# Charge Order Stabilized Quantum Spin Liquid in Hollandite K2V8O16

Ola Kenji Forslund<EMAIL_ADDRESS>Elisabetta Nocerino  Department of Applied
Physics, KTH Royal Institute of Technology, SE-106 91 Stockholm, Sweden
Masahiko Isobe  Max Planck Institute for Solid State Research,
Heisenbergstraße 1, 70569 Stuttgart, Germany  Daniel Andreica  Faculty of
Physics, Babes-Bolyai University, 400084 Cluj-Napoca, Romania  Stephen
Cottrell  ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot Oxon
OX11 0QX, United Kingdom  Hidenori Takagi  Max Planck Institute for Solid
State Research, Heisenbergstraße 1, 70569 Stuttgart, Germany  Yasmine Sassa
Department of Physics, Chalmers University of Technology, SE-41296 Göteborg,
Sweden  Jun Sugiyama  Neutron Science and Technology Center, Comprehensive
Research Organization for Science and Society (CROSS),Tokai, Ibaraki 319-1106,
Japan  Martin Månsson<EMAIL_ADDRESS>Department of Applied Physics, KTH Royal
Institute of Technology, SE-106 91 Stockholm, Sweden

###### Abstract

Quantum spin liquid is an elusive state that display strong many-body
entanglement with potential applications in future quantum computing. This
study reports muon spin relaxation ($\mu^{+}$SR) measurements on a novel high-
pressure synthesized material, the Hollandite K2V8O16. In this quasi-one-
dimensional compound, charge ordering (CO) at $T_{\rm MIT}\approx 160$ K
effectively isolates half of the vanadium chains and model-like Heisenberg
spin-1/2 chains are realized. Our zero field $\mu^{+}$SR measurements show
exponential like relaxation down to the lowest temperature $T=100$ mK and the
absence of long range ordering is confirmed. The relaxation rate is found to
be temperature independent below $T_{\rm QSL}\approx 2$ K and measurements in
longitudinal field confirms a highly dynamic ground state. These results
represents the first confirmation of quantum spin liquid (QSL) behavior within
the Hollandite family, stabilized by the CO. Finally, the presence of strong
local electron correlation and one dimensional Fermi surface suggest this QSL
to be a gapless Tomonaga-Luttinger liquid (TLL), which here uniquely presents
itself in a stoichiometric compound under zero applied magnetic field and at
ambient pressure.

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††preprint: APS/PRB

Low dimensional magnetism is a field that has developed tremendously over the
last decades. From theoretical point of view, exactly solvable solution were
more easily obtained without the need to consider the complicated models in 3D
Bethe (1931). It was later realised that these model were not only purely
theoretical but some compound actually exhibit signatures of low dimensional
magnetism. This has driven low dimensional spin systems to be an ideal
playground in order to study and find new states of matter and general
concepts of many body physics, given that quantum fluctuations are more
prominent in low dimensional systems Parkinson and Farnell (2010); Diep  _et
al._ (2013).

One of the more elusive quantum states is the so-called quantum spin liquid
(QSL) that was first proposed by Anderson Anderson (1973). The QSL embodies
strong entanglement that generates a virtual smörgåsbord of unique physical
properties that are interesting for both fundamental understanding Savary and
Balents (2016) as well as future applications in quantum computing and
spintronics Tokura  _et al._ (2017). In general, QSL may arise from
geometrical frustration in higher dimensions and result in resonance valance
bonds liquids or possible Kitaev quantum spin liquid Winter  _et al._ (2017)
in honeycomb lattices Kanyolo  _et al._ (2021). For 1D systems however, spinon
excitation were clarified for antiferromagnetic spin-1/2 chains Faddeev and
Takhtajan (1981); Takhtajan (1982) while Haldane Haldane (1983a, b) discovered
the fundamental difference between integer and half integer spin chains.
Realisation of Spin-1 chains include CsNiF3Ramirez and Wolf (1982), which
demonstrated magnetic solitons. KCuF3 Tennant  _et al._ (1993) and Sr2CuO3
Zaliznyak  _et al._ (2004) on the other hand have shown to be a realisation of
an antiferromagnetic spin-1/2 chains for which inelastic neutron scattering
confirmed a QSL via the presence of spinon excitations.

