{F}_{\leq 0}\simeq S^{0}$ and
$\widetilde{E}\mathcal{F}_{\leq|G|}\simeq*$, the maps above induces a
decreasing filtration of the equivariant slice spectral sequence of $X$ given
by the tower

$\left\\{\widetilde{E}\mathcal{F}_{\leq\bullet}\wedge\textup{SliceSS}(X),\,0\leq\bullet\leq|G|\right\\}$
(2.1)

of localized slice spectral sequences. This is called the  stratification
tower.

######  Definition 2.2.

Suppose $V\in RO(G)$, and $S$ is a collection of subgroups of $G$. Define

$\tau_{V}(S)=\max_{H\in S}\left(|V^{H}|\cdot|H|-|V|\right).$

######  Definition 2.3.

For a fixed $V\in RO(G)$ and $h\geq 1$, let $\mathcal{L}_{h-1}^{V}$ represent
the line of slope $(h-1)$ on the $(V+t-s,s)$-graded page that is defined by
the equation

$s=(h-1)(t-s)+\tau_{V}(\mathcal{F}_{\leq h}).$

In 2.3, there are two special cases worth noting:

  1. (1)

When $V=0$, we have $\tau_{0}(\mathcal{F}_{\leq h})=0$, and
$\mathcal{L}_{h-1}^{0}$ is the line of slope $(h-1)$ through the origin on the
integer-graded page. In this case, we will denote this line by
$\mathcal{L}_{h-1}$.

  2. (2)

When $h=1$, we have $\tau_{V}(\mathcal{F}_{\leq 1})=|V^{e}|\cdot 1-V=0$. In
this case, the line $L^{V}_{0}$ is the horizontal line $s=0$.

######  Theorem 2.4 (Slice Recovery Theorem, [MSZ23a] Theorem 3.3).

Let $X$ be a $G$-spectrum, and let $h\geq 1$. On the $(V+t-s,s)$-graded page
where $t-s\geq 0$, the map

$\varphi_{h}:\textup{SliceSS}(X)\longrightarrow\widetilde{E}\mathcal{F}_{\leq
h}\wedge\textup{SliceSS}(X)$

induces an isomorphism of spectral sequences on or above the line
$\mathcal{L}_{h-1}^{V}$. In the region where $t-s<0$, the map $\varphi_{h}$
induces an isomorphism of spectral sequences on or above the horizontal line
$s=\tau_{V}(\mathcal{F}_{\leq h})$.

######  Corollary 2.5 ([MSZ23a] Corollary 3.4).

The map

$\varphi_{1}:\textup{SliceSS}(X)\longrightarrow\widetilde{E}G\wedge\textup{SliceSS}(X)$

induces an isomorphism of spectral sequences on or above the horizontal line
$s=0$ on all the $(V+t-s,s)$-graded pages.

When $G=C_{2^{n+1}}$, the subgroups of $G$ are linearly ordered by inclusion.
They are of the form $C_{2^{k}}$ for $0\leq k\leq n+1$. For all $k$ and $\ell$
such that $1\leq k\leq n+1$ and $2^{k-1}\leq\ell\leq 2^{k}-1$, we have the
equality $\mathcal{F}_{\leq\ell}=\mathcal{F}[C_{2^{k}}]$, where
$\mathcal{F}[C_{2^{k}}]$ is the family consisting of all subgroups of
$C_{2^{n+1}}$ that do not contain $C_{2^{k}}$. The chain of inclusions

$\mathcal{F}_{\leq 0}\subset\mathcal{F}_{\leq
1}\subset\cdots\mathcal{F}_{\leq|G|-1}$

is essentially

$\varnothing\subset\mathcal{F}[C_{2}]\subset\mathcal{F}[C_{4}]\subset\cdots\subset\mathcal{F}[C_{2^{n+1}}].$

For any $C_{2^{n+1}}$-spectrum $X$ and a subgroup $C_{2^{k}}\subseteq
C_{2^{n+1}}$, there is a residual $(C_{2^{n+1}}/C_{2^{k}})$-action on the
$C_{2^{k}}$-geometric fixed points $\Phi^{C_{2^{k}}}(X)$, and we also have the
equivalence

$(\widetilde{E}\mathcal{F}[C_{2^{k}}]\wedge
X)^{C_{2^{n+1}}}\simeq\left(\Phi^{C_{2^{k}}}(X)\right)^{C_{2^{n+1}}/C_{2^{k}}}.$

The stratification tower (2.1) becomes the following:

${\textup{SliceSS}(X)}$${\widetilde{E}\mathcal{F}[C_{2}]\wedge\textup{SliceSS}(X)}$${{\underline{\pi}}_{\star}\Phi^{C_{2}}(X)}$${\widetilde{E}\mathcal{F}[C_{4}]\wedge\textup{SliceSS}(X)}$${{\underline{\pi}}_{\star}\Phi^{C_{4}}(X)}$${\vdots}$${\vdots}$${\widetilde{E}\mathcal{F}[C_{2^{n}}]\wedge\textup{SliceSS}(X)}$${{\underline{\pi}}_{\star}\Phi^{C_{2^{n}}}(X)}$${\widetilde{E}\mathcal{F}[C_{2^{n+1}}]\wedge\textup{SliceSS}(X)}$${\pi_{*}\,\Phi^{C_{2^{n+1}}}(X).}$$\scriptstyle{\mathcal{L}^{V}_{0}}$$\scriptstyle{\mathcal{L}^{V}_{1}}$$\scriptstyle{\mathcal{L}^{V}_{2^{n-2}-1}}$$\scriptstyle{\mathcal{L}^{V}_{2^{n-1}-1}}$$\scriptstyle{\mathcal{L}^{V}_{1}}$$\scriptstyle{\mathcal{L}^{V}_{3}}$$\scriptstyle{\mathcal{L}^{V}_{2^{n-1}-1}}$$\scriptstyle{\mathcal{L}^{V}_{2^{n}-1}}$
(2.2)

Figure 3. The stratification tower of the slice spectral sequence.

Figure 3 shows a picture of the stratification tower when $V=0$. In the
stratification tower above, the maps are labeled by the lines
$\mathcal{L}^{V}_{h-1}$, indicating the isomorphism regions. The slice
spectral sequence of $X$ is stratified into different regions separated by the
lines $\mathcal{L}^{V}_{1}$, $\mathcal{L}^{V}_{3}$, $\ldots$, and
$\mathcal{L}^{V}_{2^{n}-1}$. The differentials within these regions can be
recovered from the localized slice spectral sequences, which compute the
geometric fixed points equipped with the residue
$(C_{2^{n+1}}/C_{2^{k}})$-action. As we move up the tower, 2.4 shows that each
localized slice spectral sequence contains increasingly more information about
the original slice spectral sequence.

##  3\. Shearing isomorphisms of spectral sequences

In this section, we will discuss the pullback functor, the dual slice tower,
and the shearing isomorphism. We will prove a result (3.7) on shearing
isomorphism between two interrelated spectral sequences. This relationship
emerges through the interplay of a pullback operation and a modification of
filtrations.

###  3.1. The pullback functor

Suppose $G$ is a finite group and $N\subseteq G$ is a normal subgroup. The
$N$-geometric fixed points functor

$\Phi^{N}:\textup{Sp}_{G}\longrightarrow\textup{Sp}_{G/H}$

is the functor that sends a $G$-spectrum $X$ to
$\Phi^{N}(X)=(\widetilde{E}\mathcal{F}[N]\wedge X)^{N}$, its $N$-geometric
fixed points. This functor is a left adjoint functor.

######  Definition 3.1.

The pullback functor

$\mathcal{P}^{*}_{G/N}:\textup{Sp}_{G/N}\longrightarrow\textup{Sp}_{G}$

is the right adjoint to the $N$-geometric fixed points functor $\Phi^{N}$.

The following explicit construction of the pullback functor
$\mathcal{P}^{*}_{G/N}$ can be found in [LMSM86, II. $\S$9] and [Hil12,
Section 4]: let $q^{*}:\textup{Sp}_{G/N}\to\textup{Sp}_{G}$ denote the
inflation functor that is associated to the quotient map $q:G\to G/N$. Then
the pullback functor sends a $G/N$-spectrum $X$ to
$\mathcal{P}^{*}_{G/N}(X)=\widetilde{E}\mathcal{F}[N]\wedge q^{*}X$.

Several pleasant properties of the pullback functor $\mathcal{P}^{*}_{G/H}$
are shown in [Hil12, Section 4.1] (where $\mathcal{P}^{*}_{G/N}$ is denoted by
$\phi_{N}^{*}$). In particular, the $N$-fixed points functor $(-)^{N}$
establishes an equivalence between $G/N$-spectra and the image of
$\mathcal{P}^{*}_{G/N}$, which are $G$-spectra of the form
$\widetilde{E}\mathcal{F}[N]\wedge X$ ([Hil12, Proposition 4.3]).

There are two lemmas that will be useful for our subsequent discussions in
later sections.

######  Lemma 3.2.

For a $G$-spectrum $X$ and a normal subgroup $N$ of $G$, the following
equivalence holds:

$\widetilde{E}\mathcal{F}[N]\wedge X\simeq\mathcal{P}^{*}_{G/N}\Phi^{N}(X).$

###### Proof.

This follows immediately from the discussion above. Since
$\widetilde{E}\mathcal{F}[N]\wedge X$ is in the image of
$\mathcal{P}^{*}_{G/N}$,

$\displaystyle\widetilde{E}\mathcal{F}[N]\wedge X$
$\displaystyle\simeq\mathcal{P}^{*}_{G/N}(\widetilde{E}\mathcal{F}[N]\wedge
X)^{N}$ $\displaystyle\simeq\mathcal{P}^{*}_{G/N}\Phi^{N}(X).\qed$

######  Definition 3.3 ([GM92, Hil12]).

Suppose $\underline{M}$ is a Mackey functor for $G/N$, then
$\mathcal{P}^{*}_{G/N}\underline{M}$ is the Mackey functor on $G$ obtained by
composing $\underline{M}$ with the $N$-fixed points functor (that sends finite
$G$-sets to finite $G/N$-sets). More precisely,

$\mathcal{P}^{*}_{G/N}\underline{M}(G/H)=\left\\{\begin{array}[]{ll}0&\text{if
}N\not\subseteq H\,\,\,(H\in\mathcal{F}[N]),\\\
\underline{M}((G/N)/(H/N))&\text{if }N\subseteq
H\,\,\,(H\notin\mathcal{F}[N]).\end{array}\right.$

######  Lemma 3.4.

Suppose $X$ is a $G/N$-spectrum. Then for every $V\in RO(G)$, there is a
canonical isomorphism

${\underline{\pi}}_{V}\mathcal{P}^{*}_{G/N}(X)\cong\mathcal{P}^{*}_{G/N}({\underline{\pi}}_{V^{N}}X),$

where $V^{N}$ is the $N$-fixed points of $V$, considered as an element in
$RO(G/N)$.

###### Proof.

Let $H$ be a subgroup of $G$. If $N\not\subseteq H$, then
$i_{H}^{*}\widetilde{E}\mathcal{F}[N]$ is contractible, and so
$i_{H}^{*}\mathcal{P}^{*}_{G/N}(X)$ is also contractible. In this case, we
have $\pi_{V}^{H}\mathcal{P}^{*}_{G/N}(X)\cong 0$.

If $N\subseteq H$, then
$i_{H}^{*}\mathcal{P}^{*}_{G/N}(X)=\mathcal{P}^{*}_{H/N}(i_{H/N}^{*}X)$, and
we have

$\displaystyle\pi_{V}^{H}\mathcal{P}^{*}_{G/N}(X)$
$\displaystyle\cong\left[i_{H}^{*}S^{V},\mathcal{P}^{*}_{H/N}(i_{H/N}^{*}X)\right]^{H}$
$\displaystyle\cong\left[\Phi^{N}(S^{i_{H}^{*}V}),i_{H/N}^{*}X\right]^{H/N}$
$\displaystyle\cong\left[S^{(i_{H}^{*}V)^{N}},i_{H/N}^{*}X\right]^{H/N}$
$\displaystyle\cong\left[S^{i_{H/N}^{*}V^{N}},i_{H/N}^{*}X\right]^{H/N}$
$\displaystyle\cong\pi_{V^{N}}^{H/N}X.\qed$

###  3.2. The dual slice tower

For a $G$-spectrum $X$, let $P_{d+1}X$ be the fiber of $X\to P^{d}X$. There is
a functorial fiber sequence

$P_{d+1}X\longrightarrow X\longrightarrow P^{d}X.$

The tower

$P_{\bullet}X=\left\\{\cdots\longrightarrow P_{d+1}X\longrightarrow
P_{d}X\longrightarrow P_{d-1}X\longrightarrow\cdots\right\\}$

is the dual slice tower of $X$. It is a consequence of [HHR16, Theorem 4.42]
that $\varinjlim P_{\bullet}X\simeq X$ and $\varprojlim P_{\bullet}X\simeq*$.
Just like the slice tower is an equivariant refinement of the Postnikov tower
of $X$, the dual slice tower is an equivariant refinement of the Whitehead
tower of $X$.

From the definition, the cofiber of $P_{d+1}X\to P_{d}X$ and the fiber of
$P^{d}X\to P^{d-1}X$ are both $P_{d}^{d}X$, the $d$-slice of $X$. The spectral
sequence associated to $P_{\bullet}X$ has $\mathcal{E}_{2}$-page

$\underline{\mathcal{E}}_{2}^{s,V}={\underline{\pi}}_{V-s}P^{|V|}_{|V|}X\Longrightarrow{\underline{\pi}}_{V-s}X,$

and is the exact same spectral sequence as the slice spectral sequence of $X$
(which is associated to $P^{\bullet}X$). All the results and constructions
that we have established for the slice tower carry over directly to the dual
slice tower.

Moving forward, we will be working with the dual slice tower. This is just a
stylistic preference, as all of our theorem statements and arguments will work
analogously for the slice tower as well.

######  Definition 3.5.

The double of a tower $P_{\bullet}$ is the tower $\mathscr{D}P_{\bullet}$
defined by setting $\mathscr{D}P_{2d-\varepsilon}=P_{d}$ for all $d$ and
$\varepsilon=0,1$. For $k\geq 1$, the $k$-fold double of $P_{\bullet}$ is the
tower $\mathscr{D}^{k}P_{\bullet}$, obtained by taking the double $k$-times.
In particular, we have $(\mathscr{D}^{k}P)^{2^{k}d}_{2^{k}d}=P^{d}_{d}$, and
$(\mathscr{D}^{k}P)^{2^{k}d-j}_{2^{k}d-j}$ are contractible for all $k\geq 1$
and $1\leq j\leq 2^{k}-1$.

