# Dynamic control of 2D non-Hermitian photonic corner states in synthetic
dimensions

Xinyuan Zheng<EMAIL_ADDRESS>Equally contributing authors
Institute for Research in Electronics and Applied Physics, University of
Maryland, College Park, Maryland 20742, USA  Mahmoud Jalali Mehrabad
<EMAIL_ADDRESS>Equally contributing authors  Joint Quantum Institute,
University of Maryland, College Park, MD 20742, USA  Jonathan Vannucci  Joint
Quantum Institute, University of Maryland, College Park, MD 20742, USA  Kevin
Li  Joint Quantum Institute, University of Maryland, College Park, MD 20742,
USA  Avik Dutt  Department of Mechanical Engineering, and Institute for
Physical Science and Technology, University of Maryland, College Park,
Maryland 20742, USA  Mohammad Hafezi  Joint Quantum Institute, University of
Maryland, College Park, MD 20742, USA  Sunil Mittal<EMAIL_ADDRESS>Department of Electrical and Computer Engineering, Northeastern University,
Boston, MA, USA  Edo Waks<EMAIL_ADDRESS>Institute for Research in
Electronics and Applied Physics, University of Maryland, College Park,
Maryland 20742, USA

###### Abstract

Non-Hermitian models describe the physics of ubiquitous open systems with gain
and loss. One intriguing aspect of non-Hermitian models is their inherent
topology that can produce intriguing boundary phenomena like resilient higher-
order topological insulators (HOTIs) and non-Hermitian skin effects (NHSE).
Recently, time-multiplexed lattices in synthetic dimensions have emerged as a
versatile platform for the investigation of these effects free of geometric
restrictions. Despite holding broad applications, studies of these effects
have been limited to static cases so far, and full dynamical control over the
non-Hermitian effects has remained elusive. Here, we demonstrate the emergence
of topological non-Hermitian corner states with remarkable temporal
controllability and robustness in a two-dimensional photonic synthetic time
lattice. Specifically, we showcase various dynamic control mechanisms for
light confinement and flow, including spatial mode tapering, sequential non-
Hermiticity on-off switching, dynamical corner state relocation, and light
steering. Moreover, we establish the corner state’s robustness in the presence
of intensity modulation randomness and quantitatively determine its breakdown
regime. Our findings extend non-Hermitian and topological photonic effects
into higher synthetic dimensions, offering remarkable flexibility and real-
time control possibilities. This opens avenues for topological classification,
quantum walk simulations of many-body dynamics, and robust Floquet
engineering, free from the limitations of physical geometries.

##  I Introduction

Non-Hermitian systems host a range of intriguing phenomena in physics, such as
reconfigurable light routing [1], potential for enhanced sensitivity [2, 3]
and unidirectional invisibility [4], that are deeply rooted in symmetry and
topology. One such phenomenon is the non-Hermitian skin effect (NHSE) where a
macroscopic fraction of the eigenmodes of a finite system become exponentially
localized at its boundary [5, 6]. This localization is a direct consequence of
the nontrivial (topological) winding of the system’s eigenvalues in the
complex energy plane [7, 8, 9]. Disorder and small variations in the system do
not change the winding number which is a topological invariant [9].

Over the last few years, the NHSE has been demonstrated on a variety of
platforms [5, 10, 11]. Exemplary platforms include acoustics and phononics
[12], topo-electric circuits [13], and photonics [14]. These developments are
in part motivated by the profound impact of NHSE on band topology [7, 15, 16],
spectral symmetry [17], and dynamics [18, 19]. Particularly in photonics,
recently the NHSE has enabled intriguing demonstrations of the tuneable
directional flow of light [20], near-field beam steering [21], engineering
arbitrary band topology [22] and topological funneling of light [14].
Nevertheless, these demonstrations have been limited to systems that can be
effectively described by time-independent Hamiltonians [23]. The introduction
of time-dependent non-Hermitian Hamiltonians can lead to a dynamic control
over the skin effect and also lead to fundamental advances in novel non-
Hermitian topological phases that are not accessible using time-independent
systems.

Here we demonstrate dynamical control of the two-dimensional non-Hermitian
photonic skin effect, that is, corner states, using purely synthetic temporal
dimensions. Specifically, using time-multiplexed light pulses in fiber loops,
we show manipulation of the gain/loss in the system at a scale that is faster
than the dynamics of light pulses in the lattice. Using this dynamical
manipulation, we demonstrate adiabatic control over the degree of localization
of the corner states, adiabatic tweezing of light where we move the corner
states along a predefined trajectory in the lattice, and 2D funneling of light
where photons always funnel to the corner state irrespective of their initial
position in the 2D lattice. Finally, by introducing controlled disorder in the
system in the form of random variations in gain and loss, we quantitatively
investigate the robustness of the corner states against such disorders. Our
work opens up an avenue to explore the rich physics of time-dependent non-
Hermitian models such as non-Hermitian Floquet systems.

##  II 2D quantum walk with non-Hermitian topology

Figure 1: Two-dimensional quantum walk in non-Hermitian synthetic lattice. (a)
Example of photonic quantum walk in a 2D synthetic lattice. The blue and red
curved arrows show the direction-dependent loss and gain. (b) Winding of
effective eigenenergies $\epsilon_{up/down}(k_{x},k_{y})$ in the complex
energy plane for a single bulk non-Hermitian lattice with periodic boundary
condition, showing line-gapped topology (indicated by the green line). Here we
choose five different values $k_{y}\equiv 0,\pm\pi/4,\pm 3\pi/8$. (c) Four
bulk lattices with different gain-loss patterns are glued along their edges to
form a corner. Note that $\delta_{x}>0$ implies gain for a step towards $-X$
and loss for a step toward $+X$. For $\delta_{x}<0$ the gain-loss is inverted.
A similar rule applies for $\delta_{y}$. (d) Averaged spatial profile of
corner states formed in the system shown in (c), by taking non-Hermitian
parameters $\delta_{x}=\delta_{y}=0.175$. The lattice size is 30$\times$30\.
(e) The time-multiplexed experimental scheme, with which the lattice
parameters can be (dynamically) controlled by the intensity modulators.

Our system simulates a discrete-time quantum walk of photons on a two-
dimensional non-Hermitian square lattice, as illustrated schematically in
Figure 1(a). Specifically, we implement a split-step walk where the walker
first randomly steps to either left or right (corresponding to the $X$
direction) with equal probability, then up or down (corresponding to the $Y$
direction). To introduce non-Hermiticity, we introduce an additional gain
$e^{+\delta_{x}}$ when the walker steps to the left, and an additional loss
$e^{-\delta_{x}}$ when the walker steps to the right. Similarly, the walker
experiences a gain $e^{+\delta_{y}}$ when moving down and a loss of
$e^{-\delta_{y}}$ when moving up. The parameters $\delta_{x}$ and $\delta_{y}$
then indicate the degree of non-Hermiticity of the walk.

For this quantum walk, a concept of complex energy can analogously be defined,
by solving for the eigenmodes of the non-unitary quantum walk evolution
operator $\hat{U}$ and taking the logarithm of the corresponding eigenvalue
$u_{j}$. Namely, this can be formulated as $\epsilon_{j}=ilog(u_{j})$, where
$\hat{U}\ket{u_{j}}=u_{j}\ket{u_{j}}$. If we further impose periodic boundary
conditions (PBCs) in both $x$ and $y$ for the bulk in Figure 1(a), we can
apply the Bloch theorem for the walk and obtain the complex energy bands
$\epsilon_{up/down}(k_{x},k_{y})$. (The two bands seen in Figure 1(b) arise
due to the up/down channel configuration of our experiment, see supplementary
information (SI) for derivation details). The non-unitary time evolution of
the walk leads to a non-trivial winding of $\epsilon(k_{x},k_{y})$ for each
band in the complex energy plane as one continuously varies Bloch vector
$(k_{x},k_{y})$ along a certain curve in the Brillouin zone. To illustrate
this, in Figure 1(b), we plot the complex energies
$\epsilon_{up/down}(k_{x},k_{y})$ of the bulk lattice shown in Figure 1(a) as
we vary $k_{x}$ from $-\pi$ to $\pi$ while keeping $k_{y}$ fixed to different
values $0,\pm\pi/4,\pm 3\pi/8$. As $(k_{x},k_{y})$ varies along each of these
directed horizontal curves in the Brillouin zone, both
$\epsilon_{up}(k_{x},k_{y})$ and $\epsilon_{down}(k_{x},k_{y})$ winds one loop
in the counterclockwise direction, thus exhibiting an integer-valued winding
number $-1$. This is a topological invariant for our non-Hermitian quantum
walk. Also, the two winding loops contributed from the two bands
$\epsilon_{up/down}$ exhibit a line-gapped topology [24], such that the two
winding loops never cross the line $Re(\epsilon)=0$ in the complex plane.
Windings of complex energy along other curves in the Brillouin zone are shown
in the SI.

In a finite system, the non-trivial winding of the complex energies and the
associated 2D non-Hermitian skin effect[24] is manifested as corner states,
that is, localization of the walker can happen at an interface between regions
with opposite windings (or bulk band topologies). Figure 1(c) shows one
exemplary case which consists of four distinct regions, represented by the
four different color patches. The gray patch is identical to the system
described in Figure 1(a). The other three regions exhibit an inverted gain-
loss relation (indicated by a change in the sign of the gain parameter) either
along the $x$ or $y$-axis, or both. This inversion of gain-loss leads to
different windings for each region. Non-Hermitian skin effect occurs in such a
system, and we numerically verify in Figure 1(d) that the averaged eigenmodes
of the quantum walk exhibit clustering at the junction between the four
regions - as indicated by the red dot in Figure 1(c).

To simulate the quantum walk described above, we use classical light pulses in
a time-multiplexed setup shown in Figure 1(e). We note that for this linear
system, the evolution of classical light pulses in the lattice exactly follows
that of the quantum walk of single photons in the lattice. We map the state
space of the 2D square lattice into different time-delays in two fiber
feedback loops, as introduced in previous works [25, 26]. To introduce non-
Hermiticity, we use four intensity modulators that introduce individually
controllable loss when the walker moves along any direction. We also use two
erbium-doped fiber amplifiers (EDFAs) that provide gain in the system, and
together with the intensity modulators, introduce a gain-loss mechanism that
can be controlled at each step of the walker. We specifically choose electro-
optic modulators with a high bandwidth to allow reconfigurability of the
system’s topology at each step of the quantum walk. A full discussion of the
experimental setup is provided in the SI.

Figure 2: Light localization and light funneling for static control. (a) Light
localization at the corner state located at $(x,y)=(0,0)$ for non-Hermitian
parameter $|\delta_{x}|=|\delta_{y}|=0.175$. Here a single pulse is
initialized at $(x,y)=(0,0)$ in the up channel. (b) Light funneling for the
same lattice parameter and pulse initialization, but the corner state is
located at $(x,y)=(-10,10)$. Here the skin effect allows light to flow to the
corner state and localize there. In both (a) and (b), from left to right the
snapshots are shown for time steps $1,9,21$, respectively.