Figure 1: (a) The CO pattern revealed by XRD measurements presented in Ref.
Komarek  _et al._ , 2011. It is composed of dimerized V3+ (orange, \small{1}⃝)
and V4+ (blue, \small{2}⃝) chains and a non-dimerzied V4+ (green, \small{3}⃝)
chains. (b) DC-Magnetic susceptibility measured in ZFC protocol with $H=1$ kOe
revealing the 50% decrease with the CO formation at $T_{\rm MIT}\approx 160$ K
Isobe  _et al._ (2006, 2008). (c) Combining the XRD results Komarek  _et al._
(2011) with theoretical predictions of spin singlet formation suggest that the
\small{3}⃝ chains are effectively isolated spin-1/2 chains [see also panel
(a)], which are available to form the QSL state (as confirmed by the current
$\mu^{+}$SR investigation).

In search for novel physical properties and phases, the use of high-pressure
synthesis is a effective route for stabilizing otherwise inaccessible crystal
structures and materials. Among such compounds we find K2V8O16, which belongs
to the Hollandite family. This group of materials has attracted a lot of
attention during the recent years for their intriguing electronic and magnetic
properties, both under pressure Yamauchi  _et al._ (2011, 2015); Forslund  _et
al._ (2018, 2019) and ambient conditions Shimizu  _et al._ (2011); Sugiyama
_et al._ (2012); Ohta  _et al._ (2012); Toriyama  _et al._ (2012); Bhobe  _et
al._ (2015); Kim  _et al._ (2016). Hollandites can be described by a general
chemical formula AxM8O16 (A = alkali/alkaline-earth metal, x = 1-2, and M =
transition metal) with edge shared M2O6 octahedra zigzag double chain
structure [see Fig. 1(a)]. The chains run along the crystallographic c-axis
and are connected via corner shared oxygen atoms and with the A cation at the
tunnel site Tamada  _et al._ (1996). K2V8O16 undergoes $T_{\rm MIT}=160$ K,
driven by a charge order (CO) formation Isobe  _et al._ (2006); Horiuchi  _et
al._ (2008), which is also visible in the bulk magnetic susceptibility that
display a sudden and strong reduction (but not to zero!) at $T_{\rm MIT}=160$
K [Fig. 1(b)]. Previous zero field (ZF) muon spin relaxation ($\mu^{+}$SR)
measurement suggested that this CO formation is accompanied with imperfect
spin singlet formation, $i.e.$ some V spins are still unpaired even at lower
temperatures (2 K) Chow  _et al._ (2012). Indeed, single crystal X-ray
diffraction (XRD) Komarek  _et al._ (2011) observed a dimerization in half of
the V chains [chains \small{1}⃝ and \small{2}⃝ in Fig. 1(a,c)], causing a 50%
decrease in the magnetic susceptibility [Fig. 1(b)] due to the formation of
spin singlets Isobe  _et al._ (2006); Komarek  _et al._ (2011); Kim  _et al._
(2016), followed by a structural change from a tetragonal to a monoclinic
symmetry. Consequently, the remaining chains [\small{3}⃝ in Fig. 1(a,c)] can
be considered as isolated spin$-1/2$ Heisenberg chains, sustained within a
’sea’ of spin-dimerized chains Kim  _et al._ (2016).

In order to elucidate the detailed properties and behavior of the proposed
isolated \small{3}⃝-chains we have conducted a muon spin relaxation
($\mu^{+}$SR) study of the K2V8O16 compound down to $100$ mK. $\mu^{+}$SR is a
very sensitive technique for both long- and short-range correlations in both
static and dynamical forms. Clearly, $\mu^{+}$SR is a local probe and lacks
the detailed Q-resolution of neutron scattering. However, the muon’s larger
gyromagnetic ratio gives it a uniquely high sensitivity to even small magnetic
moments using small sample volumes. Further, similar to neutron scattering,
$\mu^{+}$SR is also able to conduct investigations at both ultra-low
temperatures as well as under zero-applied magnetic fields. Hence, $\mu^{+}$SR
is an optimal technique to investigate subtle spin properties (like the QSL)
also in powder samples available in small amounts yielded by high-pressure
synthesis.