######  Proposition 3.6 ([Ull13a] Corollary 4.5, [Hil12] Remark 4.13).

Let $G$ be a finite group and $N\subseteq G$ a normal subgroup. For $X$ a
$G/N$-spectrum, the following equivalence holds:

$P_{m}(\mathcal{P}^{*}_{G/N}X)\simeq\mathcal{P}^{*}_{G/N}\left(P_{\lceil
m/|N|\rceil}X\right).$

In the special case that $N=C_{2^{k}}$, there is an equivalence

$P_{\bullet}\left(\mathcal{P}^{*}_{G/C_{2^{k}}}X\right)\simeq\mathcal{P}^{*}_{G/C_{2^{k}}}\left(\mathscr{D}^{k}P_{\bullet}X\right).$

###  3.3. Shearing isomorphism

Let $G$ be a finite group, and let $N=C_{2^{k}}$ be a normal subgroup of $G$.
Suppose $P_{\bullet}$ is a $G/N$-equivariant tower. Define the $G$-equivariant
tower $Q_{\bullet}$ as follows:

$Q_{\bullet}=\mathcal{P}^{*}_{G/N}(\mathscr{D}^{k}P_{\bullet}).$

The following theorem establishes the relationship between the $G$-equivariant
spectral sequence $\underline{\mathcal{E}}_{Q_{\bullet}}$ that is associated
to $Q_{\bullet}$ and the $G/N$-equivariant spectral sequence
$\underline{\mathcal{E}}_{P_{\bullet}}$ that is associated to $P_{\bullet}$.

######  Theorem 3.7.

Suppose $V\in RO(G)$, and let $V^{N}\in RO(G/N)$ be the $N$-fixed points of
$V$. There is a shearing isomorphism of spectral sequences

$\underline{\mathcal{E}}_{Q_{\bullet},2^{k}r-(2^{k}-1)}^{s^{\prime},V+t^{\prime}}\cong\mathcal{P}^{*}_{G/N}\left(\underline{\mathcal{E}}_{P_{\bullet},r}^{s,V^{N}+t}\right)\,\,\,\,\,(r\geq
2),$

where

$\displaystyle t^{\prime}$ $\displaystyle=\left(|V^{N}|\cdot
2^{k}-|V|\right)+2^{k}t,$ $\displaystyle s^{\prime}$
$\displaystyle=\left(|V^{N}|\cdot 2^{k}-|V|\right)+(2^{k}-1)(t-s)+2^{k}s.$

In particular, this shearing isomorphism induces a one-to-one correspondence
between the the $d_{2^{k}r-(2^{k}-1)}$-differentials in
$\underline{\mathcal{E}}_{Q_{\bullet}}$ and the $d_{r}$-differentials in
$\underline{\mathcal{E}}_{P_{\bullet}}$.

###### Proof.

Set $W=V+t$. As elaborated in [MSZ23b, Section 3.3], the
$\underline{\mathcal{E}}_{r}$-page of the spectral sequence associated with
$P_{\bullet}$ is given by

$\underline{\mathcal{E}}_{P_{\bullet},r}^{s,W^{N}}=\textup{im}\left({\underline{\pi}}_{W^{N}-s}P^{|W^{N}|+(r-2)}_{|W^{N}|}\longrightarrow{\underline{\pi}}_{W^{N}-s}P^{|W^{N}|}_{|W^{N}|-(r-2)}\right).$

By the construction of the tower $Q_{\bullet}$ and 3.4, the Mackey functor
$P^{*}_{G/N}\underline{\mathcal{E}}_{P_{\bullet},r}^{s,W^{N}}$ is isomorphic
to

$\textup{im}\left({\underline{\pi}}_{W-s}Q^{|W^{N}|\cdot
2^{k}+2^{k}(r-2)}_{|W^{N}|\cdot
2^{k}}\longrightarrow{\underline{\pi}}_{W-s}Q^{|W^{N}|\cdot
2^{k}}_{|W^{N}|\cdot 2^{k}-2^{k}(r-2)}\right).$ (3.1)

Note that we could replace $2^{k}(r-2)$ here by any larger number smaller than
$2^{k}(r-1)$ since the terms in $Q_{\bullet}$ repeat $2^{k}$ times. The term
(3.1) is isomorphic to
$\underline{\mathcal{E}}_{Q_{\bullet},r^{\prime}}^{s^{\prime},V+t^{\prime}}=\underline{\mathcal{E}}_{Q_{\bullet},r^{\prime}}^{s^{\prime},W-t+t^{\prime}}$
if $2^{k}(r-2)\leq r^{\prime}-2\leq 2^{k}(r-1)-1$ and the pair
$(t^{\prime},s^{\prime})$ satisfies the following equalities:

$\displaystyle|W|-t+t^{\prime}$ $\displaystyle=|W^{N}|\cdot 2^{k},$
$\displaystyle W-t+t^{\prime}-s^{\prime}$ $\displaystyle=W-s.$

Equivalently, if $2^{k}r-(2^{k+1}-2)\leq r^{\prime}\leq 2^{k}r-(2^{k}-1)$ and

$\displaystyle t^{\prime}$ $\displaystyle=|W^{N}|\cdot
2^{k}-|W|+t=\left(|V^{N}|\cdot 2^{k}-|V|\right)+2^{k}t$ $\displaystyle
s^{\prime}$ $\displaystyle=t^{\prime}-t+s=\left(|V^{N}|\cdot
2^{k}-|V|\right)+(2^{k}-1)(t-s)+2^{k}s.$

Therefore,
$\underline{\mathcal{E}}_{Q_{\bullet},r^{\prime}}^{s^{\prime},V+t^{\prime}}\cong\mathcal{P}^{*}_{G/N}\left(\underline{\mathcal{E}}_{P_{\bullet},r}^{s,V^{N}+t}\right)$
for $r\geq 2$ with $r^{\prime}$ falling within the specified range.

The previous paragraph shows that unless $r^{\prime}$ is of the form
$2^{k}r-(2^{k}-1)$,
$\underline{\mathcal{E}}_{Q_{\bullet},r^{\prime}}^{s^{\prime},V+t^{\prime}}=\underline{\mathcal{E}}_{Q_{\bullet},r^{\prime}+1}^{s^{\prime},V+t^{\prime}}$
and so $d_{r^{\prime}}=0$ in $\underline{\mathcal{E}}_{Q_{\bullet}}$. We claim
that when ${r^{\prime}=2^{k}r-(2^{k}-1)}$, the differential $d_{r^{\prime}}$
in $\underline{\mathcal{E}}_{Q_{\bullet}}$ corresponds to the differential
$d_{r}\colon\underline{\mathcal{E}}_{P_{\bullet},r}^{s,W^{N}}\to\underline{\mathcal{E}}_{P_{\bullet},r}^{s+r,W^{N}+(r-1)}$.
This $d_{r}$-differential is induced by the boundary map

${\underline{\pi}}_{W^{N}-s}P^{|W^{N}|}_{|W^{N}|-(r-2)}\longrightarrow{\underline{\pi}}_{W^{N}-s-1}P^{|W^{N}|+(r-1)}_{|W^{N}|+1}.$

Upon applying $P^{*}_{G/N}$, this map becomes isomorphic to

${\underline{\pi}}_{W-s}Q^{|W^{N}|\cdot 2^{k}}_{|W^{N}|\cdot
2^{k}-2^{k}(r-2)}\longrightarrow{\underline{\pi}}_{W-s-1}Q^{|W^{N}|\cdot
2^{k}+2^{k}(r-1)}_{|W^{N}|\cdot 2^{k}+1}.$

Using the formula for $t^{\prime}$ and $s^{\prime}$ above, this indeed yields
the differential

$d_{r^{\prime}}\colon\underline{\mathcal{E}}_{Q_{\bullet},r^{\prime}}^{s^{\prime},W+t^{\prime}-t}\longrightarrow\underline{\mathcal{E}}_{Q_{\bullet},r^{\prime}}^{s^{\prime}+r^{\prime},W+t^{\prime}-t+(r^{\prime}-1)}.\qed$

##  4\. Isomorphism of localized slice towers

In this section, we will compare the localized dual slice towers of
$BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle$ and
$BP^{(\\!(C_{2^{n}})\\!)}\langle I\rangle$. Our main result in this section
the following theorem, which establishes an isomorphism between the various
$C_{2^{n+1}}$-equivariant and $C_{2^{n}}$-equivariant localized dual slice
towers via the pullback functor $\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}(-)$ and
the doubling operation $\mathscr{D}(-)$.

######  Theorem 4.1.

Suppose $n$ is a positive integer, and $I\subseteq\mathbb{N}$. For all $1\leq
k\leq n$, we have the following equivalence of towers:

$\widetilde{E}\mathcal{F}[C_{2^{k+1}}]\wedge
P_{\bullet}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle\right)\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}\mathscr{D}\left(\widetilde{E}\mathcal{F}[C_{2^{k}}]\wedge
P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)}\langle I\rangle)\right).$

Moreover, by applying this equivalence iteratively $k$-times, we have the
following equivalence of towers:

$\widetilde{E}\mathcal{F}[C_{2^{k+1}}]\wedge
P_{\bullet}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle\right)\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2^{k}}}\mathscr{D}^{k}\left(\widetilde{E}\mathcal{F}[C_{2}]\wedge
P_{\bullet}(BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle I\rangle)\right).$

To prove 4.1, we will begin by proving a lemma that establishes a connection
between the geometric fixed points of the $\bar{t}_{i}^{C_{2^{n}}}$-generators
for $\pi_{*\rho_{2}}^{C_{2}}BP^{(\\!(C_{2^{n}})\\!)}$ and the
$\bar{t}_{i}^{C_{2^{n+1}}}$-generators for
$\pi_{*\rho_{2}}^{C_{2}}BP^{(\\!(C_{2^{n+1}})\\!)}$. First, consider the
$C_{2}$-equivariant map

$\bar{t}_{i}^{C_{2^{n}}}:S^{(2^{i}-1)\rho_{2}}\longrightarrow
i_{C_{2}}^{*}BP^{(\\!(C_{2^{n}})\\!)}.$

After taking $C_{2}$-geometric fixed points $\Phi^{C_{2}}(-)$, we obtain the
map

$\Phi^{C_{2}}(\bar{t}_{i}^{C_{2^{n}}}):S^{2^{i}-1}\longrightarrow\Phi^{C_{2}}i_{C_{2}}^{*}BP^{(\\!(C_{2^{n}})\\!)}\simeq
i_{e}^{*}\Phi^{C_{2}}N_{C_{2}}^{C_{n}}(BP_{\mathbb{R}})\simeq
i_{e}^{*}N_{e}^{C_{2^{n-1}}}H\mathbb{F}_{2}.$

Now, consider the $C_{2}$-equivariant map

$\bar{t}_{i}^{C_{2^{n+1}}}:S^{(2^{i}-1)\rho_{2}}\longrightarrow
i_{C_{2}}^{*}BP^{(\\!(C_{2^{n+1}})\\!)}.$

After taking the norm $N_{C_{2}}^{C_{4}}(-)$ and applying the norm-restriction
adjunction, we obtain a $C_{4}$-equivariant map

$N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}}):S^{(2^{i}-1)\rho_{4}}\longrightarrow
N_{C_{2}}^{C_{4}}i_{C_{2}}^{*}BP^{(\\!(C_{2^{n+1}})\\!)}\longrightarrow
i_{C_{4}}^{*}BP^{(\\!(C_{2^{n+1}})\\!)}.$

Here, we would like to note that since $MU^{(\\!(C_{2^{n+1}})\\!)}$ splits as
a wedge of suspensions of $BP^{(\\!(C_{2^{n+1}})\\!)}$, the norm-restriction
adjunction for $BP^{(\\!(C_{2^{n+1}})\\!)}$ is defined to be the composition
map

$N_{C_{2}}^{C_{4}}i_{C_{2}}^{*}BP^{(\\!(C_{2^{n+1}})\\!)}\longrightarrow
N_{C_{2}}^{C_{4}}i_{C_{2}}^{*}MU^{(\\!(C_{2^{n+1}})\\!)}\longrightarrow
i_{C_{4}}^{*}MU^{(\\!(C_{2^{n+1}})\\!)}\longrightarrow
i_{C_{4}}^{*}BP^{(\\!(C_{2^{n+1}})\\!)}.$

Taking the $C_{4}$-geometric fixed points $\Phi^{C_{4}}(-)$ of the map
$N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})$ defined above produces the map

$\begin{split}\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}}):S^{2^{i}-1}\longrightarrow\Phi^{C_{4}}i_{C_{4}}^{*}BP^{(\\!(C_{2^{n+1}})\\!)}&\simeq
i_{e}^{*}\Phi^{C_{4}}N_{C_{4}}^{C_{2^{n+1}}}(BP^{(\\!(C_{4})\\!)})\\\ &\simeq
i_{e}^{*}N_{e}^{C_{2^{n-1}}}H\mathbb{F}_{2}.\end{split}$

######  Lemma 4.2.

For all $n\geq 1$ and $i\geq 1$, we have the equality

$\Phi^{C_{2}}(\bar{t}_{i}^{C_{2^{n}}})=\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}}).$

###### Proof.

When $n=1$, it is a consequence of [HHR16, Proposition 5.50] that the maps
$\Phi^{C_{2}}(\bar{t}_{i}^{C_{2^{n}}})$ and
$\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{4}})$ are both 0 for all $i\geq
1$, and so the equality holds. For $n\geq 2$, first consider the Real
orientations

$\bar{x}_{1},\bar{x}_{2}:\mathbf{CP}^{\infty}\longrightarrow\Sigma^{\rho_{2}}i_{C_{2}}^{*}BP^{(\\!(C_{2^{n}})\\!)}$

that define the Real formal group laws $\bar{F}_{1}$ and $\bar{F}_{2}$
corresponding to the maps

$\bar{\iota}_{1},\bar{\iota}_{2}:BP_{\mathbb{R}}\longrightarrow
i_{C_{2}}^{*}BP^{(\\!(C_{2^{n}})\\!)}\simeq\underbrace{BP_{\mathbb{R}}\wedge\cdots\wedge
BP_{\mathbb{R}}}_{2^{n-1}},$

where $\bar{\iota}_{1}$ and $\bar{\iota}_{2}$ are inclusions into the first
and the second factors of $i_{C_{2}}^{*}BP^{(\\!(C_{2^{n}})\\!)}$,
respectively. By definition, the $\bar{t}_{i}^{C_{2^{n}}}$-generators are the
coefficients of the power series relating $\bar{x}_{1}$ and $\bar{x}_{2}$:

$\bar{x}_{2}=\bar{x}_{1}+{\sum_{i\geq
1}}^{\bar{F}_{2}}\bar{t}_{i}^{C_{2^{n}}}\bar{x}_{1}^{2^{i}}.$ (4.1)

Note that $\Phi^{C_{2}}(\bar{\iota}_{1})=\iota_{1}$ and
$\Phi^{C_{2}}(\bar{\iota}_{2})=\iota_{2}$, where

$\iota_{1},\iota_{2}:H\mathbb{F}_{2}\longrightarrow
i_{e}^{*}N_{e}^{C_{2^{n-1}}}H\mathbb{F}_{2}\simeq\underbrace{H\mathbb{F}_{2}\wedge\cdots\wedge
H\mathbb{F}_{2}}_{2^{n-1}}$

are again the inclusions into the first and the second factors, respectively.
It follows that $\Phi^{C_{2}}(\bar{x}_{1})=a_{1}$ and
$\Phi^{C_{2}}(\bar{x}_{2})=a_{2}$, where $a_{1}$ and $a_{2}$ are the
orientations

$a_{1},a_{2}:\mathbf{RP}^{\infty}\to\Sigma^{1}i_{e}^{*}N_{e}^{C_{2^{n-1}}}H\mathbb{F}_{2}$

corresponding to the formal group laws with $MO$-orientations $\iota_{1}$ and
$\iota_{2}$. Since applying $\Phi^{C_{2}}(-)$ is a ring homomorphism [HHR16,
Proposition 2.59], applying $\Phi^{C_{2}}(-)$ to (4.1) shows that the power
series relating $a_{1}$ and $a_{2}$ is

$a_{2}=f(a_{1})=a_{1}+{\sum_{i\geq
1}}^{F_{2}}\Phi^{C_{2}}(\bar{t}_{i}^{C_{2^{n}}})a_{1}^{2^{i}}.$ (4.2)

Here, $F_{2}$ is the formal group law corresponding to $\iota_{2}$.