##  III Results

##  IV Skin effect under static control

To show the presence of non-Hermitian corner states, we first construct the
model as shown in Figure 1(c), with the corner located at the lattice origin
$(x=0,y=0)$. We inject a single light pulse into the time bin corresponding to
the lattice origin and choose non-Hermitian parameters $|\delta_{x}|$ and
$|\delta_{y}|$ to be 0.175 as in Figure 1(c). In Figure 2(a), we plot the
snapshots of the light distribution in the lattice for different time steps
$1$, $9$, and $21$, which are obtained by measuring the pulse power at each
time bin. The evolution of distribution shows that the walker stays localized
at the origin, confirming the presence of a corner state. In sharp contrast,
when we set $\delta_{x}$ and $\delta_{y}$ to 0, we observe a significant
spreading of the intensity distribution, indicating the absence of any corner
states (see the SI for the experimental data).

Having shown the localization of light at the corner state, we next
demonstrate the skin-effect-induced funneling of light. Namely, the system
dynamics bring any initial state towards the corner states. We set the corner
state to be at the lattice site $(x=-10,y=10)$ while light pulses are still
injected at $x=0,y=0$, which is now in the bulk of the lattice (Figure 2(b)).
As the system evolves, initially light spreads in bulk, but finally converges
to the corner site. As shown in the SI for several different lattice
configurations, light pulses always converge to the corner regardless of the
initialization location. This funneling of light to the corner state is a
manifestation of the skin effect where all the eigenmodes of the system are
localized at the corner. Our experimental results are in good agreement with
our theoretical prediction shown in Figure 2(b).

Figure 3: Dynamical control of the corner state. (a) Dynamical control of
corner state spatial profile. As the non-Hermitian parameter is adiabatically
reduced from $|\delta_{x,max}|=|\delta_{y,max}|=0.175$ to
$|\delta_{x,max}|=|\delta_{y,max}|=0.02$ and back to
$|\delta_{x,max}|=|\delta_{y,max}|=0.175$, the corner state becomes
delocalized and then localized. From left to right the snapshots are shown for
time steps $1,9,17,25,37$, respectively. (b) Dynamically tweezing localized
light along a designed “L”-shaped trajectory using the skin effect. Localized
light is first moved in the $+Y$ direction for 8 steps and then to the $-X$
direction for 10 steps.

##  V Dynamically controlling the non-Hermitian lattice and skin effect

The use of time as a synthetic dimension allows us to dynamically reconfigure
our non-Hermitian lattice as a function of time. Specifically, by controlling
the intensity modulators at each time step of the quantum walk, we achieve
temporal modulation of the gain/loss parameters $\delta_{x}(t)$ and
$\delta_{y}(t)$ such that they are time-dependent. Using this time dependence,
first, we demonstrate dynamical control over the degree of localization of the
corner states. At the start of the evolution, we adopt the configuration as in
Figure 1(c) and set $|\delta_{x}(0)|=|\delta_{y}(0)|=0.175$, and inject a
single light pulse at the corner state situated at the origin. As the system
evolves, we reduce both $|\delta_{x}|,|\delta_{y}|$ by $50\%$ for every four
time-steps and continue doing so until step $16$ (Figure 3(a)). Because of
this reduction, we observe that the corner states become less confined to the
origin. This is because the smaller non-Hermitian parameter exhibits
eigenmodes distributed over a larger area, as predicted theoretically (see the
SI). Thereafter, starting from step $17$, we reverse the process, that is, we
increase the gain /loss parameters $|\delta_{x}|,|\delta_{y}|$ back to its
original value at the same rate. We now observe a relocalization of light at
the origin.

Next, we demonstrate adiabatic repositioning of the corner states in the
lattice. We use the same lattice geometry shown in Figure 1(c) and fix the
non-Hermitian parameter to $\delta_{x}=\delta_{y}=0.175$. As the system
evolves, we adiabatically move the interface between the four distinct
topological regions, repositioning the corner state as a function of time. We
first move the position of the corner state upwards for 8 unit cells, and then
leftward for 10 unit cells. As before, we inject light pulses at the corner
state. As the system evolves, we observe that the center of the intensity
distribution follows the position of the corner state as it adiabatically
moves along the given ”L”-shaped trajectory from its initial location
$(x=0,y=0)$ to its final location at $(x=-10,y=8)$. Furthermore, during this
process, the intensity distribution remains tightly localized close to the
corner state. Evidently, the corner serves as a non-Hermitian tweezer of
light, which allows us to adiabatically move trapped photons along a given
trajectory in the synthetic lattice. Note that non-Hermitian light steering
has been demonstrated in real-space lattices [1], and our demonstration in
synthetic time dimensions portends the potential for such photonic control
using the temporal degree of freedom of light.

Figure 4: Robustness of the skin effect at the presence of different degrees
of lattice disorder. The randomness is introduced to the intensity modulation
of the lattice and the pulse is injected at $(x,y)=(0,0)$. (a-b) Experimental
observation of robustness of the corner state and skin effect in a lattice
with moderate disorder $(\eta_{y}=\eta_{x}=0.5,1)$. Here the disorder leads to
a relaxed spatial confinement of light without breaking the localization of
light. (c-d) Breakdown of the localization in the presence of strong disorder
$(\eta_{y}=\eta_{x}=1.5,2)$. Light can diffuse arbitrarily far away from the
original location. In (a-d), from left to right the snapshots are shown for
time steps $1,5,13$, respectively.

##  VI Robustness of the skin effect

The topological nature of the non-Hermitian skin effect ensures its robustness
against disorder in gain/loss parameters $\delta_{x},\delta_{y}$. To
quantitatively investigate this robustness, we introduce a disorder on the
gain/loss. At each lattice site, we randomly pick both $\delta_{x},\delta_{y}$
from a uniform distribution on the interval
$[\delta_{max}(1-\eta),\delta_{max}]$, where max is the maximum gain parameter
and is the disorder parameter which quantifies the variance of the gain
parameter. In our experiment, we vary the disorder parameter between 0 (no
disorder) and 2 (max disorder).

Figure 5: (a) Theory and (b) Experimental evolution of space-averaged
displacement $<r^{2}>=<x^{2}+y^{2}>$ as a function of step number under
different lattice disorders. For disorder strength lower than the threshold
$\eta=1$, the disorder increases the effective spatial diameter of the corner
state, as shown in the evolution of average displacement with time. For
disorders higher than the threshold, light diffusively spreads to large
distances on the lattice.

We find that the skin effect is robust when the disorder parameter $\eta<1$.
In Figure 4(a-b), we plot the evolution of light pulses in the lattice for two
different values $\eta=0.5$ and $\eta=1$. For both cases, we inject light
pulses at the corner state located at the origin. We observe that, even though
the localization of the intensity distribution reduces as disorder increases,
the distribution still stays confined around the origin, indicating the
existence of corner states even in the presence of disorder. Nevertheless,
once we increase the disorder parameter to 1.5 and 2 (Figure 4(c-d)), the
intensity distribution diffusively spreads away from the origin, indicating
the breakdown of corner states. Our experimental observation agrees with the
intuitively expected behavior that, for $\eta<1$, even though there is a
disorder in the modulation amplitudes, the gain for the step towards the
corner is always larger than that of the outwards direction. Thus the four
regions maintain their distinct non-Hermiticity and the corner state exists.
But, when $\eta>1$, a direction-dependent gain for the time steps is no longer
always valid and therefore the four regions are no longer distinct and the
corner state ceases to exist.

To better characterize the robustness and breakdown of the skin effect, we
compute the evolution of the mean-square displacement of the intensity
distribution in the lattice as a function of time. The mean-square
displacement is quantified as $<r^{2}>(n)=\sum_{x,y}P_{x,y}(n)(x^{2}+y^{2})$,
where $P_{x,y}(n)$ is the time-varying intensity distribution of light. Figure
5 shows the calculated $<r^{2}>(n)$ for several values of the disorder
parameter for both theoretical calculations and experimentally measured
values. Each experimental curve corresponds to an average of four independent
experimental realizations of disorder, while the theory corresponds to four
averages. Due to the limited size of the lattice $30\times 30$, we only
collect data from step $1$ to step $15$, and plot $<r^{2}>(n)$ for the odd
steps. The violet, red, and green curves correspond to the weak disorder, with
disorder parameters being $0$, $0.5$, and $1$, respectively. All three curves
saturate to a fixed value which is well below a certain threshold. This
behavior thus implies the robustness of the skin effect. But for larger
disorders of $1.5$ and $2$, corresponding to the yellow and blue curve, the
mean squared distance does not converge. Instead, it spreads out until it
becomes limited by the finite size of the lattice, indicating a complete
breakdown of the skin effect.

##  VII Summary and outlook

In conclusion, we demonstrated robust dynamical control over the photonic non-
Hermitian skin effect in a 2D synthetic lattice. We created a corner state
that localizes light and dynamically tuned the degree of light localization.
Moreover, we dynamically steered trapped light along any given trajectory in
the 2D lattice. We also demonstrate the robustness of the skin effect under
lattice disorder below a certain threshold.

Looking forward, the dynamic techniques developed in this work can be further
applied to investigate Floquet non-Hermitian models [27, 28], in particular in
synthetic dimensions [29]. Further, one can create an analogue of on-site
interaction by imposing a nonlinear phase shifter after the linear optical
transformations, and investigate non-Hermitian models of interacting particles
[30]. Such nonlinearities could also have implications in the recently
discovered regime of topological frequency combs [31, 32, 33] due to the
periodic temporal pulses that define our platform. Moreover, the two-fold spin
characteristics in our system can potentially be extended to non-Hermitian
models for lattice gauge theories with higher spins and non-Abelian
statistics, by increasing the number of loops [34]. Finally, our non-hermitian
lattice can be enriched with engineered synthetic gauge fields as demonstrated
recently for both Hermitian [35] and non-Hermitian models[20], to explore
intriguing proposals such as the quantum Skin Hall effect [36].

##  VIII Acknowledgements

The authors wish to acknowledge fruitful discussions with Shanhui Fan and
Nathan Schine. This research was supported by The Office of Naval Research
ONR-MURI grant N00014-20-1-2325, AFOSR FA95502010223, NSF PHY1820938, NSF
DMR-2019444, the Army Research Laboratory grant W911NF1920181 and EPSRC grant
EP/V026496/1.

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##  IX Supplementary Information: Dynamic control of 2D non-Hermitian photonic
corner states in synthetic dimensions

###  IX.1 Experimental Setup

Here we show the details of the experimental setup.

Figure S1:  Sketch of the complete experimental setup and encoding scheme. (a)
Details of the experiment, with continuous wave (CW) laser, polarization
control (PC), wavelength division multiplexer (WDM), photodiode (PD), beam
splitter (BS), intensity modulators (X/Y-mod), erbium doped fiber amplifier
(EDFA) and semiconductor optical amplifier (SOA). (b) Encoding the 2D lattice
in time bins in two fiber loops. The two loops are named “up channel” and
“down channel”, respectively.