The current powder samples of K2V8O16 were synthesized through a high-
pressure, high-temperature, solid state reaction of KVO3, V2O3 and V2O5, which
is described in greater detail in Ref. Isobe  _et al._ , 2006. The $\mu^{+}$SR
measurements were carried out at the MUSR surface muon beamline at ISIS pulsed
muon source, UK. Approximately 600 mg of sample was mounted onto a silver
holder and a dilution fridge cryostat was utilized to reach temperatures down
to $100$ mK. The $\mu^{+}$SR data has been analysed with the musrfit A. Suter
and B. M. Wojek (2012) software suite.

Zero field (ZF) $\mu^{+}$SR time spectra for selected temperatures are
displayed in Fig. 2. At 4 K, an exponentially damped Kubo-Toyabe (KT)
depolarisation is observed, which become more exponential-like at lower
temperatures. Therefore, the fitting function was divided into two temperature
regions: for $T>2$ K, the measured ZF time spectrum was fitted with an
exponentially relaxing static KT together with a background component to
account for the sample holder, whereas a stretched exponential together with
the same sample holder contribution was fitted for $T\leq 2~{}$K:

$\displaystyle A_{0}\,P_{\rm ZF}(t)=A_{\rm KT}G^{\rm SGKT}(\Delta_{\rm
KT},t)e^{-\lambda_{\rm KT}t}$ (1) $\displaystyle+A_{\rm BG}e^{-\lambda_{\rm
BG}t}>2~{}{\rm K},$ (2)

$\displaystyle A_{0}\,P_{\rm ZF}(t)=A_{\rm S}e^{-(\lambda_{\rm
S}t)^{\beta}}+A_{\rm BG}e^{-\lambda_{\rm BG}t}\leq 2~{}{\rm K},$ (3)

where $A_{0}$ is the initial $t=0$ asymmetry that depends on the detector
geometry of the instrument and $P_{\rm ZF}(t)$ represents the muon
polarisation function under ZF configuration. $A_{\rm KT}$ and $\lambda_{\rm
KT}$ are the respective asymmetries and the associated depolarisation rate.
The asymmetries represent the magnetic volumic fraction, $i.e.$ the fraction
of muon experiencing that particular field distribution descried by the
associated polarisation function. $G^{\rm SGKT}$ represents a static Gaussian-
KT and has the form

$\displaystyle P_{\rm
SGKT}(t)=\frac{1}{3}+\frac{2}{3}(1-t^{2}\Delta^{2})e^{-\frac{t^{2}\Delta^{2}}{2}},$
(4)

which is derived assuming isotropic randomly Gaussian-distributed magnetic
moments with a field distribution width $\Delta_{\rm KT}$. The ZF $\mu^{+}$SR
time spectrum at $4$ K was found to fully decouple (not shown) even for small
values of externally applied longitudinal fields, LF = 5-20 G (here, LF refers
to a magnetic field direction parallel to the initial muon spin polarisation).
Therefore, the isotropically distributed Gaussian field distribution at higher
temperatures is understood to be originated from static nuclear moments,
composed mostly of $I_{{}^{51}\rm V}=7/2$. The exponential ($\lambda_{\rm
KT}$) on the other hand accounts for fluctuations stemming from localised V4+
electron moments, that depolarises the muon spin polarisation on top of the
nuclear depolarisation.

Figure 2: Zero field (ZF) time spectra at selected temperatures. The solid
lines are the best fits obtained using Eq. (2) and Eq. (3), depending on the
temperature.

$A_{\rm X}$ and $\lambda_{\rm X}$ are the respective asymmetry and relaxation
rates where the index indicates if it is from the sample ($X=$ S) or the
sample holder ($X=$ BG). Indeed, very low values of $\lambda_{\rm
BG}=0.0056(5)\mu$s-1 are obtained at base temperature, as expected for Ag
contribution. Moreover, since $A_{\rm BG}$ is expected to be temperature
independent, the value was fixed to $A_{\rm BG}=0.18462$ for the whole
temperature range, a value estimated from the 4 K measurement for which the
sample polarisation function is known to be exponential KT function. $\beta$
on the other hand is the stretched exponent where values $<1$ implies the
presence of distributions of relaxation rates Forslund  _et al._ (2021).