Now, consider the Real orientations

$\widetilde{x}_{1},\widetilde{x}_{2}:\mathbf{CP}^{\infty}\longrightarrow\Sigma^{\rho_{2}}i_{C_{2}}^{*}BP^{(\\!(C_{2^{n+1}})\\!)}$

defining the Real formal group laws $\widetilde{F}_{1}$ and
$\widetilde{F}_{2}$ corresponding to the maps

$\widetilde{\iota}_{1},\widetilde{\iota}_{2}:BP_{\mathbb{R}}\longrightarrow
i_{C_{2}}^{*}BP^{(\\!(C_{2^{n+1}})\\!)}\simeq\underbrace{BP_{\mathbb{R}}\wedge\cdots\wedge
BP_{\mathbb{R}}}_{2^{n}},$

where $\widetilde{\iota}_{1}$ and $\widetilde{\iota}_{2}$ are inclusions into
the first and second factors of $i_{C_{2}}^{*}BP^{(\\!(C_{2^{n+1}})\\!)}$
respectively. The $\bar{t}_{i}^{C_{2^{n+1}}}$-generators are defined to be the
coefficients of the power series relating $\widetilde{x}_{1}$ and
$\widetilde{x}_{2}$:

$\widetilde{x}_{2}=\widetilde{x}_{1}+{\sum_{i\geq
1}}^{\widetilde{F}_{2}}\bar{t}_{i}^{C_{2^{n+1}}}\widetilde{x}_{1}^{2^{i}}.$
(4.3)

After taking the norm $N_{C_{2}}^{C_{4}}(-)$ and applying the norm-restriction
adjunction to obtain the maps

$N_{C_{2}}^{C_{4}}(\widetilde{\iota}_{1}),N_{C_{2}}^{C_{4}}(\widetilde{\iota}_{2}):BP^{(\\!(C_{4})\\!)}\longrightarrow
N_{C_{2}}^{C_{4}}i_{C_{2}}^{*}BP^{(\\!(C_{2^{n+1}})\\!)}\longrightarrow
i_{C_{4}}^{*}BP^{(\\!(C_{2^{n+1}})\\!)}\\\
\simeq\underbrace{BP^{(\\!(C_{4})\\!)}\wedge\cdots\wedge
BP^{(\\!(C_{4})\\!)}}_{2^{n-1}},$

the observation is that $N_{C_{2}}^{C_{4}}(\widetilde{\iota}_{1})$ and
$N_{C_{2}}^{C_{4}}(\widetilde{\iota}_{2})$ are inclusions into the first and
second factors of $i_{C_{4}}^{*}BP^{(\\!(C_{2^{n+1}})\\!)}$, respectively.
This implies that after taking $\Phi^{C_{4}}(-)$, the maps

$\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(\widetilde{\iota}_{1}),\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(\widetilde{\iota}_{2}):H\mathbb{F}_{2}\longrightarrow
i_{e}^{*}N_{e}^{C_{2^{n-1}}}H\mathbb{F}_{2}$

are again $\iota_{1}$ and $\iota_{2}$, respectively. It follows that
$\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(\widetilde{x}_{1})=a_{1}$ and
$\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(\widetilde{x}_{2})=a_{2}$. Since applying
$\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(-)$ is a ring homomorphism [HHR16, Proposition
2.59], applying $\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(-)$ to (4.3) shows that the
power series relating $a_{1}$ and $a_{2}$ is

$a_{2}=f(a_{1})=a_{1}+{\sum_{i\geq
1}}^{F_{2}}\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})a_{1}^{2^{i}}.$
(4.4)

Comparing the coefficients of the power series in (4.2) and (4.4) implies the
equality
$\Phi^{C_{2}}(\bar{t}_{i}^{C_{2^{n}}})=\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})$,
as desired. ∎

######  Proposition 4.3.

For all $n\geq 1$ and $I\subseteq\mathbb{N}$, the following equivalence holds:

$\widetilde{E}\mathcal{F}[C_{4}]\wedge BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle\simeq\widetilde{E}\mathcal{F}[C_{4}]\wedge\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle.$

###### Proof.

By 3.2, the left hand side is equal to

$\displaystyle\quad\,\,\mathcal{P}^{*}_{C_{2^{n+1}}/C_{4}}\left(\Phi^{C_{4}}BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle\right)$
$\displaystyle\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{4}}\Phi^{C_{4}}\left(BP^{(\\!(C_{2^{n+1}})\\!)}{\underset{S^{0}\left[C_{2^{n+1}}\cdot\bar{t}_{i}^{C_{2^{n+1}}}\,|\,i\geq
1\right]}{\wedge}}S^{0}\left[C_{2^{n+1}}\cdot\bar{t}_{i}^{C_{2^{n+1}}}\,|\,i\in
I\right]\right)$
$\displaystyle\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{4}}\left(N_{e}^{C_{2^{n-1}}}H\mathbb{F}_{2}{\underset{S^{0}\left[C_{2^{n-1}}\cdot\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})\,|\,i\geq
1\right]}{\wedge}}S^{0}\left[C_{2^{n-1}}\cdot\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})\,|\,i\in
I\right]\right).$

Here, for the last equivalence, we have used the fact that the geometric fixed
points functor $\Phi^{H}(-)$ is a symmetric monoidal functor. Furthermore, if
$H$ is a normal subgroup of $G$ that contains $C_{2}$, then

$\Phi^{H}S^{0}[G\cdot\bar{x}]\simeq
S^{0}\left[G/H\cdot\Phi^{H}N_{C_{2}}^{H}(\bar{x})\right]$

according to the properties of the functor (see [HHR16, Section 2] and
[Ull13a, Lemma 4.3]).

By 3.2, the right hand side is equal to

$\displaystyle\quad\,\,\mathcal{P}^{*}_{C_{2^{n+1}}/C_{4}}\left(\Phi^{C_{4}}\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle)\right)$
$\displaystyle\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{4}}\left(\Phi^{C_{2}}BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle\right)$
$\displaystyle\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{4}}\Phi^{C_{2}}\left(BP^{(\\!(C_{2^{n}})\\!)}{\underset{S^{0}\left[C_{2^{n}}\cdot\bar{t}_{i}^{C_{2^{n}}}\,|\,i\geq
1\right]}{\wedge}}S^{0}\left[C_{2^{n}}\cdot\bar{t}_{i}^{C_{2^{n}}}\,|\,i\in
I\right]\right)$
$\displaystyle\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{4}}\left(N_{e}^{C_{2^{n-1}}}H\mathbb{F}_{2}{\underset{S^{0}\left[C_{2^{n-1}}\cdot\Phi^{C_{2}}(\bar{t}_{i}^{C_{2^{n}}})\,|\,i\geq
1\right]}{\wedge}}S^{0}\left[C_{2^{n-1}}\cdot\Phi^{C_{2}}(\bar{t}_{i}^{C_{2^{n}}})\,|\,i\in
I\right]\right).$

The equivalence now follows from 4.2 and the fact that the maps

$\displaystyle
S^{0}\left[C_{2^{n-1}}\cdot\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})\,|\,i\geq
1\right]$ $\displaystyle\longrightarrow
S^{0}\left[C_{2^{n-1}}\cdot\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})\,|\,i\in
I\right]$ $\displaystyle
S^{0}\left[C_{2^{n-1}}\cdot\Phi^{C_{2}}(\bar{t}_{i}^{C_{2^{n}}})\,|\,i\geq
1\right]$ $\displaystyle\longrightarrow
S^{0}\left[C_{2^{n-1}}\cdot\Phi^{C_{2}}(\bar{t}_{i}^{C_{2^{n}}})\,|\,i\in
I\right]$

are the same quotient maps via the identification
$\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})\leftrightsquigarrow\Phi^{C_{2}}(\bar{t}_{i}^{C_{2^{n}}})$.
∎

To proceed further, we will briefly recall the dual slice towers of
$BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle$ and
$BP^{(\\!(C_{2^{n}})\\!)}\langle I\rangle$, following the construction in
[HHR16, Section 6]. For more details on the slice towers of
$MU^{(\\!(C_{2^{n}})\\!)}$ and its quotients, see [HHR16, Section 6] and
[BHL+22, Section 2].

Let

$S^{0}\left[C_{2^{n+1}}\cdot\bar{t}_{i}^{C_{2^{n+1}}}\,|\,i\geq
1\right]\longrightarrow BP^{(\\!(C_{2^{n+1}})\\!)}$

be the refinement of $BP^{(\\!(C_{2^{n+1}})\\!)}$ with respect to the
generators $\bar{t}_{i}^{C_{2^{n+1}}}$, and let
$A=S^{0}\left[C_{2^{n+1}}\cdot\bar{t}_{i}^{C_{2^{n+1}}}\,|\,i\in I\right]$.
Then $A$ is a wedge summand of
$S^{0}\left[C_{2^{n+1}}\cdot\bar{t}_{i}^{C_{2^{n+1}}}\,|\,i\geq 1\right]$.
Applying $(-)\wedge_{S^{0}[C_{2^{n+1}}\cdot\bar{t}_{i}^{C_{2^{n+1}}}\,|\,i\geq
1]}A$ to the refinement above produces the refinement

$A\longrightarrow BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle$

for $BP^{(\\!(C_{2^{n}})\\!)}\langle I\rangle$.

The spectrum $BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle$ is a
$S^{0}\left[C_{2^{n+1}}\cdot\bar{t}_{i}^{C_{2^{n+1}}}\,|\,i\geq
1\right]$-module, and we can further view it as an $A$-module via the
associative algebra map

$A\longrightarrow
S^{0}\left[C_{2^{n+1}}\cdot\bar{t}_{i}^{C_{2^{n+1}}}\,|\,i\geq 1\right].$

Define $M_{\geq d}\subset A$ to be the monomial ideal consisting of all the
spheres of dimension $\geq d$. These monomial ideals produces an increasing
filtration

$\cdots\hookrightarrow M_{\geq d+1}\hookrightarrow M_{\geq d}\hookrightarrow
M_{\geq d-1}\hookrightarrow\cdots$

on $A$. Define

$P_{d}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle\right):=BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle\wedge_{A}M_{\geq
d}.$ (4.5)

Then the tower

$\cdots\longrightarrow P_{d+1}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle\right)\longrightarrow P_{d}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle\right)\longrightarrow P_{d-1}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle\right)\longrightarrow\cdots$

is the dual slice tower of $BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle$.

Smashing the dual slice tower with $\widetilde{E}\mathcal{F}[C_{4}]$ produces
the localized dual slice tower $\widetilde{E}\mathcal{F}[C_{4}]\wedge
P_{\bullet}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle\right)$. By 3.2,
the following equivalence holds:

$\displaystyle\quad\,\,\widetilde{E}\mathcal{F}[C_{4}]\wedge
P_{d}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle\right)$
$\displaystyle\simeq\widetilde{E}\mathcal{F}[C_{4}]\wedge\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle\wedge_{A}M_{\geq d}\right)$
$\displaystyle\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{4}}\left(\Phi^{C_{4}}(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle){\underset{\Phi^{C_{4}}(A)}{\wedge}}\Phi^{C_{4}}(M_{\geq d})\right).$
(4.6)

The construction of the dual slice tower of $BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle$ follows a similar method. Let $B\to BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle$ be the refinement of $BP^{(\\!(C_{2^{n}})\\!)}\langle I\rangle$ with
respect to the generators $\bar{t}_{i}^{C_{2^{n}}}$, where

$B=S^{0}\left[C_{2^{n}}\cdot\bar{t}_{i}^{C_{2^{n}}}\,|\,i\in I\right].$

Define $N_{\geq d}\subset B$ to be the monomial ideal consisting of all the
spheres of dimension $\geq d$ and let

$P_{d}\left(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle\right):=BP^{(\\!(C_{2^{n}})\\!)}\langle I\rangle\wedge_{B}N_{\geq
d}.$

Then the tower

$\cdots\longrightarrow P_{d+1}\left(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle\right)\longrightarrow P_{d}\left(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle\right)\longrightarrow P_{d-1}\left(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle\right)\longrightarrow\cdots$

is the dual slice tower of $BP^{(\\!(C_{2^{n}})\\!)}\langle I\rangle$.
Smashing this dual slice tower with $\widetilde{E}\mathcal{F}[C_{2}]$ produces
the localized dual slice tower $\widetilde{E}\mathcal{F}[C_{2}]\wedge
P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)}\langle I\rangle)$. By 3.2, we have the
following equivalence:

$\displaystyle\widetilde{E}\mathcal{F}[C_{2}]\wedge
P_{d}\left(BP^{(\\!(C_{2^{n}})\\!)}\langle I\rangle\right)$
$\displaystyle\simeq\widetilde{E}\mathcal{F}[C_{2}]\wedge\left(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle\wedge_{B}N_{\geq d}\right)$
$\displaystyle\simeq\mathcal{P}^{*}_{C_{2^{n}}/C_{2}}\left(\Phi^{C_{2}}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle){\underset{\Phi^{C_{2}}(B)}{\wedge}}\Phi^{C_{2}}(N_{\geq d})\right)$
(4.7)

######  Proposition 4.4.

We have the following equivalence of towers:

$\widetilde{E}\mathcal{F}[C_{4}]\wedge
P_{\bullet}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle\right)\simeq\widetilde{E}\mathcal{F}[C_{4}]\wedge
P_{\bullet}\left(\mathcal{P}_{C_{2^{n+1}}/C_{2}}^{*}BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle\right).$

###### Proof.

For any $N\subseteq G$ a normal subgroup and $H\subseteq G$ a subgroup, there
is a natural equivalence (see [Hil12, Section 4.2])

$\Phi^{N}(G_{+}\wedge_{H}S^{k\rho_{H}})\simeq\left\\{\begin{array}[]{ll}*&\text{if
}N\not\subseteq H,\\\ (G/N)_{+}\wedge_{H/N}S^{k\rho_{H/N}}&\text{if
}N\subseteq H.\end{array}\right.$

This fact, combined with (4.6), shows that for all $d$ and $0\leq i\leq 3$,
the localized dual slice sections

$\widetilde{E}\mathcal{F}[C_{4}]\wedge
P_{4d-i}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle\right)$

are equivalent to each other. Similarly, by (4.7) and 4.5,

$\widetilde{E}\mathcal{F}[C_{4}]\wedge
P_{4d-i}\left(\mathcal{P}_{C_{2^{n+1}}/C_{2}}^{*}BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle\right)$

are also equivalent to each other for all $d$ and $0\leq i\leq 3$. Therefore,
in order to prove the desired equivalence of towers, it suffices to show that
for all $d$, we have the equivalence

$\widetilde{E}\mathcal{F}[C_{4}]\wedge
P_{4d}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle\right)\simeq\widetilde{E}\mathcal{F}[C_{4}]\wedge
P_{4d}\left(\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle\right),$

and that the maps

$\displaystyle\widetilde{E}\mathcal{F}[C_{4}]\wedge
P_{4d+4}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle\right)$
$\displaystyle\longrightarrow\widetilde{E}\mathcal{F}[C_{4}]\wedge
P_{4d}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle\right),$
$\displaystyle\widetilde{E}\mathcal{F}[C_{4}]\wedge
P_{4d+4}\left(\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle\right)$
$\displaystyle\longrightarrow\widetilde{E}\mathcal{F}[C_{4}]\wedge
P_{4d}\left(\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle\right)$

are the same. By (4.6), 4.5, and (4.7), this is equivalent to showing that

$\Phi^{C_{4}}(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle){\underset{\Phi^{C_{4}}(A)}{\wedge}}\Phi^{C_{4}}(M_{\geq
4d})\simeq\Phi^{C_{2}}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle){\underset{\Phi^{C_{2}}(B)}{\wedge}}\Phi^{C_{2}}(N_{\geq 2d})$ (4.8)

and that the two maps

$\Phi^{C_{4}}(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle){\underset{\Phi^{C_{4}}(A)}{\wedge}}\Phi^{C_{4}}(M_{\geq
4d+4})\longrightarrow\Phi^{C_{4}}(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle){\underset{\Phi^{C_{4}}(A)}{\wedge}}\Phi^{C_{4}}(M_{\geq 4d})$ (4.9)
$\Phi^{C_{2}}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle){\underset{\Phi^{C_{2}}(B)}{\wedge}}\Phi^{C_{2}}(N_{\geq
2d+2})\longrightarrow\Phi^{C_{2}}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle){\underset{\Phi^{C_{2}}(B)}{\wedge}}\Phi^{C_{2}}(N_{\geq 2d})$ (4.10)

are the same.