To encode the 2D lattice in time we consider two fiber loops shown in Figure
S1(a), labeled up channel and down channel. The length of each fiber loop is
$\sim$ 3 (km), and one circulation of light in the loop is equivalent to one
step of the walk. Hence, we can encode the entire 2D lattice within a time-
duration (or time-delay) of $\sim$ 15000 (ns) without mixing time-bins in step
$n$ and step $(n+1)$. As in Figure S1(b), we first encode $30$ “Y”-time bins
in both the up and down channel, each of time duration 250 (ns) in a total
time duration of 7500 (ns). Each “Y”-time bin is then occupied by $30$
“X”-time bins, each of time duration 7.5 (ns). At any time, the state of the
system is thus represented by a complex vector $(U_{x,y},D_{x,y})$, encoded in
the phase and amplitude of the light pulse circulating in the two fiber loops.

To initialize the system, we inject a single pulse into the down channel of
the fiber loop. We use a continuous wave CW laser with 1550 (nm) wavelength
(Optilab DFB-1551-SM-10) and by modulation of this laser using a Thorlabs SOA
(SOA1013SXS), we have generated pulses of width $\sim$ 6 (ns) at a repetition
rate of 1 (pulse/ms). We then control the polarization of the laser with an
inline fiber polarization control (PC) before injecting the light into the
down channel with a $90/10$ beam splitter. Note that we use two identical
$90/10$ beam splitter, one for each channel. The $90/10$ beam splitter in the
down channel is used to inject light into the quantum walk, whereas the
$90/10$ beam splitter in the up channel is used to weakly couple light pulses
out of the quantum walk so that we can measure the pulse power after $n$ steps
of evolution.

As a pulse enters the system, by default we recognize it as entering the
$(x=0,y=0)$ time bin, and thus the initial state is $D_{0,0}=1$. The pulse
then sequentially passes through a $50/50$ beam splitter denoted as $\pm
X$-beam splitter, a pair of time-varying intensity modulators (Optilab
IMP-1550-20-PM) is used to impose the correct gain/loss as each time bin
$(x,y)$ passes through it, controlled by RF signal generated from Teledyne
Lecroy arbitrary waveform generator (T3AWG3252). We then impose a delay of 3
(m) in the up channel and no delay in the down channel. The same procedure
then repeats for $Y$, as shown in Figure S1(a), except the difference in delay
between the up and down channels is 100 (m).

To combat photon loss in the walk, we use two Thorlabs erbium-doped fiber
amplifiers (EDFA) (EDFA100S), one for each channel. Before amplifying the
pulse, we use wavelength division multiplexers (WDM) (DWDM-SM-1-34-L-1-2) to
couple a 1543 (nm) CW laser (DFB-1543-SM-30) to the pulses so that the
spontaneous emission noise during the amplification is reduced. We decouple
the 1550 (nm) pulses from the 1543 (nm) CW laser with the same WDM after the
amplification is done. Finally, we use PC to ensure the correct linear
polarization for the 1550 (nm) signal pulses. After this, a complete quantum
walk step is finished.

###  IX.2 Mathematical formulation of the quantum walk experiment

Here we give the most general mathematical formulation of our quantum walk.
Figure S2(a-g) shows this evolution when the input state is $D_{0,0}=1$.

Figure S2:  Example of full evolution in a complete step. Both the up and down
channels of the fiber loops are shown. (a-g) The complete evolution within one
step given that the initial state is $D_{x=0,y=0}=1$ and all other $U_{x,y}$
and $D_{x,y}$ are $0$.

As mentioned in the previous section, the state at each step is given by a
complex vector $(U_{x,y},D_{x,y})$ where $x$ and $y$ ranges from $-15$ to
$+15$. After the first beam splitter ($\pm X$), the state is updated to
(Figure S2(b):

$\displaystyle U_{x,y}^{{}^{\prime}}=\frac{1}{\sqrt{2}}(U_{x,y}-D_{x,y})$ (S1)
$\displaystyle D_{x,y}^{{}^{\prime}}=\frac{1}{\sqrt{2}}(U_{x,y}+D_{x,y})$

After the first pair of modulators ($\pm X$ modulators), we obtain:

$\displaystyle U_{x,y}^{{}^{\prime\prime}}=U_{x,y}^{{}^{\prime}}f^{(U)}_{x,y}$
(S2) $\displaystyle
D_{x,y}^{{}^{\prime\prime}}=D_{x,y}^{{}^{\prime}}f^{(D)}_{x,y}$

where $f^{(U/D)}_{x,y}$ is the gain/loss applied to each time bin in the
up/down channel by the $X$ modulators.

After the delay:

$\displaystyle
U_{x,y}^{{}^{\prime\prime\prime}}=U_{x+1,y}^{{}^{\prime\prime}}$ (S3)
$\displaystyle
D_{x,y}^{{}^{\prime\prime\prime}}=D_{x-1,y}^{{}^{\prime\prime}}$

This yields:

$\displaystyle
U_{x,y}^{{}^{\prime\prime\prime}}=\frac{1}{\sqrt{2}}(U_{x+1,y}-D_{x+1,y})f^{(U)}_{x+1,y}$
(S4) $\displaystyle
D_{x,y}^{{}^{\prime\prime\prime}}=\frac{1}{\sqrt{2}}(U_{x-1,y}+D_{x-1,y})f^{(D)}_{x-1,y}$

Then, we enter the second beamsplitter ($\pm Y$):

$\displaystyle
W_{x,y}^{{}^{\prime}}=\frac{1}{\sqrt{2}}(U_{x,y}^{{}^{\prime\prime\prime}}-D_{x,y}^{{}^{\prime\prime\prime}})$
(S5) $\displaystyle
F_{x,y}^{{}^{\prime}}=\frac{1}{\sqrt{2}}(U_{x,y}^{{}^{\prime\prime\prime}}+D_{x,y}^{{}^{\prime\prime\prime}})$

After modulation ($\pm Y$ modulators):

$\displaystyle W_{x,y}^{{}^{\prime\prime}}=W_{x,y}^{{}^{\prime}}c^{(U)}_{x,y}$
(S6) $\displaystyle
F_{x,y}^{{}^{\prime\prime}}=F_{x,y}^{{}^{\prime}}c^{(D)}_{x,y}$

where $c^{(U/D)}_{x,y}$ is the gain/loss applied to each time bin in the
up/down channel by the $Y$ modulators.

After delay:

$\displaystyle
W_{x,y}^{{}^{\prime\prime\prime}}=W_{x,y+1}^{{}^{\prime\prime}}=\frac{1}{\sqrt{2}}(U_{x,y+1}^{{}^{\prime\prime\prime}}-D_{x,y+1}^{{}^{\prime\prime\prime}})\times
c^{(U)}_{x,y+1}$ (S7) $\displaystyle
F_{x,y}^{{}^{\prime\prime\prime}}=F_{x,y-1}^{{}^{\prime\prime}}=\frac{1}{\sqrt{2}}(U_{x,y-1}^{{}^{\prime\prime\prime}}+D_{x,y-1}^{{}^{\prime\prime\prime}})\times
c^{(D)}_{x,y-1}$

$(W_{x,y}^{{}^{\prime\prime\prime}},F_{x,y}^{{}^{\prime\prime\prime}})$ is
thus the output state given the input state $(U_{x,y},D_{x,y})$. Consider a
pulse ending up in time bin $(x,y)$ at step $(n+1)$. Denoting it as
$U^{(n+1)}_{x,y}$ and $D^{(n+1)}_{x,y}$, we have:

$\displaystyle U_{x,y}^{(n+1)}$
$\displaystyle=\frac{1}{2}[(U^{(n)}_{x+1,y+1}-D^{(n)}_{x+1,y+1})f^{(U)}_{x+1,y+1}c^{(U)}_{x+1,y+1}-(U^{(n)}_{x-1,y+1}+D^{(n)}_{x-1,y+1})f^{(D)}_{x-1,y+1}c^{(U)}_{x-1,y+1}]$
(S8) $\displaystyle D_{x,y}^{(n+1)}$
$\displaystyle=\frac{1}{2}[(U^{(n)}_{x+1,y-1}-D^{(n)}_{x+1,y-1})f^{(U)}_{x+1,y-1}c^{(D)}_{x+1,y-1}+(U^{(n)}_{x-1,y-1}+D^{(n)}_{x-1,y-1})f^{(D)}_{x-1,y-1}c^{(D)}_{x-1,y-1}]$

The above equation thus describes the full evolution of the state within one
step. We can obtain lattice gain/loss pattern as shown in Figure 1(a) or in
Figure 1(c) of the main text, by properly choosing $f^{(U/D)}$ and $c^{(U/D)}$
as a function of $(x,y)$. For example, for a bulk lattice as shown in Figure
1(a) of the main text, we choose $f^{(U)}_{x,y}=\alpha e^{0.175}$,
$f^{(D)}_{x,y}=\alpha e^{-0.175}$, $c^{(U)}_{x,y}=\alpha e^{0.175}$,
$c^{(D)}_{x,y}=\alpha e^{-0.175}$. Here $\alpha$ is the additional loss
imposed by the modulators, and we use EDFA to compensate for the loss such
that effectively, $\alpha=1$. In the experiment, we are only measuring light
in the up channel, and thus the power of pulses, or the probability
distribution of the walker in the up channel, is
$P_{x,y}\propto|U_{x,y}|^{2}$. We normalize this probability distribution for
all experiment data.

###  IX.3 Effective band and energy winding of the bulk model

Figure S3:  Bulk band non-Hermitian topology. (a) For each bulk patch, we
choose two oriented loops in the Brillouin zone:
$l_{h}=\\{k_{y}\equiv\pi/4,k_{x}\in[-\pi\rightarrow\pi]\\}$ and
$l_{v}=\\{k_{x}\equiv\pi/4,k_{y}\in[-\pi\rightarrow\pi]\\}$. Each loop $l_{i}$
then contributes to two individual complex energy winding loops
$\epsilon_{up}(l_{i})$ and $\epsilon_{down}(l_{i})$ winding in the same
direction. We calculate this winding for all four patches in (b),
corresponding to positive or negative $\delta_{x}$ and $\delta_{y}$. (c-f) The
topological invariant is the winding direction of the directed curve
$\epsilon(l_{h})$ and $\epsilon(l_{v})$ in the complex energy plane, which can
either be clockwise or counterclockwise.

Here we impose periodic boundary conditions for the bulk lattice as shown in
Figure 1(a) of the main text, in both $X$ and $Y$ directions. The modulations
are $f^{(U)}_{x,y}=e^{0.175}$, $f^{(D)}_{x,y}=e^{-0.175}$,
$c^{(U)}_{x,y}=e^{0.175}$, $c^{(D)}_{x,y}=e^{-0.175}$, and thus
$\delta_{x}=0.175$ and $\delta_{y}=0.175$. With this assumption, we can
therefore apply the Bloch theorem to the quantum walk evolution equation in
the previous section and introduce the Bloch vector $(k_{x},k_{y})$. We use
the ansatz $U_{x,y}=e^{ik_{x}x+ik_{y}y}\tilde{U}_{k_{x},k_{y}}$ and
$D_{x,y}=e^{ik_{x}x+ik_{y}y}\tilde{D}_{k_{x},k_{y}}$ for the eigenmodes of the
walk. The evolution equation can now be simplified to:

$\displaystyle\begin{bmatrix}\tilde{U}_{k_{x},k_{y}}\\\
\tilde{D}_{k_{x},k_{y}}\end{bmatrix}=\frac{1}{2}\begin{bmatrix}e^{ik_{x}+ik_{y}}f^{(U)}c^{(U)}-e^{-ik_{x}+ik_{y}}f^{(D)}c^{(U)}&&-e^{ik_{x}+ik_{y}}f^{(U)}c^{(U)}-e^{-ik_{x}+ik_{y}}f^{(D)}c^{(U)}\\\
e^{ik_{x}-ik_{y}}f^{(U)}c^{(D)}+e^{-ik_{x}-ik_{y}}f^{(D)}c^{(D)}&&-e^{ik_{x}-ik_{y}}f^{(U)}c^{(D)}+e^{-ik_{x}-ik_{y}}f^{(D)}c^{(D)}\\\
\end{bmatrix}\begin{bmatrix}\tilde{U}_{k_{x},k_{y}}\\\
\tilde{D}_{k_{x},k_{y}}\end{bmatrix}$ (S9)

Since we have two discrete degree of freedom $U$ and $D$, we always obtain two
different eigenvalues for each $(k_{x},k_{y})$ as we diagonalize the above
$2\times 2$ matrix. We call the two eigenvalues $u_{k_{x},k_{y}}$ and
$d_{k_{x},k_{y}}$. The two effective energies are thus defined as
$\epsilon_{up}(k_{x},k_{y})=ilog(u_{k_{x},k_{y}})$ and
$\epsilon_{down}(k_{x},k_{y})=ilog(d_{k_{x},k_{y}})$.

We further consider the four bulk patches in Figure S3(b) and show that they
exhibit different non-Hermitian topological invariants, namely the winding of
the effective energy $\epsilon_{up/down}(k_{x},k_{y})$ in the complex plane.
Without loss of generality, we always pick two loops in the Brillouin zone:
$l_{v}=\\{k_{x}\equiv\pi/4,k_{y}\in[-\pi\rightarrow\pi]\\}$ and
$l_{h}=\\{k_{y}\equiv\pi/4,k_{x}\in[-\pi\rightarrow\pi]\\}$, as shown in
Figure S3(a). For each bulk panel in Figure S3(b), as one varies
$(k_{x},k_{y})$ along $l_{v}$ and $l_{h}$, the corresponding complex energy
$\epsilon(k_{x},k_{y})$ can finish a single loop in the complex plane, either
in the clockwise or counterclockwise direction. The winding direction of
$\epsilon(l_{v})$ and $\epsilon(l_{h})$ forms the non-Hermitian topological
invariant of the bulk. Note that we have suppressed the unimportant label $up$
and $down$ since $\epsilon_{up}(l_{i})$ and $\epsilon_{down}(l_{i})$ always
wind in the same direction.

###  IX.4 Localized eigenmodes at the presence of boundary and corner

Figure S4:  Averaged eigenmode spatial profile for different non-Hermitian
parameters $\delta_{x}=\delta_{y}$. We adopt the lattice geometry as in Figure
S2(b). From left to right we take values $0.03,0.06,0.09,0.12,0.15$ and
$0.18$.

As shown in Figure 1(d) in the main text, the averaged spatial profile of the
eigenmodes of the walk is localized at the corner. In Figure 1(c) of the main
text we have chosen $\delta_{x}=\delta_{y}=0.175$, but the feature of the
spatial profile persists for any $\delta_{x}=\delta_{y}>0$. Here in Figure S4
of the supplementary section, we show the average eigenmode spatial profile
for $\delta_{x}=\delta_{y}=0.03,0.06,0.09,0.12,0.15$ and $0.18$. As one
increases $\delta_{x}=\delta_{y}$, we observe that the averaged spatial
profile becomes more localized. This explains the adiabatic tapering of the
walker’s probability distribution as one increases/decreases the non-Hermitian
parameter in time, as shown in Figure 3(a) in the main text.

###  IX.5 Static control supplementary data

Figure S5:  Evolution of probability distribution for non-Hermitian parameters
$\delta_{x}=\delta_{y}=0$, showing diffusive spreading. (a) Probability
distribution evolution, where the snapshots are taken at time step $1,5,9,13$,
respectively. (b) Evolution of averaged displacement
$<r^{2}>=<x^{2}+y^{2}>=\sum_{x,y}P_{x,y}(x^{2}+y^{2})$ for time step from
$1,3,5,7,9,11,13$ and $15$.

Here we present additional experimental results for the quantum walk with no
dynamical control. We first show that, with $\delta_{x}=\delta_{y}=0$, the
walker diffusively spreads into the bulk of the lattice, as shown in Figure
S5(a), where the probability distribution of the walker is plotted for step
$1,5,9,13$. The averaged displacement, defined as
$<r^{2}>(n)=\sum_{x,y}P_{x,y}(n)(x^{2}+y^{2})$, is plotted in Figure S5(b),
for step $1,3,5,7,9,11,13$ and $15$. Here $P_{x,y}$ is the probability
distribution of the walker at time step $n$.

Furthermore, as mentioned in the main text, the funneling of light happens
wherever the walker is initialized, assuming the lattice gain-loss pattern
shown in Figure 1(c) of the main text. This is manifestly shown in Figure S6,
where we always choose the initial state to be $D_{x=0,y=0}=1$, but lattice
corner is located at $(10,-10)$, $(10,10)$ and $(-10,-10)$, respectively.

Figure S6:  Funneling and stabilization of light starting from arbitrary bulk
patches. Light is always initialized at $(x,y)=(0,0)$, but the corner position
is held fixed at (a) $(10,-10)$, (b) $(10,10)$ and (c) $(-10,-10)$. For each
panel, the top row is probability distributions collected at step $1,9,17,21$
of the experiment, and the bottom row is the corresponding simulation results.


11footnotetext: The author was partially supported by an NSF grant.

# The homotopy orbit spectrum for profinite groups

Daniel G. Davis1

###### Abstract.

Let $G$ be a profinite group. We define an $S[[G]]$-module to be a
$G$-spectrum $X$ that satisfies certain conditions, and, given an
$S[[G]]$-module $X$, we define the homotopy orbit spectrum $X_{hG}$. When $G$
is countably based and $X$ satisfies a certain finiteness condition, we
construct a homotopy orbit spectral sequence whose $E_{2}$-term is the
continuous homology of $G$ with coefficients in the graded profinite
$\widehat{\mathbb{Z}}[[G]]$-module $\pi_{\ast}(X)$. Let $G_{n}$ be the
extended Morava stabilizer group and let $E_{n}$ be the Lubin-Tate spectrum.
As an application of our theory, we show that the function spectrum
$F(E_{n},L_{K(n)}(S^{0}))$ is an $S[[G_{n}]]$-module with an associated
homotopy orbit spectral sequence.

##  1\. Introduction

Let $G$ be a finite group and let $X$ be a (left) $G$-spectrum. Then the
homotopy orbit spectrum $X_{hG}$ is defined to be
$\operatorname*{hocolim}_{G}X$, the homotopy colimit of the $G$-action on $X$
(see, for example, [16, pg. 42]). Furthermore, there is a homotopy orbit
spectral sequence

$H_{p}(G,\pi_{q}(X))\Rightarrow\pi_{p+q}(X_{hG}),$

where the $E_{2}$-term is the group homology of $G$, with coefficients in the
graded $G$-module $\pi_{\ast}(X)$ ([17, 5.1]). In this paper, under certain
hypotheses, we extend these constructions to the case where $G$ is a profinite
group.

After making a few comments about notation, we summarize the contents of this
paper. We follow the convention that all of our spectra are in $\mathrm{Spt}$,
the category of Bousfield-Friedlander spectra of simplicial sets. We use
$(-)_{\mathtt{f}}$ to denote functorial fibrant replacement in the category of
spectra; for any spectrum $Z$, $Z\rightarrow Z_{\mathtt{f}}$ is a weak
equivalence with $Z_{\mathtt{f}}$ fibrant. Also, “holim” always denotes the
version of the homotopy limit of spectra that is constructed levelwise in the
category of simplicial sets, as defined in [2] and [22, 5.6].

In Section 2, for a finite group $G$, we use the fact that the homotopy
colimit of a diagram of pointed simplicial sets is the diagonal of the
simplicial replacement of the diagram, to obtain an alternative model for
homotopy orbits. Then, in Section 3, we use this model to define the homotopy
orbit spectrum for a profinite group $G$ and a certain type of $G$-spectrum
$X$, which we now define. We point out that the following definition was
essentially first formulated by Mark Behrens.

######  Definition 1.1.

Given a profinite group $G$, let $\\{N_{i}\\}$ be a collection of open normal
subgroups of $G$ such that $G\cong\lim_{i}G/N_{i}$. Then we let $S[[G]]$ be
the spectrum $\operatorname*{holim}_{i}(S[G/N_{i}])_{\mathtt{f}}$. Also, let
$\\{X_{i}\\}$ be an inverse system of $G$-spectra and $G$-equivariant maps
indexed over $\\{i\\}$, such that, for each $i$, the $G$-action on $X_{i}$
factors through $G/N_{i}$ (so that $X_{i}$ is a $G/N_{i}$-spectrum) and
$X_{i}$ is a fibrant spectrum. Then we call the associated $G$-spectrum
$X=\operatorname*{holim}_{i}X_{i}$ an $S[[G]]$-module.

After defining the homotopy orbit spectrum $X_{hG}$ for an $S[[G]]$-module
$X$, we show that this construction agrees with the classical definition when
$G$ is finite. Now let us recall a definition (see [4, Section 3] for more
detail).

######  Definition 1.2.

A discrete right $G$-spectrum $X$ is a spectrum of pointed simplicial sets
$X_{k}$, for $k\geq 0$, such that each simplicial set $X_{k}$ is a pointed
simplicial discrete right $G$-set, and each bonding map $S^{1}\wedge
X_{k}\rightarrow X_{k+1}$ is $G$-equivariant ($S^{1}$ has trivial $G$-action).

In Section 4, given a discrete right $G$-spectrum $X$ and any spectrum $Z$, we
show that the function spectrum $F(X,Z)$ is an $S[[G]]$-module, so that one
can form $F(X,Z)_{hG}$. Interesting examples of $F(X,Z)$ are the Brown-
Comenetz dual $cX$ of $X$ and the Spanier-Whitehead dual $DX$.

Now we summarize the main result of Section 5, in which we build a homotopy
orbit spectral sequence.

Recall that a topological space $Y$ is countably based if there is a countable
family $\mathcal{B}$ of open sets, such that each open set of $Y$ is a union
of members of $\mathcal{B}$. In particular, one can consider countably based
profinite groups. Then, if $G$ is a countably based profinite group, by [24,
Proposition 4.1.3], $G$ has a chain

$N_{0}\geq N_{1}\geq N_{2}\geq\cdots$

of open normal subgroups, such that $G\cong\lim_{i\geq 0}G/N_{i}$. For
example, a compact $p$-adic analytic group is a countably based profinite
group. (If a topological group $G$ is compact $p$-adic analytic, then $G$ is a
profinite group with an open subgroup $H$ of finite rank, by [7, Corollary
8.34]. Then, by [7, Exercise 1, pg. 58], $G$ has finite rank, and thus, $G$ is
finitely generated ([7, pg. 51]), so that $G$ is countably based, by [24, pg.
55].)