Note that since

$\displaystyle\Phi^{C_{4}}(A)$
$\displaystyle=S^{0}\left[C_{2^{n-1}}\cdot\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})\,|\,i\in
I\right],$ $\displaystyle\Phi^{C_{2}}(B)$
$\displaystyle=S^{0}\left[C_{2^{n-1}}\cdot\Phi^{C_{2}}(\bar{t}_{i}^{C_{2^{n}}})\,|\,i\in
I\right],$

they are isomorphic wedges of slice cells under the identification

$\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})\leftrightsquigarrow\Phi^{C_{2}}(\bar{t}_{i}^{C_{2^{n}}}).$

Also, under this identification, the quotient maps

$\Phi^{C_{4}}(A)\longrightarrow\Phi^{C_{4}}(M_{\geq 4d})$

and

$\Phi^{C_{2}}(B)\longrightarrow\Phi^{C_{2}}(N_{\geq 2d})$

are the same. Furthermore, by Lemma 4.2, and the proof of Proposition 4.3, the
maps

$\displaystyle\Phi^{C_{4}}(A)$
$\displaystyle\longrightarrow\Phi^{C_{4}}(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle),$ $\displaystyle\Phi^{C_{2}}(B)$
$\displaystyle\longrightarrow\Phi^{C_{2}}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle)$

are also the same. It follows that equivalence (4.8) holds.

To prove that the maps (4.9) and (4.10) are the same, it suffices to show that
the inclusion maps

$\displaystyle\Phi^{C_{4}}(M_{\geq 4d+4})$
$\displaystyle\longrightarrow\Phi^{C_{4}}(M_{\geq 4d}),$
$\displaystyle\Phi^{C_{2}}(N_{\geq 2d+2})$
$\displaystyle\longrightarrow\Phi^{C_{2}}(N_{\geq 2d})$

are the same. This follows directly from the same identification

$\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})\leftrightsquigarrow\Phi^{C_{2}}(\bar{t}_{i}^{C_{2^{n}}})$

earlier. ∎

######  Proposition 4.5.

We have the following equivalence of towers:

$\widetilde{E}\mathcal{F}[C_{4}]\wedge
P_{\bullet}\left(\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle\right)\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}\left(\widetilde{E}\mathcal{F}[C_{2}]\wedge\mathscr{D}P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle)\right).$

###### Proof.

By 3.6, there is an equivalence of towers

$P_{\bullet}\left(\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle\right)\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}\left(\mathscr{D}P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle)\right).$ (4.11)

More precisely, for all $d\geq 1$, we have the equivalences

$\displaystyle
P_{2d}\left(\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle\right)$ $\displaystyle\simeq
P_{2d-1}\left(\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle\right)$
$\displaystyle\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}\left(P_{d}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle)\right).$

Therefore, by (4.11), the left hand side of the equivalence in the proposition
statement is equivalent to

$\displaystyle\quad\,\,\widetilde{E}\mathcal{F}[C_{4}]\wedge\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}\left(\mathscr{D}P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle)\right)$
$\displaystyle\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{4}}\left(\Phi^{C_{4}}\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}\mathscr{D}P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle)\right)\,\,\,(\text{\lx@cref{creftype~refnum}{lem:easyFact1}})$
$\displaystyle\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{4}}\left(\Phi^{C_{2}}\mathscr{D}P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle\right).$ (4.12)

By 3.2, the right hand side is equivalent to

$\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}\left(\mathcal{P}^{*}_{C_{2^{n}}/C_{2}}\Phi^{C_{2}}\mathscr{D}P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle\right)\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{4}}\left(\Phi^{C_{2}}\mathscr{D}P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle\right).$

This proves the desired equivalence. ∎

###### Proof of 4.1.

The case $k=1$ is an immediate consequence of 4.4 and 4.5:

$\displaystyle\widetilde{E}\mathcal{F}[C_{4}]\wedge
P_{\bullet}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle\right)$
$\displaystyle\simeq\widetilde{E}\mathcal{F}[C_{4}]\wedge
P_{\bullet}\left(\mathcal{P}_{C_{2^{n+1}}/C_{2}}^{*}BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle\right)$
$\displaystyle\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}\left(\widetilde{E}\mathcal{F}[C_{2}]\wedge\mathscr{D}P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle)\right)$
$\displaystyle\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}\mathscr{D}\left(\widetilde{E}\mathcal{F}[C_{2}]\wedge
P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)}\langle I\rangle)\right).$

For the general case when $1\leq k\leq n$, the left hand side of the
equivalence is

$\displaystyle\quad\,\,\widetilde{E}\mathcal{F}[C_{2^{k+1}}]\wedge
P_{\bullet}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle\right)$
$\displaystyle\simeq\widetilde{E}\mathcal{F}[C_{2^{k+1}}]\wedge\left(\widetilde{E}\mathcal{F}[C_{4}]\wedge
P_{\bullet}(BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle)\right)$
$\displaystyle\simeq\widetilde{E}\mathcal{F}[C_{2^{k+1}}]\wedge\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}\mathscr{D}\left(\widetilde{E}\mathcal{F}[C_{2}]\wedge
P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)}\langle I\rangle)\right)\,\,\,\text{($k=1$
case)}$
$\displaystyle\simeq\widetilde{E}\mathcal{F}[C_{2^{k+1}}]\wedge\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}\left(\widetilde{E}\mathcal{F}[C_{2}]\wedge\mathscr{D}P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle)\right)$
$\displaystyle\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2^{k+1}}}\Phi^{C_{2^{k+1}}}\left(\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}\left(\widetilde{E}\mathcal{F}[C_{2}]\wedge\mathscr{D}P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle)\right)\right)\,\,\,\text{(\lx@cref{creftype~refnum}{lem:easyFact1})}$
$\displaystyle\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2^{k+1}}}\Phi^{C_{2^{k}}}\left(\widetilde{E}\mathcal{F}[C_{2}]\wedge\mathscr{D}P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle)\right)$
$\displaystyle\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2^{k+1}}}\Phi^{C_{2^{k}}}\left(\mathscr{D}P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle)\right).$

The right hand side of the equivalence is

$\displaystyle\quad\,\,\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}\mathscr{D}\left(\widetilde{E}\mathcal{F}[C_{2^{k}}]\wedge
P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)}\langle I\rangle)\right)$
$\displaystyle\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}\left(\widetilde{E}\mathcal{F}[C_{2^{k}}]\wedge\mathscr{D}P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle)\right)$
$\displaystyle\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}\left(\mathcal{P}^{*}_{C_{2^{n}}/C_{2^{k}}}\Phi^{C_{2^{k}}}\left(\mathscr{D}P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle)\right)\right)\,\,\,\text{(\lx@cref{creftype~refnum}{lem:easyFact1})}$
$\displaystyle\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2^{k+1}}}\Phi^{C_{2^{k}}}\left(\mathscr{D}P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)}\langle
I\rangle)\right).$

This establishes the desired equivalence. ∎

##  5\. Correspondence formulas

By 3.7, the equivalence of localized dual slice towers proven in 4.1 induces a
shearing isomorphism of the corresponding localized slice spectral sequences.
In this section, we will establish correspondence formulas between classes in
these localized slice spectral sequences under the shearing isomorphism.

In order to establish our formulas, we will first focus on the equivalence

$\widetilde{E}\mathcal{F}[C_{4}]\wedge
P_{\bullet}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\right)\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}\mathscr{D}\left(\widetilde{E}\mathcal{F}[C_{2}]\wedge
P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)})\right).$ (5.1)

Once we have established correspondence formulas for classes on the
$\mathcal{E}_{2}$-pages of
$\widetilde{E}\mathcal{F}[C_{4}]\wedge\textup{SliceSS}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\right)$
and
$\widetilde{E}\mathcal{F}[C_{2}]\wedge\textup{SliceSS}(BP^{(\\!(C_{2^{n}})\\!)})$,
the correspondence formulas for all the other equivalences in 4.1 will follow.

To start our analysis, we will first recall some notations. When working
2-locally, the representation spheres associated to the 2-dimensional real
$C_{2^{n}}$-representations corresponding to rotations by
$\left(\frac{2\pi\cdot k}{2^{n}}\right)$ and $\left(\frac{2\pi\cdot
k^{\prime}}{2^{n}}\right)$ are equivalent whenever $k$ and $k^{\prime}$ have
the same 2-adic valuation. Therefore, the $RO(C_{2^{n}})$-graded homotopy
groups are graded by the representations
$\\{1,\sigma,\lambda_{1},\ldots,\lambda_{n-1}\\}$, and the
$RO(C_{2^{n+1}})$-graded homotopy groups are graded by the representations
$\\{1,\sigma,\lambda_{1},\ldots,\lambda_{n}\\}$. Here, 1 is the trivial
representation, $\sigma$ is the sign representation, and $\lambda_{i}$ is the
2-dimensional real representation that corresponds to rotation by
$\left(\frac{\pi}{2^{i}}\right)$.

We will denote the $C_{2^{n}}$\- and the $C_{2^{n+1}}$-equivariant constant
Mackey functors by $\underline{\mathbb{Z}}_{C_{2^{n}}}$ and
$\underline{\mathbb{Z}}_{C_{2^{n+1}}}$, respectively.

The slice associated graded for the tower
$\widetilde{E}\mathcal{F}[C_{4}]\wedge
P_{\bullet}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\right)$ is

$\widetilde{E}\mathcal{F}[C_{4}]\wedge
H\underline{\mathbb{Z}}_{C_{2^{n+1}}}\left[C_{2^{n+1}}\cdot\bar{t}_{i}^{C_{2^{n+1}}}\,|\,i\geq
1\right]\simeq
a_{\lambda_{n-1}}^{-1}H\underline{\mathbb{Z}}_{C_{2^{n+1}}}\left[C_{2^{n}}\cdot
N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})\,|\,i\geq 1\right],$

and the slice associated graded for the tower
$\widetilde{E}\mathcal{F}[C_{2}]\wedge P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)})$
is

$\widetilde{E}\mathcal{F}[C_{2}]\wedge
H\underline{\mathbb{Z}}_{C_{2^{n}}}\left[C_{2^{n}}\cdot\bar{t}_{i}^{C_{2^{n}}}\,|\,i\geq
1\right]\simeq
a_{\lambda_{n-1}}^{-1}H\underline{\mathbb{Z}}_{C_{2^{n}}}\left[C_{2^{n}}\cdot\bar{t}_{i}^{C_{2^{n}}}\,|\,i\geq
1\right].$

For every $V\in RO(C_{2^{n+1}})$ and $C_{2}\subseteq H\subseteq C_{2^{n+1}}$,
(5.1) induces an isomorphism

$\displaystyle\,\,\pi_{V}^{H}\left(a_{\lambda_{n-1}}^{-1}H\underline{\mathbb{Z}}_{C_{2^{n+1}}}\left[C_{2^{n}}\cdot
N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})\,|\,i\geq 1\right]\right)$
$\displaystyle\cong$
$\displaystyle\,\,\pi_{V^{C_{2}}}^{H/C_{2}}\left(a_{\lambda_{n-1}}^{-1}H\underline{\mathbb{Z}}_{C_{2^{n}}}\left[C_{2^{n}}\cdot\bar{t}_{i}^{C_{2^{n}}}\,|\,i\geq
1\right]\right)$ (5.2)

by 3.4. There are distinguished generators for the homotopy groups on each
side of the above isomorphism ($a_{V}$, $u_{V}$, $\bar{t}_{i}^{C_{2^{n}}}$,
$N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})$). Our first goal is to prove
correspondence formulas for these generators.

######  Proposition 5.1.

Suppose $V\in RO(C_{2^{n+1}}/C_{2})$ is of the form

$V=c_{0}\sigma+c_{1}\lambda_{1}+\cdots+c_{n-1}\lambda_{n-1},$

where $c_{i}\geq 0$ for all $0\leq i\leq n-1$. Equivalence (5.1) and
isomorphism (5) induce the correspondence $a_{V}\leftrightsquigarrow a_{V}$.
Here, $V$ is also viewed as an element of $RO(C_{2^{n+1}})$ through pullback
along the quotient map $C_{2^{n+1}}\to C_{2^{n+1}}/C_{2}$.

###### Proof.

By definition, on the left hand side of isomorphism (5), the special class
${a_{V}\in\pi_{-V}^{C_{2^{n+1}}}(\widetilde{E}\mathcal{F}[C_{4}]\wedge
H\underline{\mathbb{Z}}_{C_{2^{n+1}}})}$ is defined to be the composite

$S^{-V}\xrightarrow{a_{V}}S^{0}\longrightarrow\widetilde{E}\mathcal{F}[C_{4}]\wedge
H\underline{\mathbb{Z}}_{C_{2^{n+1}}}\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}(\widetilde{E}\mathcal{F}[C_{2}]\wedge
H\underline{\mathbb{Z}}_{C_{2^{n}}}),$

where the first map is the Euler class $a_{V}\in a_{-V}S^{0}$ that corresponds
to the inclusion map $S^{0}\to S^{V}$, and the last equivalence can be deduced
from (5.1). By adjunction between the functors
$\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}$ and $\Phi^{C_{2}}$, the composition
above corresponds to the composition

$\Phi^{C_{2}}(S^{-V})\xrightarrow{\Phi^{C_{2}}(a_{V})}\Phi^{C_{2}}(S^{0})\longrightarrow\widetilde{E}\mathcal{F}[C_{2}]\wedge
H\underline{\mathbb{Z}}_{C_{2^{n}}}.$

By our assumption on $V$, we have the equality $V^{C_{2}}=V$. Therefore,
$\Phi^{C_{2}}(S^{-V})=S^{-V}$ and
${\Phi^{C_{2}}(a_{V})=a_{V}\in\pi_{-V}^{C_{2^{n}}}S^{0}}$. It follows that the
composition map is the class
${a_{V}\in\pi_{-V}^{C_{2^{n}}}(\widetilde{E}\mathcal{F}[C_{2}]\wedge
H\underline{\mathbb{Z}}_{C_{2^{n}}})}$. ∎

######  Proposition 5.2.

Suppose $V\in RO(C_{2^{n+1}}/C_{2})$ is of the form

$c_{0}(2\sigma)+c_{1}\lambda_{1}+\cdots+c_{n-1}\lambda_{n-1},$

where $c_{i}\geq 0$ for all $0\leq i\leq n-1$. Equivalence (5.1) and
isomorphism (5) induce the correspondence $u_{V}\leftrightsquigarrow u_{V}$,
where $V$ is also viewed as an element of $RO(C_{2^{n+1}})$ through pullback
along the quotient map $C_{2^{n+1}}\to C_{2^{n+1}}/C_{2}$.

###### Proof.

Note that by our assumption, $V$ is orientable, and the orientation class
$u_{V}$ is defined to be the class in
${\pi_{|V|-V}^{C_{2^{n+1}}}(\widetilde{E}\mathcal{F}[C_{4}]\wedge
H\underline{\mathbb{Z}}_{C_{2^{n+1}}})}$ that represents the composite map

$S^{|V|-V}\xrightarrow{u_{V}}H\underline{\mathbb{Z}}_{C_{2^{n+1}}}\longrightarrow\widetilde{E}\mathcal{F}[C_{4}]\wedge
H\underline{\mathbb{Z}}_{C_{2^{n+1}}}\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}(\widetilde{E}\mathcal{F}[C_{2}]\wedge
H\underline{\mathbb{Z}}_{C_{2^{n}}}).$

Here, the first map $u_{V}$ is the (preferred) generator for
$\pi_{|V|-V}^{C_{2^{n+1}}}H\underline{\mathbb{Z}}_{C_{2^{n+1}}}\cong\mathbb{Z}$,
and the second map is the quotient map

$\mathbb{Z}\cong\pi_{|V|-V}^{C_{2^{n+1}}}H\underline{\mathbb{Z}}_{C_{2^{n+1}}}\longrightarrow\pi_{|V|-V}^{C_{2^{n+1}}}(a_{\lambda_{n-1}}^{-1}H\underline{\mathbb{Z}}_{C_{2^{n+1}}})\cong\mathbb{Z}/2^{n}.$

By adjunction between the functors $\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}$ and
$\Phi^{C_{2}}$, the composite map $u_{V}$ corresponds to the map

$S^{|V|-V}\simeq\Phi^{C_{2}}(S^{|V|-V})\longrightarrow\widetilde{E}\mathcal{F}[C_{2}]\wedge
H\underline{\mathbb{Z}}_{C_{2^{n}}}\simeq
a_{\lambda_{n-1}}^{-1}H\underline{\mathbb{Z}}_{C_{2^{n}}}.$

This represents the generator in
$\pi_{|V|-V}^{C_{2^{n}}}(a_{\lambda_{n-1}}^{-1}H\underline{\mathbb{Z}}_{C_{2^{n}}})\cong\mathbb{Z}/2^{n}$.
Therefore, this map must be in the image of a generator $u_{V}$ for
$\pi_{|V|-V}^{C_{2^{n}}}H\underline{\mathbb{Z}}_{C_{2^{n}}}\cong\mathbb{Z}$
under the composition

$S^{|V|-V}\longrightarrow
H\underline{\mathbb{Z}}_{C_{2^{n}}}\longrightarrow\widetilde{E}\mathcal{F}[C_{2}]\wedge
H\underline{\mathbb{Z}}_{C_{2^{n}}}.$

This shows that $u_{V}\leftrightsquigarrow u_{V}$. ∎

######  Proposition 5.3.

For all $1\leq j\leq n$, Equivalence (5.1) and isomorphism (5) induce the
correspondence

$N_{C_{2}}^{C_{2^{j+1}}}(\bar{t}_{i}^{C_{2^{n+1}}})a_{\lambda_{j}}^{2^{j-1}(2^{i}-1)}\leftrightsquigarrow
N_{C_{2}}^{C_{2^{j}}}(\bar{t}_{i}^{C_{2^{n}}})$

###### Proof.

We will first prove that when $j=1$, we have the correspondence

$N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})a_{\lambda_{1}}^{2^{i}-1}\leftrightsquigarrow\bar{t}_{i}^{C_{2^{n}}}.$

The general case will follow immediately by applying the norm.

By 4.2, we have the equality

$\Phi^{C_{2}}(\bar{t}_{i}^{C_{2^{n}}})=\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}}).$

For the map

$\bar{t}_{i}^{C_{2^{n}}}:S^{(2^{i}-1)(1+\sigma)}\longrightarrow
i_{C_{2}}^{*}BP^{(\\!(C_{2^{n}})\\!)},$

the $C_{2}$-geometric fixed points $\Phi^{C_{2}}(\bar{t}_{i}^{C_{2^{n}}})$ is
represented by the composite map
$\bar{t}_{i}^{C_{2^{n}}}a_{\sigma}^{2^{i}-1}$:

$S^{2^{i}-1}\xrightarrow{a_{\sigma}^{2^{i}-1}}S^{(2^{i}-1)(1+\sigma)}\xrightarrow{\bar{t}_{i}^{C_{2^{n}}}}i_{C_{2}}^{*}BP^{(\\!(C_{2^{n}})\\!)}\longrightarrow\widetilde{E}\mathcal{F}[C_{2}]\wedge
i_{C_{2}}^{*}BP^{(\\!(C_{2^{n}})\\!)}.$

Similarly, for the map

$N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}}):S^{(2^{i}-1)(1+\sigma+\lambda_{1})}\longrightarrow
i_{C_{4}}^{*}BP^{(\\!(C_{2^{n+1}})\\!)},$

the $C_{4}$-geometric fixed points
$\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})$ is represented by
the composite map
$N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})a_{\bar{\rho}_{4}}^{2^{i}-1}:$

$S^{2^{i}-1}\xrightarrow{a_{\bar{\rho}_{4}}^{2^{i}-1}}S^{(2^{i}-1)(1+\sigma+\lambda_{1})}\xrightarrow{N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})}i_{C_{4}}^{*}BP^{(\\!(C_{2^{n+1}})\\!)}\longrightarrow\widetilde{E}\mathcal{F}[C_{4}]\wedge
i_{C_{4}}^{*}BP^{(\\!(C_{2^{n+1}})\\!)}.$

Therefore, this implies that we have the correspondence

$\bar{t}_{i}^{C_{2^{n}}}a_{\sigma}^{2^{i}-1}\leftrightsquigarrow
N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})a_{\bar{\rho}_{4}}^{2^{i}-1}=N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})a_{\sigma}^{2^{i}-1}a_{\lambda_{1}}^{2^{i}-1}.$

Since we have already established the correspondence
$a_{\sigma}\leftrightsquigarrow a_{\sigma}$ in 5.1, we obtain the
correspondence

$\bar{t}_{i}^{C_{2^{n}}}\leftrightsquigarrow
N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})a_{\lambda_{1}}^{2^{i}-1}.$

Applying the norm, we deduce that for $1\leq j\leq n$, we have the
correspondence

$N_{C_{2}}^{C_{2^{j}}}(\bar{t}_{i}^{C_{2^{n}}})\leftrightsquigarrow
N_{C_{4}}^{C_{2^{j+1}}}\left(N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n+1}}})a_{\lambda_{1}}^{2^{i}-1}\right)=N_{C_{2}}^{C_{2^{j+1}}}(\bar{t}_{i}^{C_{2^{n+1}}})a_{\lambda_{j}}^{2^{j-1}(2^{i}-1)}.$

∎

We are now ready to prove correspondence formulas for the general case of the
equivalence in 4.1, which states that for $1\leq k\leq n$, we have the
equivalence

$\widetilde{E}\mathcal{F}[C_{2^{k+1}}]\wedge
P_{\bullet}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle\right)\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2^{k}}}\mathscr{D}^{k}\left(\widetilde{E}\mathcal{F}[C_{2}]\wedge
P_{\bullet}(BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle I\rangle)\right).$ (5.3)

The slice associated graded for the tower
$\widetilde{E}\mathcal{F}[C_{2^{k+1}}]\wedge
P_{\bullet}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle\right)$ is

$\displaystyle\,\,\widetilde{E}\mathcal{F}[C_{2^{k+1}}]\wedge
H\underline{\mathbb{Z}}_{C_{2^{n+1}}}\left[C_{2^{n+1}}\cdot\bar{t}_{i}^{C_{2^{n+1}}}\,|\,i\in
I\right]$ $\displaystyle\simeq$
$\displaystyle\,\,a_{\lambda_{n-k}}^{-1}H\underline{\mathbb{Z}}_{C_{2^{n+1}}}\left[C_{2^{n-k+1}}\cdot
N_{C_{2}}^{C_{2^{k+1}}}(\bar{t}_{i}^{C_{2^{n+1}}})\,|\,i\in I\right],$

and the slice associated graded for the tower
$\widetilde{E}\mathcal{F}[C_{2}]\wedge
P_{\bullet}(BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle I\rangle)$ is

$\displaystyle\,\,\widetilde{E}\mathcal{F}[C_{2}]\wedge
H\underline{\mathbb{Z}}_{C_{2^{n-k+1}}}\left[C_{2^{n-k+1}}\cdot\bar{t}_{i}^{C_{2^{n-k+1}}}\,|\,i\in
I\right]$ $\displaystyle\simeq$
$\displaystyle\,\,a_{\lambda_{n-k}}^{-1}H\underline{\mathbb{Z}}_{C_{2^{n-k+1}}}\left[C_{2^{n-k+1}}\cdot\bar{t}_{i}^{C_{2^{n-k+1}}}\,|\,i\in
I\right].$

For every $V\in RO(C_{2^{n+1}})$ and $C_{2^{k}}\subseteq H\subseteq
C_{2^{n+1}}$, (5.3) induces an isomorphism

$\displaystyle\,\,\pi_{V}^{H}\left(a_{\lambda_{n-k}}^{-1}H\underline{\mathbb{Z}}_{C_{2^{n+1}}}\left[C_{2^{n-k+1}}\cdot
N_{C_{2}}^{C_{2^{k+1}}}(\bar{t}_{i}^{C_{2^{n+1}}})\,|\,i\in I\right]\right)$
$\displaystyle\cong$
$\displaystyle\,\,{\underline{\pi}}_{V^{C_{2^{k}}}}^{H/C_{2^{k}}}\left(a_{\lambda_{n-k}}^{-1}H\underline{\mathbb{Z}}_{C_{2^{n-k+1}}}\left[C_{2^{n-k+1}}\cdot\bar{t}_{i}^{C_{2^{n-k+1}}}\,|\,i\in
I\right]\right)$ (5.4)

by 3.4.

Recall that the reduced regular representation $\bar{\rho}_{2^{n+1}}\in
RO(C_{2^{n+1}})$ can be decomposed as

$\bar{\rho}_{2^{n+1}}=2^{n-1}\lambda_{n}+2^{n-2}\lambda_{n-1}+\cdots+\lambda_{1}+\sigma.$

######  Definition 5.4.

For $1\leq k\leq n+1$, define $\bar{\rho}_{2^{n+1}/2^{k}}\in RO(C_{2^{n+1}})$
to be the representation

$\bar{\rho}_{2^{n+1}/2^{k}}:=2^{n-k-1}\lambda_{n-k}+2^{n-k-2}\lambda_{n-k-1}+\ldots+\lambda_{1}+\sigma.$

######  Theorem 5.5 (Correspondence formulas).

Equivalence (5.3) and isomorphism (5) induce the following correspondences:

  1. (1)

Suppose $V\in RO(C_{2^{n+1}}/C_{2^{k}})=RO(C_{2^{n-k+1}})$ is of the form
$V=c_{0}\sigma+c_{1}\lambda_{1}+\cdots+c_{n-k}\lambda_{n-k}$, where $c_{i}\geq
0$ for all $0\leq i\leq n-k$. Then we have the correspondence

$a_{V}\leftrightsquigarrow a_{V},$

where $V$ is also viewed as an element of $RO(C_{2^{n+1}})$ through pullback
along the map $C_{2^{n+1}}\to C_{2^{n+1}}/C_{2^{k}}$.

  2. (2)

Suppose $V\in RO(C_{2^{n+1}}/C_{2^{k}})=RO(C_{2^{n-k+1}})$ is of the form
$V=c_{0}(2\sigma)+c_{1}\lambda_{1}+\cdots+c_{n-k}\lambda_{n-k}$, where
$c_{i}\geq 0$ for all $0\leq i\leq n-k$. Then we have the correspondence

$u_{V}\leftrightsquigarrow u_{V},$

where as above $V$ is also viewed as an element of $RO(C_{2^{n+1}})$ through
pullback along the map $C_{2^{n+1}}\to C_{2^{n+1}}/C_{2^{k}}$.

  3. (3)

For all $i\in I$ and $1\leq j\leq n-k+1$, we have the correspondence

$N_{C_{2}}^{C_{2^{k+j}}}(\bar{t}_{i}^{C_{2^{n+1}}})\cdot\left(\frac{a_{\bar{\rho}_{2^{k+j}}}}{a_{\bar{\rho}_{2^{k+j}/2^{k}}}}\right)^{2^{i}-1}\leftrightsquigarrow
N_{C_{2}}^{C_{2^{j}}}(\bar{t}_{i}^{C_{2^{n-k+1}}}).$

###### Proof.

Correspondences (1) and (2) follow immediately from 5.1 and 5.2. To prove
correspondence (3), we will apply 5.3 iteratively. For $j=1$, we have the
correspondences

$\displaystyle\bar{t}_{i}^{C_{2^{n-k+1}}}$ $\displaystyle\leftrightsquigarrow
N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{n-k+2}}})\cdot a_{\lambda_{1}}^{2^{i}-1}$
$\displaystyle\leftrightsquigarrow
N_{C_{2}}^{C_{8}}(\bar{t}_{i}^{C_{2^{n-k+3}}})\cdot
a_{\lambda_{2}}^{2(2^{i}-1)}a_{\lambda_{1}}^{2^{i}-1}$
$\displaystyle\,\,\,\,\,\vdots$ $\displaystyle\leftrightsquigarrow
N_{C_{2}}^{C_{2^{k+1}}}(\bar{t}_{i}^{C_{2^{n+1}}})\cdot
a_{\lambda_{k}}^{2^{k-1}(2^{i}-1)}a_{\lambda_{k-1}}^{2^{k-2}(2^{i}-1)}\cdots
a_{\lambda_{2}}^{2(2^{i}-1)}a_{\lambda_{1}}^{(2^{i}-1)}$

Applying the norm, it follows that for $1\leq j\leq n-k+1$, we have the
correspondence

$\displaystyle N_{C_{2}}^{C_{2^{j}}}(\bar{t}_{i}^{C_{2^{n-k+1}}})$
$\displaystyle\leftrightsquigarrow
N_{C_{2^{k+1}}}^{C_{2^{k+j}}}\left(N_{C_{2}}^{C_{2^{k+1}}}(\bar{t}_{i}^{C_{2^{n+1}}})\cdot
a_{\lambda_{k}}^{2^{k-1}(2^{i}-1)}a_{\lambda_{k-1}}^{2^{k-2}(2^{i}-1)}\cdots
a_{\lambda_{2}}^{2(2^{i}-1)}a_{\lambda_{1}}^{(2^{i}-1)}\right)$
$\displaystyle=N_{C_{2}}^{C_{2^{k+j}}}(\bar{t}_{i}^{C_{2^{n+1}}})\cdot
a_{\lambda_{k+j-1}}^{2^{k+j-2}(2^{i}-1)}a_{\lambda_{k+j-2}}^{2^{k+j-3}(2^{i}-1)}\cdots
a_{\lambda_{j}}^{2^{j-1}(2^{i}-1)}$
$\displaystyle=N_{C_{2}}^{C_{2^{k+j}}}(\bar{t}_{i}^{C_{2^{n+1}}})\cdot\left(\frac{a_{\bar{\rho}_{2^{k+j}}}}{a_{\bar{\rho}_{2^{k+j}/2^{k}}}}\right)^{2^{i}-1}.\qed$

##  6\. The Transchromatic Isomorphism Theorem

In this section, we will prove the Transchromatic Isomorphism Theorem (6.2),
which establishes shearing isomorphisms between the slice spectral sequences
of $BP^{(\\!(G)\\!)}\langle I\rangle$ for different groups $G$.