Let $G$ be a countably based profinite group, with a chain $\\{N_{i}\\}_{i\geq
0}$ of open normal subgroups, such that $G\cong\lim_{i}G/N_{i}$, and let
$\\{X_{i}\\}$ be a diagram of $G$-spectra, such that
$\operatorname*{holim}_{i}X_{i}$ is an $S[[G]]$-module. Recall from [3, pg. 5]
that a spectrum $X$ is an f-spectrum, if, for each integer $q$, the abelian
group $\pi_{q}(X)$ is finite. Let $\widehat{\mathbb{Z}}$ denote $\lim_{n\geq
1}\mathbb{Z}/n\mathbb{Z}.$ Then, if each $X_{i}$ is an $f$-spectrum, we show
that there is a homotopy orbit spectral sequence

$E_{2}^{p,q}\cong
H^{c}_{p}(G,\pi_{q}(\operatorname*{holim}_{i}X_{i}))\Rightarrow\pi_{p+q}((\operatorname*{holim}_{i}X_{i})_{hG}),$

where the $E_{2}$-term is the continuous homology of $G$ with coefficients in
the graded profinite $\widehat{\mathbb{Z}}[[G]]$-module
$\pi_{\ast}(\operatorname*{holim}_{i}X_{i})$.

We continue to assume that $G$ is a countably based profinite group in Section
6, where we prove a result about the homotopy orbits of a tower of Eilenberg-
Mac Lane spectra that was suggested to the author by Mark Behrens. To be
precise, let $A_{0}\leftarrow A_{1}\leftarrow A_{2}\leftarrow\cdots$ be a
tower of $G$-modules and $G$-equivariant maps, such that, for each $i$, the
$G$-action on $A_{i}$ factors through $G/N_{i}$, so that $A_{i}$ is a
$G/N_{i}$-module. If each abelian group $A_{i}$ is finite, then we call
$\\{A_{i}\\}$ a nice tower of $G$-modules. Given a nice tower $\\{A_{i}\\}$ of
$G$-modules, we show that $\operatorname*{holim}_{i}H(A_{i})$ is an
$S[[G]]$-module, with

$\pi_{\ast}((\operatorname*{holim}_{i}H(A_{i}))_{hG})\cong
H^{c}_{\ast}(G,\lim_{i}A_{i}),$

where $H(A_{i})$ is an Eilenberg-Mac Lane spectrum.

In Section 7, we consider an example that arises in chromatic homotopy theory.
Let $\ell$ be a prime and let $n\geq 1$. Let $E_{n}$ denote the Lubin-Tate
spectrum, so that
$\pi_{\ast}(E_{n})=W(\mathbb{F}_{\ell^{n}})[[u_{1},...,u_{n-1}]][u^{\pm 1}],$
where the degree of $u$ is $-2$ and the power series ring over the Witt
vectors has degree zero. Also, let
$G_{n}=S_{n}\rtimes\mathrm{Gal}(\mathbb{F}_{\ell^{n}}/\mathbb{F}_{\ell})$ be
the extended Morava stabilizer group, where $S_{n}$ is the $n$th Morava
stabilizer group, and recall that the profinite group $G_{n}$ is compact
$\ell$-adic analytic. By work of Paul Goerss, Mike Hopkins, and Haynes Miller
(see [18], [10], and [6]), the group $G_{n}$ acts on $E_{n}$.

Now let $K(n)$ be the $n$th Morava $K$-theory, with
$K(n)_{\ast}=\mathbb{F}_{\ell}[v_{n}^{\pm 1}]$, where the degree of $v_{n}$ is
$2(\ell^{n}-1)$. Also, let $L_{K(n)}(-)$ denote the Bousfield localization
functor with respect to $K(n)$. Then we show that the $K(n)$-local Spanier-
Whitehead dual of $E_{n}$, the function spectrum $F(E_{n},L_{K(n)}(S^{0}))$,
is an $S[[G_{n}]]$-module and there is a homotopy orbit spectral sequence

$H^{c}_{p}(G_{n},\pi_{q}(F(E_{n},L_{K(n)}(S^{0}))))\Rightarrow\pi_{p+q}(F(E_{n},L_{K(n)}(S^{0}))_{hG_{n}}).$

When $(\ell-1)\nmid n$, the use of the above spectral sequence to identify the
spectrum $F(E_{n},L_{K(n)}(S^{0}))_{hG_{n}}$ more concretely is work in
progress.

We want to point out that others have investigated homotopy orbits for
profinite groups. In [8], Halvard Fausk studies homotopy orbits in the setting
of pro-orthogonal $G$-spectra, where $G$ is profinite. As pointed out in
various places in this paper, Mark Behrens has worked on homotopy orbits for a
profinite group. Also, Dan Isaksen has thought about homotopy orbits for
profinite groups in the context of pro-spectra. Finally, [12], by Mike Hopkins
and Hal Sadofsky, contains some explorations of homotopy orbits involving
$E_{n}$ and $G_{n}$. It was the author’s study of [12] that motivated him to
begin work on the project represented by this paper.

Acknowledgements. I thank Hal Sadofsky for sparking my interest in homotopy
orbits for profinite groups. I thank Mark Behrens for helpful and encouraging
discussions about this work. His positive influence on this work is seen in
the many places where his name is mentioned. Also, I thank Paul Goerss and
Mark Hovey for their encouragement.

##  2\. Homotopy orbits when $G$ is finite

In this section, let $G$ be a finite group and let $X$ be a (left)
$G$-spectrum. We use simplicial replacement to obtain an alternative model for
the homotopy orbit spectrum $X_{hG}$, which will be useful for defining the
homotopy orbit spectrum when $G$ is a profinite group. Given a spectrum $Z$,
we let $Z_{k}$ denote the $k$th pointed simplicial set of $Z$ and we use
$Z_{k,l}$ to signify the set of $l$-simplices of $Z_{k}$.

By [2, XII, 5.2],

$(X_{hG})_{k}=\mathrm{hocolim}\,_{G}\,X_{k}=\mathrm{diag}(\,\textstyle{\coprod}_{\ast}\,(G\rightarrow
X_{k})),$

where $\mathrm{diag}(-)$ is the functor that takes the diagonal of a
bisimplicial set, and $G\rightarrow X_{k}$ is the diagram out of the groupoid
$G$ that is defined by the action of $G$ on $X_{k}$. The simplicial
replacement is defined as follows:

$\textstyle{\coprod}_{l}\,(G\rightarrow X_{k})=\textstyle{\bigvee}\,_{u_{l}\in
I_{l}}\,X_{k},$

where $I_{l}$ is the indexing set that consists of all compositions of the
form

$u_{l}=i_{0}\overset{\alpha_{1}}{\longleftarrow}\cdots\overset{\alpha_{l}}{\longleftarrow}i_{l},$

such that each $\alpha_{j}$ is a morphism in the groupoid $G$.

The face and degeneracy maps of the pointed simplicial set
$\textstyle{\coprod}_{l}\,(G\rightarrow X_{k})$ are defined in the following
way. Let $0\leq j<l$ and let $d_{j}(u_{l})$ be the map that, given a morphism
$u_{l}\in I_{l}$, is defined by the composition

$X_{k}\overset{\mathrm{id}}{\longrightarrow}X_{k}\longrightarrow\textstyle{\bigvee}\,_{I_{l-1}}\,X_{k},$

where the target of $\mathrm{id}$ is the copy of $X_{k}$ that is indexed by
the morphism

$u_{l-1}=i_{1}\overset{\alpha_{2}}{\longleftarrow}\cdots\overset{\alpha_{l}}{\longleftarrow}i_{l}\in
I_{l-1}.$

Then the $j$th face map
$d_{j}\colon\textstyle{\bigvee}\,_{I_{l}}\,X_{k}\rightarrow\textstyle{\bigvee}\,_{I_{l-1}}\,X_{k}$
is the unique map that is induced by all the $d_{j}(u_{l})$, where
$d_{j}(u_{l})$ is the canonical map out of the copy of $X_{k}$ that is indexed
by $u_{l}$. Also, let $d_{l}(u_{l})$ be the map that, given a morphism
$u_{l}\in I_{l}$, is defined by the composition

$X_{k}\overset{\alpha_{l}}{\longrightarrow}X_{k}\longrightarrow\textstyle{\bigvee}\,_{I_{l-1}}\,X_{k},$

where the target of $\alpha_{l}$ is the copy of $X_{k}$ that is indexed by the
morphism

$u_{l-1}=i_{0}\overset{\alpha_{1}}{\longleftarrow}\cdots\overset{\alpha_{l-1}}{\longleftarrow}i_{l-1}\in
I_{l-1}.$

Then the $l$th face map
$d_{l}\colon\textstyle{\bigvee}\,_{I_{l}}\,X_{k}\rightarrow\textstyle{\bigvee}\,_{I_{l-1}}\,X_{k}$
is the unique map that is induced by all the $d_{l}(u_{l})$, where
$d_{l}(u_{l})$ is the canonical map out of the copy of $X_{k}$ that is indexed
by $u_{l}$.

For $0\leq j\leq l$, let $s_{j}(u_{l})$ be the map that, given a morphism
$u_{l}\in I_{l}$, is defined by the composition

$X_{k}\overset{\mathrm{id}}{\longrightarrow}X_{k}\longrightarrow\textstyle{\bigvee}\,_{I_{l+1}}\,X_{k},$

where the target of $\mathrm{id}$ is the copy of $X_{k}$ that is indexed by
the morphism

$u_{l+1}=i_{0}\overset{\alpha_{1}}{\longleftarrow}\cdots\overset{\alpha_{l}}{\longleftarrow}i_{l}\overset{\mathrm{id}}{\longleftarrow}i_{l}\in
I_{l+1}.$

Then the $j$th degeneracy
$s_{j}\colon\textstyle{\bigvee}\,_{I_{l}}\,X_{k}\rightarrow\textstyle{\bigvee}\,_{I_{l+1}}\,X_{k}$
is the unique map that is induced by all the $s_{j}(u_{l})$, where
$s_{j}(u_{l})$ is the canonical map out of the copy of $X_{k}$ that is indexed
by $u_{l}$.

######  Definition 2.1.

Let $K$ be a pointed simplicial set and let $L$ be a set. Then

$K[L]=K\wedge L_{+},$

where $L_{+}$ is the constant simplicial set on $L$, together with a disjoint
basepoint. Similarly, if $Z$ is a spectrum, then $Z[L]$ is the spectrum with
each $(Z[L])_{k}$ equal to $Z_{k}\wedge L_{+}.$

Since each $u_{l}$ can be identified with an element of $G^{l},$ the $l$-fold
product of $G$, $I_{l}=G^{l}$ and

$\textstyle{\coprod}_{l}(G\rightarrow
X_{k})=\textstyle{\bigvee}\,_{G^{l}}\,X_{k}\cong
X_{k}\wedge(G^{l})_{+}=X_{k}[G^{l}].$

Thus,

$(X_{hG})_{k}\cong\mathrm{diag}(X_{k}[G^{\bullet}]),$

where the $l$-simplices of the simplicial pointed simplicial set
$X_{k}[G^{\bullet}]$ are the pointed simplicial set $X_{k}[G^{l}]$.

We introduce some notation that organizes the above, allowing us to summarize
it in a theorem.

######  Definition 2.2.

Let $X$ be a $G$-spectrum. Then $X[G^{\bullet}]$ is the simplicial spectrum
that is defined above, with $(X[G^{\bullet}])_{k}=X_{k}[G^{\bullet}]$ and
$l$-simplices equal to the spectrum $X[G^{l}]$. Thus, $X[G^{\bullet}]$ is the
simplicial spectrum

$\textstyle{X\cong
X[\ast]}$$\textstyle{X[G]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X[G^{2}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots.\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

We let $\pi_{q}(X[G^{\bullet}])$ denote the simplicial abelian group
associated to $X[G^{\bullet}]$.

######  Definition 2.3.

Let $d(Z_{\bullet})$ denote the spectrum that is the diagonal of the
simplicial spectrum $Z_{\bullet}$ (see [14, pg. 100]). Thus, for all $k\geq
0,$

$(d(Z_{\bullet}))_{k}=\mathrm{diag}((Z_{\bullet})_{k})$

and, for all $l\geq 0,$

$(d(Z_{\bullet}))_{k,l}=(Z_{l})_{k,l}.$

######  Theorem 2.4.

If $G$ is a finite group and $X$ is a $G$-spectrum, then

$X_{hG}\cong d(X[G^{\bullet}]).$

##  3\. The homotopy orbit spectrum $X_{hG}$

In this section, we use the model for the homotopy orbit spectrum that is
given in Theorem 2.4 to help us define homotopy orbits for $G$ a profinite
group. We begin with a comment about Definition 1.1.

######  Remark 3.1.

Let $X=\operatorname*{holim}_{i}X_{i}$ be an $S[[G]]$-module. The choice of
the term “$S[[G]]$-module” is motivated by the fact that, for each $i$, the
action of $G/N_{i}$ on $X_{i}$ yields a function
$G/N_{i}\rightarrow\mathrm{Hom}_{\mathrm{Spt}}(X_{i},X_{i})$ in

$\displaystyle\mathrm{Hom}_{\mathbf{Sets}}(G/N_{i},\mathrm{Hom}_{\mathrm{Spt}}(X_{i},X_{i}))$
$\displaystyle\cong\textstyle{\prod}_{G/N_{i}}\mathrm{Hom}_{\mathrm{Spt}}(X_{i},X_{i})$
$\displaystyle\cong\mathrm{Hom}_{\mathrm{Spt}}(X_{i}[G/N_{i}],X_{i})$

that corresponds to a map $X_{i}\wedge(G/N_{i})_{+}\rightarrow X_{i}$. The
analogy with actual modules over the spectrum $S[[G]]$ could be pursued
further (as Mark Behrens has done); however, for simplicity, we do not do
this.

Using the author’s preliminary work on homotopy orbits as a reference point,
the following definition was basically given by Behrens, and then later
independently formulated by the author.

######  Definition 3.2.

Let $X=\operatorname*{holim}_{i}X_{i}$ be an $S[[G]]$-module. For each $l\geq
0$, the diagrams $\\{X_{i}\\}$ and $\\{G/N_{i}\\}$ induce the diagram
$\\{X_{i}[(G/N_{i})^{l}]\\}$, so that one can form
$\operatorname*{holim}_{i}(X_{i}[(G/N_{i})^{l}])_{\mathtt{f}}$, which gives
the simplicial spectrum
$\operatorname*{holim}_{i}(X_{i}[(G/N_{i})^{\bullet}])_{\mathtt{f}}$. Then we
define $X_{hG}$, the homotopy orbit spectrum of $X$ with respect to the
$G$-action, to be the spectrum

$X_{hG}=d(\operatorname*{holim}_{i}(X_{i}[(G/N_{i})^{\bullet}])_{\mathtt{f}}).$

Henceforth, we use the notation $X_{h^{\prime}G}$ to denote the classical
homotopy orbit spectrum $\operatorname*{hocolim}_{G}X$, when $G$ is finite.

The definition of the homotopy orbit spectrum comes from imitating the model
in Theorem 2.4 and from the demands of the homotopy orbit spectral sequence
(see the proof of Theorem 5.3). For example, the reader might expect that the
definition involve a homotopy colimit of $G$-spectra, instead of the homotopy
limit of such; however, the homotopy limit is necessitated by the nature of
continuous group homology, which is the $E_{2}$-term of the homotopy orbit
spectral sequence.

######  Remark 3.3.

The functor $(-)_{\mathtt{f}}$ appears in Definition 3.2 so that the homotopy
limit is well-behaved. Let us suppose that the functor $(-)_{\mathtt{f}}$ does
not appear in the definition of $X_{hG}$, so that it reads instead as
${X_{hG}=d(\operatorname*{holim}_{i}X_{i}[(G/N_{i})^{\bullet}])}.$ Recall that
$X_{i}[(G/N_{i})^{\bullet}]$ is involved in the construction of
$(X_{i})_{hG/N_{i}}\cong d(X_{i}[(G/N_{i})^{\bullet}])$, by Theorem 2.4. Thus,
one might wonder if there is an isomorphism

$d(\operatorname*{holim}_{i}X_{i}[(G/N_{i})^{\bullet}])\cong\operatorname*{holim}_{i}d(X_{i}[(G/N_{i})^{\bullet}]),$

so that, morally, $X_{hG}=\operatorname*{holim}_{i}(X_{i})_{hG/N_{i}}.$
However, it is not hard to see that this is not the case, since

$\displaystyle(d(\operatorname*{holim}_{i}X_{i}[(G/N_{i})^{\bullet}]))_{k,l}$
$\displaystyle=(\mathrm{diag}(\operatorname*{holim}_{i}(X_{i})_{k}[(G/N_{i})^{\bullet}]))_{l}$
$\displaystyle=(\operatorname*{holim}_{i}(X_{i})_{k}[(G/N_{i})^{l}])_{l}$
$\displaystyle=\mathrm{Hom}_{\mathcal{S}^{(\\{i\\}^{\mathrm{op}})}}(\Delta[l]\times(\\{i\\}^{\mathrm{op}})/-,\\{(X_{i})_{k}[(G/N_{i})^{l}]\\}_{i})$

(see [2, Chapter XI, 2.2, 3.2]), where $\mathcal{S}$ is the category of
simplicial sets, is not equivalent to

$\displaystyle(\operatorname*{holim}_{i}d(X_{i}[(G/N_{i})$
${}^{\bullet}]))_{k,l}=(\operatorname*{holim}_{i}\mathrm{diag}((X_{i})_{k}[(G/N_{i})^{\bullet}]))_{l}$
$\displaystyle=\mathrm{Hom}_{\mathcal{S}^{(\\{i\\}^{\mathrm{op}})}}(\Delta[l]\times(\\{i\\}^{\mathrm{op}})/-,\\{\mathrm{diag}((X_{i})_{k}[(G/N_{i})^{\bullet}])\\}_{i}).$

We recall the following useful result.

######  Theorem 3.4 ([14, Corollary 4.22]).

If $X_{\bullet}$ is a simplicial spectrum, then there is a spectral sequence

$E_{2}^{p,q}=H_{p}(\pi_{q}(X_{\ast}))\Rightarrow\pi_{p+q}(d(X_{\bullet})),$

where $H_{\ast}(\pi_{q}(X_{\ast}))$ is the homology of the Moore complex of
the simplicial abelian group $\pi_{q}(X_{\bullet}).$

######  Lemma 3.5.

Let $X_{\bullet}\rightarrow Y_{\bullet}$ be a map between simplicial spectra,
such that, for each $n\geq 0$, the map $X_{n}\rightarrow Y_{n}$ is a weak
equivalence between the $n$-simplices. Then the induced map
$d(X_{\bullet})\rightarrow d(Y_{\bullet})$ is a weak equivalence of spectra.

###### Proof.

There is a spectral sequence

$H_{p}(\pi_{q}(X_{\ast}))\Rightarrow\pi_{p+q}(d(X_{\bullet})),$

and a map to the spectral sequence

$H_{p}(\pi_{q}(Y_{\ast}))\Rightarrow\pi_{p+q}(d(Y_{\bullet})).$

Since $\pi_{q}(X_{n})\cong\pi_{q}(Y_{n})$, for each $n\geq 0$, and
$\pi_{q}(X_{\ast})$ and $\pi_{q}(Y_{\ast})$ are chain complexes, there is an
isomorphism
$H_{p}(\pi_{q}(X_{\ast}))\overset{\cong}{\rightarrow}H_{p}(\pi_{q}(Y_{\ast}))$
of $E_{2}$-terms. Therefore, the abutments of the above two spectral sequences
are isomorphic, giving the conclusion of the lemma. ∎

######  Remark 3.6.

Let $G$ be finite and let $X$ be any $G$-spectrum. Let $\\{i\\}=\\{0\\},$
$N_{0}=\\{e\\},$ and $X_{0}=X_{\mathtt{f}}$. Then
$\operatorname*{holim}_{\\{0\\}}X_{0}\cong X_{\mathtt{f}}$ is an
$S[[G]]$-module and

$(\operatorname*{holim}_{\\{0\\}}X_{0})_{hG}=d(\operatorname*{holim}_{\\{0\\}}(X_{\mathtt{f}}[G^{\bullet}])_{\mathtt{f}})\cong
d((X_{\mathtt{f}}[G^{\bullet}])_{\mathtt{f}}).$

Thus, there is a weak equivalence

$X_{h^{\prime}G}\cong
d(X[G^{\bullet}])\overset{\simeq}{\longrightarrow}d((X_{\mathtt{f}}[G^{\bullet}])_{\mathtt{f}})\cong(\operatorname*{holim}_{\\{0\\}}X_{0})_{hG},$

so that Definition 3.2 recovers the classical definition of homotopy orbits.

Now we show that $(-)_{hG}$ preserves weak equivalences of $S[[G]]$-modules,
as defined below.

######  Definition 3.7.

Let $\\{X_{i}\\}\rightarrow\\{Y_{i}\\}$ be a natural transformation of
diagrams of $G$-spectra, such that $X_{i}\rightarrow Y_{i}$ is a weak
equivalence, for each $i$, and the induced map
$X=\operatorname*{holim}_{i}X_{i}\rightarrow\operatorname*{holim}_{i}Y_{i}=Y$
is a map between $S[[G]]$-modules. Then we say that the weak equivalence
$X\rightarrow Y$ is a weak equivalence of $S[[G]]$-modules.

######  Theorem 3.8.

If $X\rightarrow Y$, as in Definition 3.7, is a weak equivalence of
$S[[G]]$-modules, then $X_{hG}\rightarrow Y_{hG}$ is a weak equivalence.

###### Proof.

Let $i\geq 0$. Since $X_{i}\rightarrow Y_{i}$ is a weak equivalence, the
induced map

$(X_{i}[(G/N_{i})^{l}])_{\mathtt{f}}\rightarrow(Y_{i}[(G/N_{i})^{l}])_{\mathtt{f}}$

is a weak equivalence between fibrant spectra, for each $l\geq 0$. Thus,

$\operatorname*{holim}_{i}(X_{i}[(G/N_{i})^{l}])_{\mathtt{f}}\rightarrow\operatorname*{holim}_{i}(Y_{i}[(G/N_{i})^{l}])_{\mathtt{f}}$

is a weak equivalence, so that, by Lemma 3.5, $X_{hG}\rightarrow Y_{hG}$ is a
weak equivalence. ∎

##  4\. Examples of $S[[G]]$-modules

In this section, we consider a way that $S[[G]]$-modules arise naturally from
discrete right $G$-spectra.