Recall that if $V$ is an element in $RO(C_{2^{n+1}})$ and $1\leq k\leq n+1$,
then $V^{C_{2^{k}}}\in RO(C_{2^{n+1}}/C_{2^{k}})$ is the fixed points of $V$
with respect to the subgroup $C_{2^{k}}\subseteq C_{2^{n+1}}$. Also, recall
from 2.3 that the line $\mathcal{L}^{V}_{2^{k}-1}$ on the $(V+t-s,s)$-graded
page of an $RO(C_{2^{n+1}})$-graded spectral sequence is defined by the
equation

$s=(2^{k}-1)(t-s)+\tau_{V}(\mathcal{F}_{\leq 2^{k}}),$

where by 2.2,

$\tau_{V}(\mathcal{F}_{\leq 2^{k}})=\max_{0\leq j\leq
k}\left(|V^{C_{2^{j}}}|\cdot 2^{j}-|V|\right).$

######  Definition 6.1.

For $V\in RO(C_{2^{n+1}})$ and $1\leq k\leq n$, let $C_{V,k}$ be the non-
negative constant defined by the equation

$C_{V,k}=\frac{\tau_{V}(\mathcal{F}_{\leq 2^{k}})-\left(|V^{C_{2^{k}}}|\cdot
2^{k}-|V|\right)}{2^{k}}\geq 0.$

######  Theorem 6.2 (Shearing Isomorphism).

Suppose $I\subseteq\mathbb{N}$. For all $V\in RO(C_{2^{n+1}})$ and $1\leq
k\leq n$, there is a shearing isomorphism
$d_{2^{k}r-(2^{k}-1)}\leftrightsquigarrow d_{r}$ between the following regions
of spectral sequences:

  1. (1)

The $(V+t-s,s)$-graded page of
${\textup{SliceSS}(BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle)}$ on or above
the line $\mathcal{L}^{V}_{2^{k}-1}$ within the region $t-s\geq 0$; and

  2. (2)

The $(V^{C_{2^{k}}}+t-s,s)$-graded page of
$\textup{SliceSS}(BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle I\rangle)$ on or above
the horizontal line $s=C_{V,k}$ within the region $t-s\geq 0$.

More precisely, let $\underline{\mathcal{E}}_{C_{2^{n+1}}}$ and
$\underline{\mathcal{E}}_{C_{2^{n-k+1}}}$ represent the slice spectral
sequences of $BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle$ and
$BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle I\rangle$, respectively. Then the
following statements hold:

  1. (1)

In the specified regions of their respective $\mathcal{E}_{2}$-pages, there is
an isomorphism

$\underline{\mathcal{E}}_{C_{2^{n+1}},2}^{s^{\prime},V+t^{\prime}}\cong\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2^{k}}}\left(\underline{\mathcal{E}}_{C_{2^{n-k+1}},2}^{s,V^{C_{2^{k}}}+t}\right),$
(6.1)

where

$\displaystyle t^{\prime}$ $\displaystyle=\left(|V^{C_{2^{k}}}|\cdot
2^{k}-|V|\right)+2^{k}t,$ $\displaystyle s^{\prime}$
$\displaystyle=\left(|V^{C_{2^{k}}}|\cdot
2^{k}-|V|\right)+(2^{k}-1)(t-s)+2^{k}s.$

  2. (2)

In the specified region for $\underline{\mathcal{E}}_{C_{2^{n+1}}}$, all the
differentials have lengths of the form

$2^{k}r-(2^{k}-1),\,\,\,r\geq 2.$

  3. (3)

The isomorphism (6.1) induces a one-to-one correspondence between the
$d_{2^{k}r-(2^{k}-1)}$-differentials in
$\underline{\mathcal{E}}_{C_{2^{n+1}}}$ originating on or above the line
$\mathcal{L}^{V}_{2^{k}-1}$ and the $d_{r}$-differentials in
$\underline{\mathcal{E}}_{C_{2^{n-k+1}}}$ originating on or above the
horizontal line $s=C_{V,k}$.

In 6.2, for every $V\in RO(C_{2^{n+1}})$, we have shearing isomorphisms
between $RO(G)$-graded regions of spectral sequences. Certain choices of $V$
will be particularly useful for our applications. Before presenting the proof
of 6.2, we will state some immediate corollaries that will be useful for our
discussions in subsequent sections.

######  Corollary 6.3.

Suppose $I\subseteq\mathbb{N}$ and $V\in RO(C_{2^{n+1}}/C_{2^{k}})$ with
$|V|=0$. We can view $V$ as an element in $RO(C_{2^{n+1}})$ through pullback
along the map $C_{2^{n+1}}\to C_{2^{n+1}}/C_{2^{k}}$. There is a shearing
isomorphism ${d_{2^{k}r-(2^{k}-1)}\leftrightsquigarrow d_{r}}$ between the
following regions of spectral sequences:

  1. (1)

The $(V+t-s,s)$-graded page of
$\textup{SliceSS}(BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle)$ on or above the
line $\mathcal{L}_{2^{k}-1}$ (the line of slope $(2^{k}-1)$ through the origin
$(V,0)$) within the region $t-s\geq 0$; and

  2. (2)

The $(V+t-s,s)$-graded page of
$\textup{SliceSS}(BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle I\rangle)$ on or above
the line $s=0$ within the region $t-s\geq 0$.

###### Proof.

As an element in $RO(C_{2^{n+1}})$, $V$ is fixed by $C_{j}$ for all $0\leq
j\leq k$. Therefore $|V^{C_{j}}|=|V|=0$ for all $0\leq j\leq k$. This implies
that the constants $\tau_{V}(\mathcal{F}_{\leq 2^{k}})$ and $C_{V,k}$ are both
0. The claim now follows directly from 6.2. ∎

######  Corollary 6.4.

Suppose $I\subseteq\mathbb{N}$. On the integer-graded page, there is a
shearing isomorphism ${d_{2^{k}r-(2^{k}-1)}\leftrightsquigarrow d_{r}}$
between the following regions of spectral sequences:

  1. (1)

The slice spectral sequence of $BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle$ on
or above the line $\mathcal{L}_{2^{k}-1}$ (the line of slope $(2^{k}-1)$
through the origin); and

  2. (2)

The entire slice spectral sequence of $BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle
I\rangle$.

###### Proof.

By [MSZ23a, Theorem C], both slice spectral sequences are concentrated in the
region $t-s\geq 0$. The claim now follows directly from 6.3 by setting $V=0$.
∎

######  Example 6.5.

Consider the slice spectral sequence of $BP^{(\\!(C_{4})\\!)}\langle
1\rangle$, as shown in Figure 4. Letting $V=0$ and $I=\\{1\\}$ in 6.2 shows
that there is a shearing isomorphism $d_{2r-1}\leftrightsquigarrow d_{r}$
between the integer-graded page of
$\textup{SliceSS}(BP^{(\\!(C_{4})\\!)}\langle 1\rangle)$ on or above the line
$\mathcal{L}_{1}$ and the integer-graded page of
$\textup{SliceSS}(BP_{\mathbb{R}}\langle 1\rangle)$ (Figure 5) on or above the
line $s=0$. This is the first instance of the shearing isomorphism that was
observed by Hill–Hopkins–Ravenel that motivated the authors to embark on the
current project.

Figure 4. The slice spectral sequence of $BP^{(\\!(C_{4})\\!)}\langle
1\rangle$.

Figure 5. The slice spectral sequence of $BP_{\mathbb{R}}\langle 1\rangle$.

######  Corollary 6.6.

Suppose $I\subseteq\mathbb{N}$ and $V\in RO(C_{2^{n+1}}/C_{2^{k}})$ is of the
form $V=c_{0}(2\sigma)+c_{1}\lambda_{1}+\cdots+c_{n-k}\lambda_{n-k}$, where
$c_{i}\geq 0$ for all $0\leq i\leq n-k$. Then the orientation class $u_{V}$ is
a permanent cycle in the slice spectral sequence of
$BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle$ if and only if $u_{V}$ is a
permanent cycle in the slice spectral sequence of
$BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle I\rangle$.

###### Proof.

Since $|u_{V}|=|V|-V$, it is in position $(|V|-V,0)$ on the
$(|V|-V+t-s,s)$-graded page. The claim now follows from applying 6.3 for the
${\left(|V|-V+t-s,s\right)}$-graded page and 5.5 (2). ∎

As a consequence of 6.2 and its corollaries, we give an alternative proof of
the Slice Differentials Theorem by Hill, Hopkins, and Ravenel [HHR16, Theorem
9.9]. In our proof, we utilize the $C_{2}$-slice differentials from the slice
spectral sequence of $BP_{\mathbb{R}}$ as an initial input. By applying the
shearing isomorphism, we then establish a family of differentials in the slice
spectral sequence of $BP^{(\\!(C_{2^{n+1}})\\!)}$ for all $n\geq 1$.

######  Theorem 6.7 (Hill–Hopkins–Ravenel Slice Differentials Theorem).

For all $n\geq 0$, the following family of differentials exist in the slice
spectral sequence for $BP^{(\\!(C_{2^{n+1}})\\!)}$:

$d_{2^{n+1}(2^{i}-1)+1}(u_{2\sigma}^{2^{i-1}})=N_{C_{2}}^{C_{2^{n+1}}}(\bar{t}_{i}^{C_{2^{n+1}}})a_{\bar{\rho}_{2^{n+1}}}^{2^{i}-1}a_{\sigma}^{2^{i}},\,\,\,i\geq
1.$

###### Proof.

When $n=0$, it is a consequence of the computation in [HK01] that in the slice
spectral sequence for $BP_{\mathbb{R}}$, we have the differentials

$d_{2^{i+1}-1}(u_{2\sigma}^{2^{i-1}})=\bar{t}_{i}^{C_{2}}a_{\sigma}^{2^{i+1}-1},\,\,\,i\geq
1.$

Letting $I=\mathbb{N}$, $k=n$, and
$V=|u_{2\sigma}^{2^{i-1}}|=(2^{i}-2^{i}\sigma)\in RO(C_{2^{n+1}}/C_{2^{n}})$
in 6.3 shows that there is a shearing isomorphism
$d_{2^{n}r-(2^{n}-1)}\leftrightsquigarrow d_{r}$ between the following regions
of spectral sequences:

  1. (1)

The $(2^{i}-2^{i}\sigma+t-s,s)$-graded page of
$\textup{SliceSS}(BP^{(\\!(C_{2^{n+1}})\\!)})$ on or above the line
$\mathcal{L}_{2^{n}-1}$ (the line of slope $(2^{n}-1)$ through the origin
$(2^{i}-2^{i}\sigma,0)$) within the region $t-s\geq 0$; and

  2. (2)

The $(2^{i}-2^{i}\sigma+t-s,s)$-graded page of
$\textup{SliceSS}(BP_{\mathbb{R}})$ on or above the line $s=0$ within the
region $t-s\geq 0$.

Since the $C_{2}$-differential on $u_{2\sigma}^{2^{i-1}}$ is within this
isomorphism region, it corresponds to a $C_{2^{n+1}}$-differential in
$\textup{SliceSS}(BP^{(\\!(C_{2^{n+1}})\\!)})$ of length

$2^{n}\cdot(2^{i+1}-1)-(2^{n}-1)=2^{n+1}(2^{i}-1)-1.$

Using the correspondence formulas in 5.5, we have the following
correspondences:

  1. (1)

$a_{\sigma}^{2^{i+1}-1}\leftrightsquigarrow a_{\sigma}^{2^{i+1}-1}$;

  2. (2)

$u_{2\sigma}^{2^{i-1}}\leftrightsquigarrow u_{2\sigma}^{2^{i-1}}$;

  3. (3)

$\bar{t}_{i}^{C_{2}}\leftrightsquigarrow
N_{C_{2}}^{C_{2^{n+1}}}(\bar{t}_{i}^{C_{2^{n+1}}})\cdot\left(\frac{a_{\bar{\rho}_{2^{n+1}}}}{a_{\sigma}}\right)^{2^{i}-1}$.

Therefore, the shearing isomorphism produces the $C_{2^{n+1}}$-differentials

$\displaystyle d_{2^{n+1}(2^{i}-1)+1}(u_{2\sigma}^{2^{i-1}})$
$\displaystyle=N_{C_{2}}^{C_{2^{n+1}}}(\bar{t}_{i}^{C_{2^{n+1}}})\cdot\left(\frac{a_{\bar{\rho}_{2^{n+1}}}}{a_{\sigma}}\right)^{2^{i}-1}\cdot
a_{\sigma}^{2^{i+1}-1}$
$\displaystyle=N_{C_{2}}^{C_{2^{n+1}}}(\bar{t}_{i}^{C_{2^{n+1}}})a_{\bar{\rho}_{2^{n+1}}}^{2^{i}-1}a_{\sigma}^{2^{i}}$

for all $i\geq 1$. ∎

###### Proof of 6.2.

The following diagram outlines the steps of our proof:

${P_{\bullet}\left(BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle
I\rangle\right)}$${\widetilde{E}\mathcal{F}[C_{2}]\wedge
P_{\bullet}\left(BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle
I\rangle\right)}$${\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2^{k}}}\mathscr{D}^{k}\left(\widetilde{E}\mathcal{F}[C_{2}]\wedge
P_{\bullet}(BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle
I\rangle)\right)}$${P_{\bullet}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle\right)}$${\widetilde{E}\mathcal{F}[C_{2^{k+1}}]\wedge
P_{\bullet}(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle).}$$\scriptstyle{d_{r}\leftrightsquigarrow d_{2^{k}r-(2^{k}-1)}}$
shearing
isom.$\scriptstyle{\varphi_{1}}$$\scriptstyle{\text{Corollary}~{}\ref{cor:sliceRecoveryExample1}\,\,(s=0)}$
shearing isom.Theorem 3.7$\scriptstyle{d_{r}\leftrightsquigarrow
d_{2^{k}r-(2^{k}-1)}}$$\scriptstyle{\,\simeq}$Theorem
4.1$\scriptstyle{\varphi_{2^{k}}}$$\scriptstyle{\text{Theorem}~{}\ref{thm:SliceRecovery1}\,\,\left(\mathcal{L}^{V}_{2^{k}-1}\right)}$
(6.2)

By 2.5, the map $\varphi_{1}$ induces an isomorphism of spectral sequences on
or above the horizontal line $s=0$ on the $(V^{C_{2^{k}}}+t-s,s)$-graded page.
3.7 then establishes a shearing isomorphism $d_{r}\leftrightsquigarrow
d_{2^{k}r-(2^{k}-1)}$ between the spectral sequence
$\mathcal{E}_{P_{\bullet}}$ that is associated to the tower

$\widetilde{E}\mathcal{F}[C_{2}]\wedge
P_{\bullet}\left(BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle I\rangle\right)$

and the spectral sequence $\mathcal{E}_{Q_{\bullet}}$ that is associated to
the tower

$\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2^{k}}}\mathscr{D}^{k}\left(\widetilde{E}\mathcal{F}[C_{2}]\wedge
P_{\bullet}(BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle I\rangle)\right),$

which, by 4.1, is also the spectral sequence that is associated to the tower

$\widetilde{E}\mathcal{F}[C_{2^{k+1}}]\wedge
P_{\bullet}(BP^{(\\!(C_{2^{n+1}})\\!)}\langle I\rangle).$

More explicitly, there is an isomorphism

$\underline{\mathcal{E}}_{Q_{\bullet},2}^{s^{\prime},V+t^{\prime}}\cong\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2^{k}}}\left(\underline{\mathcal{E}}_{P_{\bullet},2}^{s,V^{C_{2^{k}}}+t}\right)\,\,\,\,\,(r\geq
2),$

where

$\displaystyle t^{\prime}$ $\displaystyle=\left(|V^{C_{2^{k}}}|\cdot
2^{k}-|V|\right)+2^{k}t,$ $\displaystyle s^{\prime}$
$\displaystyle=\left(|V^{C_{2^{k}}}|\cdot
2^{k}-|V|\right)+(2^{k}-1)(t-s)+2^{k}s.$

This shearing isomorphism induces a one-to-one correspondence between the
$d_{r}$-differentials in $\mathcal{E}_{P_{\bullet}}$ and the
$d_{2^{k}r-(2^{k}-1)}$-differentials in $\mathcal{E}_{Q_{\bullet}}$. Moreover,
under this shearing isomorphism, the horizontal line $s=C_{V,k}$ is sheared to
the line

$\displaystyle s^{\prime}$ $\displaystyle=\left(|V^{C_{2^{k}}}|\cdot
2^{k}-|V|\right)+(2^{k}-1)(t^{\prime}-s^{\prime})+2^{k}\cdot C_{V,k}$
$\displaystyle=(2^{k}-1)(t^{\prime}-s^{\prime})+\tau_{V}(\mathcal{F}_{\leq
2^{k}}),$

which is exactly $\mathcal{L}^{V}_{2^{k}-1}$.