The following recollection will be helpful. There is a functor
$(-)_{c}\colon\mathrm{Spt}\rightarrow\mathrm{Spt},$ such that, given $Y$ in
$\mathrm{Spt}$, $Y_{c}$ is a cofibrant spectrum, and there is a natural
transformation $(-)_{c}\rightarrow\mathrm{id}_{\mathrm{Spt}},$ such that, for
any $Y$, the map $Y_{c}\rightarrow Y$ is a trivial fibration. For example, if
$Y$ is a right $K$-spectrum for some group $K$, then $Y_{c}$ is also a right
$K$-spectrum, and the map $Y_{c}\rightarrow Y$ is $K$-equivariant.

Let $Z$ be any spectrum and let $X$ be a discrete right $G$-spectrum. Then

$F(X,Z)\simeq F(\operatorname*{colim}_{i}X^{N_{i}},Z_{\mathtt{f}})\simeq
F(\operatorname*{hocolim}_{i}X^{N_{i}},Z_{\mathtt{f}})\simeq\operatorname*{holim}_{i}F((X^{N_{i}})_{c},Z_{\mathtt{f}}).$

Thus, when $X$ is a discrete right $G$-spectrum, we identify the left
$G$-spectrum $F(X,Z)$ with the $G$-spectrum
$\operatorname*{holim}_{i}F((X^{N_{i}})_{c},Z_{\mathtt{f}}).$ Under this
identification, we make the following observation.

######  Theorem 4.1.

If $X$ is a discrete right $G$-spectrum, then $F(X,Z)$ is an $S[[G]]$-module.

###### Proof.

The spectrum $(X^{N_{i}})_{c}$ is a right $G/N_{i}$-spectrum, because
$X^{N_{i}}$ is a right $G/N_{i}$-spectrum. Thus,
$F((X^{N_{i}})_{c},Z_{\mathtt{f}})$ is a $G/N_{i}$-spectrum. Also, since the
source and target of the function spectrum $F((X^{N_{i}})_{c},Z_{\mathtt{f}})$
are cofibrant and fibrant, respectively, the function spectrum itself is
fibrant. These facts imply that the spectrum
$\operatorname*{holim}_{i}F((X^{N_{i}})_{c},Z_{\mathtt{f}})$ is an
$S[[G]]$-module. ∎

We give two examples of $F(X,Z)$ as an $S[[G]]$-module.

######  Example 4.2.

Recall from [3] that, if $cS^{0}$ is the Brown-Comenetz dual of $S^{0}$, then
$cX$, the Brown-Comenetz dual of $X$, is $F(X,cS^{0})$. Thus, if $X$ is a
discrete right $G$-spectrum, then

$cX=\operatorname*{holim}_{i}F((X^{N_{i}})_{c},(cS^{0})_{\mathtt{f}})$

is an $S[[G]]$-module.

The following example is due to Mark Behrens.

######  Example 4.3.

If $X$ is a discrete right $G$-spectrum, then the Spanier-Whitehead dual of
$X$,

$DX=F(X,S^{0})=\operatorname*{holim}_{i}F((X^{N_{i}})_{c},(S^{0})_{\mathtt{f}}),$

is an $S[[G]]$-module.

##  5\. The homotopy orbit spectral sequence for an $S[[G]]$-module

In this section, we construct a homotopy orbit spectral sequence for a certain
type of $S[[G]]$-module. We attempted to construct such a spectral sequence
for any profinite group $G$. However, there were difficulties that we were
able to get around only by putting an additional hypothesis on the group $G$,
that is, by assuming that $G$ is countably based.

Now we recall the construction of the classical homotopy orbit spectral
sequence for $X_{h^{\prime}G}$; the brief analysis below of its $E_{2}$-term
will be useful. Thus, for the duration of this paragraph, let $G$ be a finite
group and let $X$ be a $G$-spectrum. By Theorem 3.4, there is a spectral
sequence

$E_{2}^{p,q}\Rightarrow\pi_{p+q}(d(X[G^{\bullet}]))=\pi_{p+q}(X_{h^{\prime}G}),$

where

(5.1) $E_{2}^{p,q}=H_{p}(\pi_{q}(X[G^{\ast}]))\cong H_{p}(G,\pi_{q}(X)),$

the $p$th group homology of $G$, with coefficients in $\pi_{q}(X)$ (see, for
example, [14, (7.9)]). Note that, by (5.1),

(5.2) $H_{p}(G,\pi_{q}(X))\cong H_{p}(\pi_{q}(X)[G^{\ast}]),$

where, if $A$ is an abelian group and $K$ is a group, we use $A[K]$ to denote
$\bigoplus_{K}A.$

Now we are ready to construct the homotopy orbit spectral sequence for a
countably based profinite group.

######  Theorem 5.3.

Let $G$ be a countably based profinite group, and let

$N_{0}\geq N_{1}\geq N_{2}\geq\cdots$

be a chain of open normal subgroups of $G$, such that $G\cong\lim_{i}G/N_{i}$.
Also, let

$X_{0}\leftarrow X_{1}\leftarrow X_{2}\leftarrow\cdots$

be a tower of $G$-spectra and $G$-equivariant maps such that
$X=\operatorname*{holim}_{i}X_{i}$ is an $S[[G]]$-module. If, for all $i$,
$X_{i}$ is an $f$-spectrum, then there is a spectral sequence of the form

$E_{2}^{p,q}\cong H^{c}_{p}(G,\pi_{q}(X))\Rightarrow\pi_{p+q}(X_{hG}),$

where the $E_{2}$-term is the continuous homology of $G$ with coefficients in
the graded profinite $\widehat{\mathbb{Z}}[[G]]$-module $\pi_{\ast}(X)$.

###### Proof.

By Theorem 3.4, there is a spectral sequence

$E_{2}^{p,q}\Rightarrow\pi_{p+q}(d(\operatorname*{holim}_{i}(X_{i}[(G/N_{i})^{\bullet}])_{\mathtt{f}}))=\pi_{p+q}(X_{hG}),$

where

$E_{2}^{p,q}=H_{p}(\pi_{q}(\operatorname*{holim}_{i}(X_{i}[(G/N_{i})^{\ast}])_{\mathtt{f}})).$

We spend the rest of the proof identifying the $E_{2}$-term as continuous
group homology.

Let $l\geq 0$. Since $\\{\pi_{q+1}(X_{i})[(G/N_{i})^{l}]\\}$ is a tower of
finite abelian groups,

$\operatorname*{lim^{1}}_{i}\pi_{q+1}(X_{i})[(G/N_{i})^{l}]=0.$

Thus,

$\pi_{q}(\operatorname*{holim}_{i}(X_{i}[(G/N_{i})^{l}])_{\mathtt{f}})\cong\lim_{i}\pi_{q}(X_{i})[(G/N_{i})^{l}],$

which implies that

$E_{2}^{p,q}\cong H_{p}(\lim_{i}(\pi_{q}(X_{i})[(G/N_{i})^{\ast}])).$

Since, for any $l\geq 0$, the tower of abelian groups
$\\{\pi_{q}(X_{i})[(G/N_{i})^{l}]\\}_{i}$ satisfies the Mittag-Leffler
condition, by [23, Theorem 3.5.8], there is a short exact sequence

$0\rightarrow\operatorname*{lim^{1}}_{i}H_{p+1}(\pi_{q}(X_{i})[(G/N_{i})^{\ast}])\rightarrow
E_{2}^{p,q}\rightarrow\lim_{i}H_{p}(\pi_{q}(X_{i})[(G/N_{i})^{\ast}])\rightarrow
0.$

By (5.2), this short exact sequence can be written as

$0\rightarrow\operatorname*{lim^{1}}_{i}H_{p+1}(G/N_{i},\pi_{q}(X_{i}))\rightarrow
E_{2}^{p,q}\rightarrow\lim_{i}H_{p}(G/N_{i},\pi_{q}(X_{i}))\rightarrow 0.$

Since $G/N_{i}$ and $\pi_{q}(X_{i})$ are finite,
$H_{p+1}(G/N_{i},\pi_{q}(X_{i}))$ is a finite abelian group, for each $p\geq
0$ ([23, Corollary 6.5.10]). Thus,
$\operatorname*{lim^{1}}_{i}H_{p+1}(G/N_{i},\pi_{q}(X_{i}))=0$. Therefore, we
obtain that

$E_{2}^{p,q}\cong\lim_{i}H_{p}(G/N_{i},\pi_{q}(X_{i}))\cong
H^{c}_{p}(G,\pi_{q}(X)),$

where the last isomorphism uses [19, Proposition 6.5.7]. ∎

##  6\. Eilenberg-Mac Lane spectra and their homotopy orbits

Let the profinite group $G$ be countably based, with a chain $\\{N_{i}\\}$ of
open normal subgroups, such that $G\cong\lim_{i}G/N_{i}$, and, as defined in
the Introduction, let $\\{A_{i}\\}$ be a nice tower of $G$-modules. Let
$\Gamma\colon\mathbf{Ch}_{+}\rightarrow s\mathbf{Ab}$ be the functor in the
Dold-Kan correspondence from $\mathbf{Ch}_{+}$, the category of chain
complexes $C_{\ast}$ with $C_{n}=0$ for $n<0$, to $s\mathbf{Ab}$, the category
of simplicial abelian groups (see, for example, [11, Chapter III, Corollary
2.3]). Also, if $A$ is an abelian group, let $A[-n]$ be the chain complex that
is $A$ in degree $n$ and zero elsewhere.

Given the tower $\\{A_{i}\\}$, we explain how to form $\\{H(A_{i})\\}$, a
tower of Eilenberg-Mac Lane spectra (we follow the construction given in
[15]), so that $\operatorname*{holim}_{i}H(A_{i})$ is an $S[[G]]$-module. By
functoriality, for each $k\geq 0$, $\\{\Gamma(A_{i}[-k])\\}$ is a tower of
simplicial $G$-modules and $G$-equivariant maps, such that, for each $i$,
$\Gamma(A_{i}[-k])$ is the Eilenberg-Mac Lane space $K(A_{i},k)$ and
$\Gamma(A_{i}[-k])$ is a simplicial $G/N_{i}$-module. Furthermore, by taking
$0$ as the basepoint, each $\Gamma(A_{i}[-k])$ is a pointed simplicial set.

For each $i$, we define the Eilenberg-Mac Lane spectrum $H(A_{i})$ by
$(H(A_{i}))_{k}=\Gamma(A_{i}[-k]),$ so that $\pi_{0}(H(A_{i}))=A_{i}$ and
$\pi_{n}(H(A_{i}))=0$, when $n\neq 0$. Then, by functoriality,
$\\{H(A_{i})\\}$ is a tower of $G$-spectra and $G$-equivariant maps, such that
each $H(A_{i})$ is a $G/N_{i}$-spectrum. Since each $(H(A_{i}))_{k}$ is a
fibrant simplicial set and each $H(A_{i})$ is an $\Omega$-spectrum (see, for
example, [15, Example 21]), $H(A_{i})$ is a fibrant spectrum. These facts
imply the following result.

######  Lemma 6.1.

The spectrum $\operatorname*{holim}_{i}H(A_{i})$ is an $S[[G]]$-module.

Now we show that the homotopy orbit spectral sequence can be used to compute
$\pi_{\ast}((\operatorname*{holim}_{i}H(A_{i}))_{hG})$.

######  Theorem 6.2.

If $G$ is a countably based profinite group with a chain $\\{N_{i}\\}$ of open
normal subgroups, such that $G\cong\lim_{i}G/N_{i}$, and, if $\\{A_{i}\\}$ is
a nice tower of $G$-modules, then

$\pi_{\ast}((\operatorname*{holim}_{i}H(A_{i}))_{hG})\cong
H^{c}_{\ast}(G,\lim_{i}A_{i}).$

###### Proof.

By hypothesis, each $H(A_{i})$ is an $f$-spectrum. Then, by Theorem 5.3, there
is a homotopy orbit spectral sequence

$E_{2}^{p,q}\cong
H^{c}_{p}(G,\pi_{q}(\operatorname*{holim}_{i}H(A_{i})))\Rightarrow\pi_{p+q}((\operatorname*{holim}_{i}H(A_{i}))_{hG}).$

Since $\operatorname*{lim^{1}}_{i}A_{i}=0$, we have:

$\displaystyle E_{2}^{p,q}$ $\displaystyle\cong
H^{c}_{p}(G,\lim_{i}\pi_{q}(H(A_{i})))=\begin{cases}H^{c}_{p}(G,0)&\mathrm{if}\
q\neq 0\\\ H^{c}_{p}(G,\lim_{i}A_{i})&\mathrm{if}\ q=0\end{cases}$
$\displaystyle=\begin{cases}0&\mathrm{if}\ q\neq 0\\\
H^{c}_{p}(G,\lim_{i}A_{i})&\mathrm{if}\ q=0.\end{cases}$

Thus, the spectral sequence collapses, giving the conclusion of the theorem. ∎

##  7\. The $G_{n}$-homotopy orbits of $F(E_{n},L_{K(n)}(S^{0}))$

Our main motivation for constructing homotopy orbits for a profinite group was
that we were interested in constructing the $G_{n}$-homotopy orbit spectrum of
$E_{n}$, as part of our effort to understand [12]. Though we have not been
able to construct such a spectrum, we are able to construct the
$G_{n}$-homotopy orbits of the closely related spectrum

$F(E_{n},L_{K(n)}(S^{0}))\simeq\Sigma^{-n^{2}}E_{n},$

where the weak equivalence applies [20, Proposition 16] and the function
spectrum $F(E_{n},L_{K(n)}(S^{0}))$ is the $K(n)$-local Spanier-Whitehead dual
of $E_{n}$. After explaining why we have not been able to construct a homotopy
orbit spectrum for $E_{n}$, we consider $F(E_{n},L_{K(n)}(S^{0}))_{hG_{n}}$.

Recall from the Introduction that, since $G_{n}$ is a compact $\ell$-adic
analytic group, it is a countably based profinite group. Then, following [6,
(1.4)], we fix a chain $G_{n}=N_{0}\gneq N_{1}\gneq N_{2}\gneq\cdots$ of open
normal subgroups of $G_{n}$, such that $\bigcap_{i}N_{i}=\\{e\\}$, so that
$G_{n}\cong\lim_{i}G_{n}/N_{i}$. Thus, an $S[[G_{n}]]$-module comes from a
tower $X_{0}\leftarrow X_{1}\leftarrow X_{2}\leftarrow\cdots$ of
$G_{n}$-spectra and $G_{n}$-equivariant maps.

To form the $G_{n}$-homotopy orbit spectrum of $E_{n}$, we have to show that
$E_{n}$ is an $S[[G_{n}]]$-module. However, we do not know how to do this and
we explain how a natural way to try to do this fails. Let $E_{n}^{dhN_{i}}$ be
the spectrum constructed by Devinatz and Hopkins (see [6]) that behaves like
the $N_{i}$-homotopy fixed points of $E_{n}$ with respect to a continuous
$N_{i}$-action (see [4]). Let

$M_{I_{0}}\leftarrow M_{I_{1}}\leftarrow M_{I_{2}}\leftarrow\cdots$

be a tower of generalized Moore spectra such that

$\operatorname*{holim}_{j}(L_{K(n)}(M_{I_{j}}))_{\mathtt{f}}\simeq
L_{K(n)}(S^{0})$

([13, Proposition 7.10 (e)]). Also, recall from [4, Theorem 6.6] that

$E_{n}\simeq\operatorname*{holim}_{j}(\operatorname*{colim}_{i}(E_{n}^{dhN_{i}}\wedge
M_{I_{j}})_{\mathtt{f}}).$

Thus, one might attempt to present $E_{n}$ as an $S[[G_{n}]]$-module by
considering the tower $\\{\operatorname*{colim}_{i}(E_{n}^{dhN_{i}}\wedge
M_{I_{j}})_{\mathtt{f}}\\}_{j}.$ However, by [4, Lemma 6.2], there is no open
normal subgroup $N$ of $G_{n}$ such that the $G_{n}$-action on
$\operatorname*{colim}_{i}(E_{n}^{dhN_{i}}\wedge M_{I_{j}})_{\mathtt{f}}$
factors through $G_{n}/N$. Thus, this attempt fails to present $E_{n}$ as an
$S[[G_{n}]]$-module.

We noted above that $F(E_{n},L_{K(n)}(S^{0}))\simeq\Sigma^{-n^{2}}E_{n}.$
Thus, one might like to assert that, since we can construct the
$G_{n}$-homotopy orbit spectrum of $F(E_{n},L_{K(n)}(S^{0}))$, as done below,
then we can also construct the homotopy orbit spectrum of $E_{n}$, by
considering the homotopy orbits of $\Sigma^{n^{2}}F(E_{n},L_{K(n)}(S^{0}))$.
However, we have been informed by Mark Behrens that, in general, this can not
be done, because the equivalence
$F(E_{n},L_{K(n)}(S^{0}))\simeq\Sigma^{-n^{2}}E_{n}$ need not be
$G_{n}$-equivariant. According to Behrens, the $S^{-n^{2}}$ in
$\Sigma^{-n^{2}}E_{n}$, when properly interpreted, can have a non-trivial
$G_{n}$-action on it.

To form $F(E_{n},L_{K(n)}(S^{0}))_{hG_{n}}$, we only need to show that
$F(E_{n},L_{K(n)}(S^{0}))$ is an $S[[G_{n}]]$-module. As in [4], let
$F_{n}=\operatorname*{colim}_{i}E_{n}^{dhN_{i}}$. Then

$\displaystyle F(E_{n},L_{K(n)}(S^{0}))$ $\displaystyle\simeq
F(L_{K(n)}(F_{n}),\operatorname*{holim}_{j}(L_{K(n)}(M_{I_{j}}))_{\mathtt{f}})$
$\displaystyle\simeq
F(F_{n},\operatorname*{holim}_{j}(L_{K(n)}(M_{I_{j}}))_{\mathtt{f}})$
$\displaystyle\simeq
F(\operatorname*{hocolim}_{i}(E_{n}^{dhN_{i}})_{c},\operatorname*{holim}_{j}(L_{K(n)}(M_{I_{j}}))_{\mathtt{f}})$
$\displaystyle\cong\operatorname*{holim}_{i}\operatorname*{holim}_{j}F((E_{n}^{dhN_{i}})_{c},(L_{K(n)}(M_{I_{j}}))_{\mathtt{f}})$
$\displaystyle\simeq\operatorname*{holim}_{i}F((E_{n}^{dhN_{i}})_{c},(L_{K(n)}(M_{I_{i}}))_{\mathtt{f}}),$

where the first weak equivalence applies [6, Definition 1.5] and [4, Theorem
6.3], and the last weak equivalence uses the fact that $\\{(i,i)\\}_{i}$ is
cofinal in $\\{(i,j)\\}_{i,j}$. Thus, we make the identification

$F(E_{n},L_{K(n)}(S^{0}))=\operatorname*{holim}_{i}F((E_{n}^{dhN_{i}})_{c},(L_{K(n)}(M_{I_{i}}))_{\mathtt{f}}).$

Since $E_{n}^{dhN_{i}}$ is a right $G_{n}/N_{i}$-spectrum, as in Section 4,
the above identification implies the following result.

######  Lemma 7.1.

The spectrum $F(E_{n},L_{K(n)}(S^{0}))$ is an $S[[G_{n}]]$-module.

The next result allows us to build the homotopy orbit spectral sequence for
$F(E_{n},L_{K(n)}(S^{0}))_{hG_{n}}$.

######  Lemma 7.2.

For each $i\geq 0$, the spectrum $F(E_{n}^{dhN_{i}},L_{K(n)}(M_{I_{i}}))$ is
an $f$-spectrum.

###### Proof.

Our proof follows [5, proof of Lemma 3.5]. Since

$L_{K(n)}(M_{I_{i}})\simeq L_{K(n)}(S^{0})\wedge M_{I_{i}}\simeq
E_{n}^{dhG_{n}}\wedge M_{I_{i}}$

(see [6, Theorem 1(iii)]), [6, Theorem 2(ii)] gives a strongly convergent
$K(n)$-local $E_{n}$-Adams spectral sequence

$H^{s}_{c}(G_{n},(E_{n}\wedge
M_{I_{i}})^{-t}(E_{n}^{dhN_{i}}))\Rightarrow\pi_{t-s}(F(E_{n}^{dhN_{i}},L_{K(n)}(M_{I_{i}}))),$

where the $E_{2}$-term is continuous cohomology for profinite continuous
$\mathbb{Z}_{\ell}[[G_{n}]]$-modules (see [6, Remark 1.3]). By [9, Proposition
2.5],

$(E_{n}\wedge
M_{I_{i}})^{-t}(E_{n}^{dhN_{i}})\cong\textstyle\bigoplus_{G_{n}/N_{i}}\pi_{t}(E_{n}\wedge
M_{I_{i}})$

is a finite abelian group. Thus, by [21, Proposition 4.2.2], the abelian group
$H^{s}_{c}(G_{n},(E_{n}\wedge M_{I_{i}})^{-t}(E_{n}^{dhN_{i}}))$ is finite.
The fact that $E_{\infty}^{\ast,\ast}$ has a horizontal vanishing line (see
[1, Theorem 6.10], [5, Proposition 2.3, proof of Lemma 3.5], and [6,
Proposition A.3]) implies that
$\pi_{\ast}(F(E_{n}^{dhN_{i}},L_{K(n)}(M_{I_{i}})))$ is finite in each degree.
∎

Theorem 5.3 and Lemmas 7.1 and 7.2 give the following.

######  Theorem 7.3.

There is a spectral sequence

$H^{c}_{p}(G_{n},\pi_{q}(F(E_{n},L_{K(n)}(S^{0}))))\Rightarrow\pi_{p+q}(F(E_{n},L_{K(n)}(S^{0}))_{hG_{n}}).$

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