By the Slice Recovery Theorem (2.4), the map $\varphi_{2^{k}}$ induces an
isomorphism of spectral sequences on or above the line
$\mathcal{L}^{V}_{2^{k}-1}$ in the region $t^{\prime}-s^{\prime}\geq 0$. This
finishes the proof of the theorem. ∎

######  Remark 6.8.

In the proof of 6.2, we observe that (6.1) induces an isomorphism for classes
that are above the lines $\mathcal{L}_{2^{k}-1}^{V}$ and $s=C_{V,k}$, and
there is some fringe behavior for classes that are on these two lines. More
specifically, for classes on the $\mathcal{E}_{2}$-pages that lie on the lines
$\mathcal{L}_{2^{k}-1}^{V}$ and $s=C_{V,k}$, (6.1) induces an isomorphism
between them if they map to nontrivial classes in the corresponding localized
slice spectral sequences under the maps

$\textup{SliceSS}(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle)\longrightarrow\widetilde{E}\mathcal{F}[C_{2^{k+1}}]\wedge\textup{SliceSS}(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
I\rangle)$

and

$\textup{SliceSS}(BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle
I\rangle)\longrightarrow\widetilde{E}\mathcal{F}[C_{2}]\wedge\textup{SliceSS}(BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle
I\rangle).$

According to [MSZ23a, Corollary 2.3], these maps induce surjections on the
$\mathcal{E}_{2}$-pages for classes lying on these two lines. Furthermore, by
2.4, the classes in the kernel of the aforementioned maps do not support
nontrivial differentials. Therefore, on the lines $\mathcal{L}_{2^{k}-1}^{V}$
and $s=C_{V,k}$, (6.1) induces an isomorphism between classes that could
potentially support nontrivial differentials.

##  7\. The transchromatic tower

In this section, we will use 6.2 to study transchromatic phenomena in the
slice spectral sequences of $BP^{(\\!(C_{2^{n+1}})\\!)}\langle m\rangle$ and
$BP^{(\\!(C_{2^{n+1}})\\!)}\langle m,m\rangle$. Recall from the introduction
that Beaudry, Hill, the second author, and Zeng [BHSZ21] used the
$C_{2^{n+1}}$-equivariant formal group law associated with
$BP^{(\\!(C_{2^{n+1}})\\!)}\langle m\rangle$ to construct explicit
$C_{2^{n+1}}$-models of $E_{2^{n}\cdot m}$. From their construction, the
equivariant orientation of [HS20] factors through the quotient
$BP^{(\\!(C_{2^{n+1}})\\!)}\langle m\rangle$:

${BP^{(\\!(C_{2^{n+1}})\\!)}}$${E_{h}}$${BP^{(\\!(C_{2^{n+1}})\\!)}\langle
m\rangle}$

In light of this construction, the spectrum $BP^{(\\!(C_{2^{n+1}})\\!)}\langle
m\rangle$ can be regarded as a connective model for the $C_{2^{n+1}}$-spectrum
$E_{h}$.

######  Construction 7.1 (Transchromatic tower).

In 2.1, the stratification tower (2.2) for $BP^{(\\!(C_{2^{n+1}})\\!)}\langle
m\rangle$ is

${\textup{SliceSS}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
m\rangle\right)}$${\widetilde{E}\mathcal{F}[C_{2}]\wedge\textup{SliceSS}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
m\rangle\right)}$${\widetilde{E}\mathcal{F}[C_{4}]\wedge\textup{SliceSS}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
m\rangle\right)}$${\vdots}$${\widetilde{E}\mathcal{F}[C_{2^{n}}]\wedge\textup{SliceSS}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
m\rangle\right)}$${\widetilde{E}\mathcal{F}[C_{2^{n+1}}]\wedge\textup{SliceSS}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
m\rangle\right).}$$\scriptstyle{\mathcal{L}_{0}^{V}}$$\scriptstyle{\mathcal{L}_{1}^{V}}$$\scriptstyle{\mathcal{L}_{2^{n-1}-1}^{V}}$$\scriptstyle{\mathcal{L}_{2^{n}-1}^{V}}$$\scriptstyle{\mathcal{L}_{1}^{V}}$$\scriptstyle{\mathcal{L}_{3}^{V}}$$\scriptstyle{\mathcal{L}_{2^{n-1}-1}^{V}}$$\scriptstyle{\mathcal{L}_{2^{n}-1}^{V}}$

For each localized slice spectral sequence

$\widetilde{E}\mathcal{F}[C_{2^{k+1}}]\wedge\textup{SliceSS}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
m\rangle\right),\,1\leq k\leq n$

in the tower above, the Transchromatic Isomorphism Theorem (6.2) uses the
shearing isomorphism

$\widetilde{E}\mathcal{F}[C_{2^{k+1}}]\wedge\textup{SliceSS}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
m\rangle\right)\leftrightsquigarrow\widetilde{E}\mathcal{F}[C_{2}]\wedge\textup{SliceSS}\left(BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle
m\rangle\right)$

from 4.1 to establish shearing isomorphisms between the corresponding slice
spectral sequences:

${\textup{SliceSS}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
m\rangle\right)}$${\textup{SliceSS}\left(BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle
m\rangle\right)}$$\scriptstyle{\mathcal{L}_{2^{k}-1}^{V}}$

Here, the label $\mathcal{L}_{2^{k}-1}^{V}$ indicates the shearing isomorphism
region in the slice spectral sequence of $BP^{(\\!(C_{2^{n+1}})\\!)}\langle
m\rangle$. The shearing isomorphisms $\mathcal{L}_{2^{k}-1}^{V}$ form the
following transchromatic tower:

${\textup{SliceSS}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
m\rangle\right)\,\,\,\left(E_{h}^{hC_{2^{n+1}}}\right)}$${\textup{SliceSS}\left(BP^{(\\!(C_{2^{n}})\\!)}\langle
m\rangle\right)\,\,\,\left(E_{h/2}^{hC_{2^{n}}}\right)}$${\textup{SliceSS}\left(BP^{(\\!(C_{2^{n-1}})\\!)}\langle
m\rangle\right)\,\,\,\left(E_{h/4}^{hC_{2^{n-1}}}\right)}$${\vdots}$${\textup{SliceSS}\left(BP^{(\\!(C_{4})\\!)}\langle
m\rangle\right)\,\,\,\left(E_{h/2^{n-1}}^{hC_{4}}\right)}$${\textup{SliceSS}\left(BP_{\mathbb{R}}\langle
m\rangle\right)\,\,\,\left(E_{h/2^{n}}^{hC_{2}}\right).}$$\scriptstyle{\mathcal{L}_{1}^{V}}$$\scriptstyle{\mathcal{L}_{3}^{V}}$$\scriptstyle{\mathcal{L}_{2^{n-1}-1}^{V}}$$\scriptstyle{\mathcal{L}_{2^{n}-1}^{V}}$
(7.1)

The transchromatic tower (7.1) provides an inductive method for computing the
slice spectral sequences of the higher height theories from the lower height
theories (see Figure 6). More specifically, when computing the slice spectral
sequence of $BP^{(\\!(C_{2^{n+1}})\\!)}\langle m\rangle$, which serves as the
$C_{2^{n+1}}$-equivariant connective model for $E_{h}$, the tower reveals that
the region on or above the line $\mathcal{L}_{2^{k}-1}^{V}$, where $1\leq
k\leq n$, is completely determined by the slice spectral sequence of
$BP^{(\\!(C_{2^{n-k+1}})\\!)}\langle m\rangle$, the
$C_{2^{n-k+1}}$-equivariant connective model for $E_{h/2^{k}}$.

Figure 6. Regions of the slice spectral sequence of $E_{h}$ corresponding to
the transchromatic tower.

######  Example 7.2.

For all $m\geq 1$, $BP_{\mathbb{R}}\langle m\rangle$ is the
$C_{2}$-equivariant connective model for $E_{m}$, and
$BP^{(\\!(C_{2^{n+1}})\\!)}\langle m\rangle$ is the $C_{2^{n+1}}$-connective
model for $E_{2^{n}\cdot m}$. The computation of the $C_{2}$-slice spectral
sequence of $BP_{\mathbb{R}}\langle m\rangle$ by Hu and Kriz [HK01] completely
determines the $RO(G)$-graded page of the slice spectral sequence of
$BP^{(\\!(C_{2^{n+1}})\\!)}\langle m\rangle$ on or above the line
$\mathcal{L}^{V}_{2^{n}-1}$.

######  Example 7.3.

For $n\geq 0$, the $C_{2^{n+1}}$-spectrum $BP^{(\\!(C_{2^{n+1}})\\!)}\langle
1\rangle$ serves as the $C_{2^{n+1}}$-equivariant connective model for
$E_{2^{n}}$. The computation of the slice spectral sequence of
$BP^{(\\!(C_{4})\\!)}\langle 1\rangle$ ($E_{2}^{hC_{4}}$) by Hill, Hopkins,
and Ravenel [HHR17] completely determines the integer-graded spectral sequence
of $BP^{(\\!(C_{2^{n+1}})\\!)}\langle 1\rangle$ ($E_{2^{n}}^{hC_{2^{n+1}}}$)
on or above the line of slope $(2^{n-1}-1)$ through the origin.

In particular, when $n=2$, in the slice spectral sequence of
$BP^{(\\!(C_{8})\\!)}\langle 1\rangle$ ($E_{4}^{hC_{8}}$), which detects all
the Kervaire invariant elements, the differentials on or above the line of
slope 1 are completely determined by the differentials in the slice spectral
sequence of $BP^{(\\!(C_{4})\\!)}\langle 1\rangle$.

######  Example 7.4.

For $n\geq 0$, the $C_{2^{n+1}}$-spectrum $BP^{(\\!(C_{2^{n+1}})\\!)}\langle
2\rangle$ serves as the $C_{2^{n+1}}$-equivariant connective model for
$E_{2^{n+1}}$. The computation of the slice spectral sequence of
$BP^{(\\!(C_{4})\\!)}\langle 2\rangle$ ($E_{4}^{hC_{4}}$) by Hill, the second
author, Wang, and Xu [HSWX23] completely determines the integer-graded
spectral sequence of $BP^{(\\!(C_{2^{n+1}})\\!)}\langle 2\rangle$
($E_{2^{n+1}}^{hC_{2^{n+1}}}$) on or above the line of slope $(2^{n-1}-1)$
through the origin.

######  Remark 7.5.

In [BHL+22], Beaudry, Hill, Lawson, the second author, and the third author
constructed and studied the $C_{2^{n+1}}$-equivariant theories
$BP^{(\\!(C_{2^{n+1}})\\!)}\langle m,m\rangle$. These theories are the
equivariant analogues of the integral connective Morava $K$-theories. One
notable feature of these theories is that while the non-equivariant connective
Morava $K$-theory $K(m)=BP\langle m,m\rangle$ has chromatic heights
concentrated at levels $0$ and $m$, the theories
$BP^{(\\!(C_{2^{n+1}})\\!)}\langle m,m\rangle$ have chromatic heights
concentrated at levels $0$, $m$, $2m$, $\ldots$, $2^{n}\cdot m$ [BHL+22,
Theorem 3.1].

6.2 establishes a transchromatic tower for the theories
$BP^{(\\!(C_{2^{n+1}})\\!)}\langle m,m\rangle$. In particular, everything on
or above the line $\mathcal{L}_{1}^{V}$ in the slice spectral sequence for
$BP^{(\\!(C_{2^{n+1}})\\!)}\langle m,m\rangle$, which captures chromatic
information at heights $0$, $m$, $\ldots$, $2^{n}\cdot m$, is completely
determined by the slice spectral sequence for $BP^{(\\!(C_{2^{n}})\\!)}\langle
m,m\rangle$, which captures chromatic information at heights $0$, $m$,
$\ldots$, $2^{n-1}\cdot m$.

##  8\. $RO(G)$-graded periodicities

In this section, we will apply the Transchromatic Isomorphism Theorem (6.2) to
establish $RO(G)$-graded periodicities of $E_{h}$ from $RO(G/{C_{2}})$-graded
periodicities of $E_{h/2}$.

For $G=C_{2^{n}}$, Beaudry–Hill–Shi–Zeng [BHSZ21] used the universal
$C_{2^{n}}$-equivariant formal group law of $BP^{(\\!(C_{2^{n}})\\!)}$ to
construct $C_{2^{n}}$-equivariant models of the height-$h$ Lubin–Tate theory
$E_{h}$, where $h=2^{n-1}\cdot m$. For our discussion in this section, we will
focus on the following two Lubin–Tate theories:

  1. (1)

The $C_{2^{n}}$-equivariant theory $E_{2^{n-1}\cdot m}$; and

  2. (2)

The $C_{2^{n+1}}$-equivariant theory $E_{2^{n}\cdot m}$.

By work of Hahn and the second author [HS20], these models are equipped with
equivariant orientations from $BP^{(\\!(C_{2^{n}})\\!)}$ and
$BP^{(\\!(C_{2^{n+1}})\\!)}$, respectively. It is a consequence of the
construction in [BHSZ21] that these equivariant orientations factor through
certain localizations and quotients of $BP^{(\\!(C_{2^{n}})\\!)}$ and
$BP^{(\\!(C_{2^{n+1}})\\!)}$, which we will now discuss.

For $1\leq k\leq n$, the $\bar{t}_{i}^{C_{2^{k}}}$-generators are originally
defined to be elements in
$\pi_{*\rho_{2}}^{C_{2}}i_{C_{2}}^{*}BP^{(\\!(C_{2^{k}})\\!)}$. The unit map

$BP^{(\\!(C_{2^{k}})\\!)}\longrightarrow
i_{C_{2^{k}}}^{*}N_{C_{2^{k}}}^{C_{2^{n}}}(BP^{(\\!(C_{2^{k}})\\!)})=i_{C_{2^{k}}}^{*}BP^{(\\!(C_{2^{n}})\\!)}$

of the norm-restriction adjunction enables us to treat the
$\bar{t}_{i}^{C_{2^{k}}}$-generators as elements in
$\pi_{*\rho_{2}}^{C_{2}}i_{C_{2}}^{*}BP^{(\\!(C_{2^{n}})\\!)}$ through the
composition

$\bar{t}_{i}^{C_{2^{k}}}:S^{(2^{i}-1)\rho_{2}}\longrightarrow
i_{C_{2}}^{*}BP^{(\\!(C_{2^{k}})\\!)}\longrightarrow
i_{C_{2}}^{*}BP^{(\\!(C_{2^{n}})\\!)}.$

Similarly, we can treat the $\bar{t}_{i}^{C_{2^{k+1}}}$-generators as elements
in $\pi_{*\rho_{2}}^{C_{2}}i_{C_{2}}^{*}BP^{(\\!(C_{2^{n+1}})\\!)}$ through
the composition

$\bar{t}_{i}^{C_{2^{k+1}}}:S^{(2^{i}-1)\rho_{2}}\longrightarrow
i_{C_{2}}^{*}BP^{(\\!(C_{2^{k+1}})\\!)}\longrightarrow
i_{C_{2}}^{*}BP^{(\\!(C_{2^{n+1}})\\!)}.$

We note that under this more general treatment, the equality

$\Phi^{C_{2}}(\bar{t}_{i}^{C_{2^{k}}})=\Phi^{C_{4}}N_{C_{2}}^{C_{4}}(\bar{t}_{i}^{C_{2^{k+1}}})$

proven in 4.2 and the correspondence formulas proven in 5.3 and 5.5 still
hold.

######  Definition 8.1.

Define
$D_{C_{2^{n}},m}\in\pi_{*\rho_{2^{n}}}^{C_{2^{n}}}BP^{(\\!(C_{2^{n}})\\!)}$ to
be the element

$\displaystyle
D_{C_{2^{n}},m}=\prod_{k=1}^{n}N_{C_{2}}^{C_{2^{n}}}(\bar{t}^{C_{2^{k}}}_{2^{n-k}\cdot
m})=N_{C_{2}}^{C_{2^{n}}}(\bar{t}^{C_{2}}_{2^{n-1}\cdot m})\cdots
N_{C_{2}}^{C_{2^{n}}}(\bar{t}^{C_{2^{n}}}_{m}).$

Furthermore, define
$\overline{D}_{C_{2^{n}},m}\in\pi_{*\rho_{2^{n}}}^{C_{2^{n}}}BP^{(\\!(C_{2^{n}})\\!)}$
to be the element

$\overline{D}_{C_{2^{n}},m}=\prod_{k=2}^{n}N_{C_{2}}^{C_{2^{n}}}(\bar{t}^{C_{2^{k}}}_{2^{n-k}\cdot
m})=N_{C_{2}}^{C_{2^{n}}}(\bar{t}^{C_{4}}_{2^{n-2}\cdot m})\cdots
N_{C_{2}}^{C_{2^{n}}}(\bar{t}^{C_{2^{n}}}_{m}).$

Note that with our definition,
$D_{C_{2^{n}},m}=N_{C_{2}}^{C_{2^{n}}}(\bar{t}^{C_{2}}_{2^{n-1}\cdot
m})\cdot\overline{D}_{C_{2^{n}},m}$.

It is a consequence from the constructions of $E_{2^{n-1}\cdot m}$ and
$E_{2^{n}\cdot m}$ in [BHSZ21] that the equivariant orientations
$BP^{(\\!(C_{2^{n}})\\!)}\to E_{2^{n-1}\cdot m}$ and
$BP^{(\\!(C_{2^{n+1}})\\!)}\to E_{2^{n}\cdot m}$ factor through the
localizations $D_{C_{2^{n}},m}^{-1}BP^{(\\!(C_{2^{n}})\\!)}$ and
$D_{C_{2^{n+1}},m}^{-1}BP^{(\\!(C_{2^{n+1}})\\!)}$, and further through the
quotients $D_{C_{2^{n}},m}^{-1}BP^{(\\!(C_{2^{n}})\\!)}\langle m\rangle$ and
$D_{C_{2^{n+1}},m}^{-1}BP^{(\\!(C_{2^{n+1}})\\!)}\langle m\rangle$. There are
diagrams

${D_{C_{2^{n}},m}^{-1}BP^{(\\!(C_{2^{n}})\\!)}}$${E_{2^{n-1}\cdot
m}}$${D_{C_{2^{n}},m}^{-1}BP^{(\\!(C_{2^{n}})\\!)}\langle m\rangle}$

and

${D_{C_{2^{n+1}},m}^{-1}BP^{(\\!(C_{2^{n+1}})\\!)}}$${E_{2^{n}\cdot
m}}$${D_{C_{2^{n+1}},m}^{-1}BP^{(\\!(C_{2^{n+1}})\\!)}\langle m\rangle}$

######  Lemma 8.2.

We have the following equivalence of towers:

$\widetilde{E}\mathcal{F}[C_{4}]\wedge
P_{\bullet}\left(\overline{D}_{C_{2^{n+1}},m}^{-1}BP^{(\\!(C_{2^{n+1}})\\!)}\langle
m\rangle\right)\\\
\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}\mathscr{D}\left(\widetilde{E}\mathcal{F}[C_{2}]\wedge
P_{\bullet}(D_{C_{2^{n}},m}^{-1}BP^{(\\!(C_{2^{n}})\\!)}\langle
m\rangle)\right).$

###### Proof.

By setting $I=\\{1,\ldots,m\\}$ and $k=1$ in 4.1, we obtain the equivalence

$\widetilde{E}\mathcal{F}[C_{4}]\wedge
P_{\bullet}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle
m\rangle\right)\simeq\mathcal{P}^{*}_{C_{2^{n+1}}/C_{2}}\mathscr{D}\left(\widetilde{E}\mathcal{F}[C_{2}]\wedge
P_{\bullet}(BP^{(\\!(C_{2^{n}})\\!)}\langle m\rangle)\right).$ (8.1)

Under this equivalence, 5.3 establishes the correspondence

$N_{C_{2}}^{C_{2^{k+1}}}(\bar{t}^{C_{2^{k+1}}}_{2^{n-k}\cdot
m})a_{\lambda_{k}}^{2^{k-1}(2^{2^{n-k}\cdot m}-1)}\leftrightsquigarrow
N_{C_{2}}^{C_{2^{k}}}(\bar{t}^{C_{2^{k}}}_{2^{n-k}\cdot m})$

for all $1\leq k\leq n$. Applying the norm, these correspondence formulas
become

$N_{C_{2}}^{C_{2^{n+1}}}(\bar{t}^{C_{2^{k+1}}}_{2^{n-k}\cdot
m})a_{\lambda_{n}}^{2^{n-1}(2^{2^{n-k}\cdot m}-1)}\leftrightsquigarrow
N_{C_{2}}^{C_{2^{n}}}(\bar{t}^{C_{2^{k}}}_{2^{n-k}\cdot m})$

for all $1\leq k\leq n$. Multiplying these formulas together yields the
correspondence

$\overline{D}_{C_{2^{n+1}},m}\cdot a_{\lambda_{n}}^{\ell}\leftrightsquigarrow
D_{C_{2^{n}},m},$

where

$\ell=\sum_{1\leq k\leq n}2^{n-1}(2^{2^{n-k}\cdot m}-1).$

Since smashing with $\widetilde{E}\mathcal{F}[C_{4}]$ inverts the class
$a_{\lambda_{n-1}}$, the class $a_{\lambda_{n}}$ is also inverted. Therefore
the term $a_{\lambda_{n}}^{\ell}$ is a unit in
${\widetilde{E}\mathcal{F}[C_{4}]\wedge
P_{\bullet}\left(BP^{(\\!(C_{2^{n+1}})\\!)}\langle m\rangle\right)}$. The
desired equivalence now follows from Equivalence (8.1) and the correspondence
formula above. ∎

For the remainder of this section, we will use the notation $G=C_{2^{n+1}}$
and $h=2^{n}\cdot m$. Suppose $V$ is an element in $RO(G)$ such that the
orientation class $u_{V}$ in the slice spectral sequence of
$D_{G,m}^{-1}BP^{(\\!(G)\\!)}\langle m\rangle$ is a permanent cycle. Then
under the map of spectral sequences

$\textup{SliceSS}(D_{G,m}^{-1}BP^{(\\!(G)\\!)}\langle
m\rangle)\longrightarrow\textup{SliceSS}(E_{h}),$

its image $u_{V}\in\textup{SliceSS}(E_{h})$ is also a permanent cycle. Let
$u\in\pi_{|V|-V}^{G}E_{h}$ denote any element in homotopy that is represented
by $u_{V}$. The composite

$S^{|V|-V}\wedge E_{h}\xrightarrow{u\wedge\textup{id}}E_{h}\wedge
E_{h}\xrightarrow{m}E_{h}$

is a $G$-equivariant map between cofree (Borel) spectra. Since the underlying
non-equivariant map induces an equivalence $i_{e}^{*}E_{h}\simeq
i_{e}^{*}E_{h}$, the composition map above induces a $G$-equivariant
equivalence. This produces a $(|V|-V)$-periodicity for $E_{h}$.

######  Theorem 8.3.

Suppose $V\in RO(G/C_{2})$. The class $u_{V}$ is a permanent cycle in the
slice spectral sequence of $D_{G,m}^{-1}BP^{(\\!(G)\\!)}\langle m\rangle$ (and
hence in the $G$-slice spectral sequence of $E_{h}$) if and only if the class
$u_{V}$ is a permanent cycle in the slice spectral sequence of
$D_{G/C_{2},m}^{-1}BP^{(\\!(G/C_{2})\\!)}\langle m\rangle$ (and therefore in
the $G/C_{2}$-slice spectral sequence of $E_{h/2}$). Here, $V$ is treated as
an element in $RO(G)$ through pullback along the map $G\to G/C_{2}$.

######  Remark 8.4.

If the class $u_{V}$ is a permanent cycle in the slice spectral sequence of
$D_{G,m}^{-1}BP^{(\\!(G)\\!)}$ or $BP^{(\\!(G)\\!)}\langle m\rangle$, then
$u_{V}$ will also be a permanent cycle in the slice spectral sequence of
$D_{G,m}^{-1}BP^{(\\!(G)\\!)}\langle m\rangle$ by naturality. Therefore, the
statement of 8.3 remains valid if we replace
$D_{G,m}^{-1}BP^{(\\!(G)\\!)}\langle m\rangle$ by
$D_{G,m}^{-1}BP^{(\\!(G)\\!)}$ or $BP^{(\\!(G)\\!)}\langle m\rangle$.

######  Remark 8.5.

In light of 8.4, we can apply 8.3 iteratively to establish that if $V\in
RO(G/C_{2^{k}})$, where $1\leq k\leq n$, and $u_{V}$ is a permanent cycle in
the slice spectral sequence of $BP^{(\\!(G/C_{2^{k}})\\!)}\langle m\rangle$
(and therefore in the $G/C_{2^{k}}$-slice spectral sequence of $E_{h/2^{k}}$),
then $u_{V}$ remains a permanent cycle in the slice spectral sequence of
$BP^{(\\!(G)\\!)}\langle m\rangle$ (and hence in the $G$-slice spectral
sequence of $E_{h}$).

###### Proof of 8.3.

The equivalence of localized dual slice towers in 8.2 establishes a shearing
isomorphism between the corresponding localized slice spectral sequences.
Consequently, this shearing isomorphism extends to the regions of the
respective slice spectral sequences:

${\textup{SliceSS}\left(\overline{D}_{G,m}^{-1}BP^{(\\!(G)\\!)}\langle
m\rangle\right)}$${\textup{SliceSS}\left(D_{G/C_{2},m}^{-1}BP^{(\\!(G/C_{2})\\!)}\langle
m\rangle\right)}$${\widetilde{E}\mathcal{F}[C_{4}]\wedge\textup{SliceSS}\left(\overline{D}_{G,m}^{-1}BP^{(\\!(G)\\!)}\langle
m\rangle\right)}$${\widetilde{E}\mathcal{F}[C_{2}]\wedge\textup{SliceSS}\left(D_{G/C_{2},m}^{-1}BP^{(\\!(G/C_{2})\\!)}\langle
m\rangle\right).}$shearingshearing

The proof of 6.6 applies to this context, showing that $u_{V}$ is a permanent
cycle in the slice spectral sequence of
$D_{G/C_{2},m}^{-1}BP^{(\\!(G/C_{2})\\!)}\langle m\rangle$ if and only if
$u_{V}$ is a permanent cycle in the slice spectral sequence of
$D_{G,m}^{-1}BP^{(\\!(G)\\!)}\langle m\rangle$. ∎

######  Remark 8.6.

Recall that the $RO(G)$-graded homotopy groups are graded by the
representations $\\{1,\sigma,\lambda_{1},\ldots,\lambda_{n}\\}$, while
$RO(G/C_{2})$-graded homotopy groups are graded by the representations
$\\{1,\sigma,\lambda_{1},\ldots,\lambda_{n-1}\\}$. By considering the
$RO(G/C_{2})$-graded periodicities of the $G/C_{2}$-spectrum $E_{h/2}$ and
applying 8.3, we can immediately establish all the $RO(G)$-graded
periodicities of the $G$-spectrum $E_{h}$ that are of the form $|V|-V$, where
$V$ is generated by the representations
$\\{1,\sigma,\lambda_{1},\ldots,\lambda_{n-1}\\}$.

######  Example 8.7.

By explicit computation of Hu and Kriz [HK01], it is shown that the class
$u_{2^{m+1}\sigma}$ is a permanent cycle in the slice spectral sequence of
$BP_{\mathbb{R}}\langle m\rangle$. This establishes a
${(2^{m+1}-2^{m+1}\sigma)}$-graded periodicity for $E_{m}$ when considered as
a $C_{2}$-spectrum. Applying 8.3, the class $u_{2^{m+1}\sigma}$ is also a
permanent cycle in the slice spectral sequence of
$BP^{(\\!(C_{2^{n+1}})\\!)}\langle m\rangle$ for all $n\geq 0$. As a
consequence, a ${(2^{m+1}-2^{m+1}\sigma)}$-graded periodicity is established
for the $C_{2^{n+1}}$-spectrum $E_{2^{n}\cdot m}$ for all $n\geq 0$.

######  Example 8.8.

The computation by Hill, Hopkins, and Ravenel [HHR17] shows that the classes
$u_{4\sigma}$, $u_{8\lambda_{1}}$, and $u_{4\lambda_{1}2\sigma}$ are permanent
cycles in the slice spectral sequence of $BP^{(\\!(C_{4})\\!)}\langle
1\rangle$. These permanent cycles establish $RO(C_{4})$-graded periodicities