Figure 3: Temperature dependencies of the obtained fit parameters using Eq.
(2) and Eq. (3): (a) The relaxation rates ($\lambda_{\rm KT}$, $\lambda_{\rm
S}$) and (b) the stretched exponent ($\beta$). For reference, the low
temperature part of the magnetic susceptibility is also included, as well as
the relaxation rate obtained in a previous $\mu^{+}$SR study at higher
temperatures Chow  _et al._ (2012). Formation of the QSL state is indicated by
the vertical dashed line, $T_{\rm QSL}\approx 2$ K.

The temperature dependencies of the obtained ZF fit parameters are displayed
in Fig. 3. In the whole temperature range, $A_{\rm KT}\simeq 0.09\simeq A_{\rm
S}$ was obtained, signifying the quality of the fits. The relaxation rates,
$\lambda_{\rm S}$ and $\lambda_{\rm KT}$, are shown together with the measured
magnetic susceptibility (down to 5 K). Indeed, the relaxation rates follow the
measured $\chi$ meaning the tail of $\chi$ (Fig. 1(b)) observed at lower
temperature is a result of imperfect spin singlet formation. These unpaired
spins are responsible for the fluctuations still present and thus the increase
in $\lambda_{\rm KT}$ and $\chi$. In other words, the tail feature observed in
$\chi$ is not due to impurities as previously speculated Isobe  _et al._
(2006). Below 2 K, the fit favors $\Delta_{\rm KT}=0$ $\mu$s-1 and Eq. (2) can
no longer reproduce the data, which is instead replaced by Eq. (3). The
crossover between the two regions has no impact on the fitting results as
smooth transition from $\lambda_{\rm KT}$ to $\lambda_{\rm S}$ is observed,
suggesting the assessments of the fit functions to be correct Fit .
Unexpectedly, a temperature independent behavior of $\lambda_{\rm S}$ is
observed at lower temperatures [Fig. 3(a)]. This would suggest that the
fluctuations are no longer thermally driven, i.e. below $T_{\rm QSL}\approx 2$
K, quantum fluctuations are instead dominating. Similar and supporting trend
is found for the temperature dependence of the stretched exponent parameter
($\beta$), which poses values close to 1 at 2 K and slowly decrease and
saturate to a value of $\beta\approx 0.7$. The value of 1 is naturally
expected at higher temperature. As thermal fluctuations is suppressed, $\beta$
takes on values $<1$. The quantum fluctuations present at low temperatures,
which by all means are uncorrelated, result in a distribution of spin-spin
correlation times. This kind of distribution forces $\beta<1$. While it is
difficult to assert any details from the specific value of $\beta\approx 0.7$,
a clear decrease from 1 is none the less observed. In the literature, a change
in $\beta$ together with temperature independent relaxation rate has been
observed in other QSL compounds, such as the 1D chain K3Cu3AlO2(SO4)4 Fujihala
_et al._ (2017), the 2D triangular antiferromagnet YbMgGaO4 Li  _et al._
(2016) or the 3D antiferromagnet PbCuTe2O6 Khuntia  _et al._ (2016).

It should of course be noted that static fields may in certain cases cause
exponential like relaxation. In order to confirm the dynamical origin of the
relaxation rate, the sample was measured under a series of longitudinal fields
(LF), as shown in Fig. 4. The initial decoupling observed at LF = 10 G comes
from the background signal along with a very small impurity fraction
(asymmetry $\sim$ 0.01). The latter was also confirmed in the basic XRD
characterisation performed after sample synthesis. Such very minor
contribution has still been readily accounted for in the analysis and could be
subtracted without any influence on the final result. What is very clear is
that the spectra are not fully decoupled even at LF = 2400 G (Fig. 4),
suggesting that the ground state is highly dynamic. The LF spectra were fitted
in accordance to the ZF model, $i.e.$ Eq. (3). Although, to reduce the number
of parameters, $A_{\rm S}$ was kept as a global (free) parameter, while
$A_{\rm BG}$ and and $\beta$ were fixed to the values as previously
determined.

Figure 4: Longitudinal field (LF) $\mu^{+}$SR spectra at selected LF collected
at $T=100$ mK. The solid lines represents fit with Eq. (3) using a global fit
procedure described in the text.

Consequently, the fit procedure effectively reduced the number of free and
field-dependent fit parameters to one, the relaxation rate ($\lambda_{\rm
S}$). This parameters is plotted as a function of the applied LF in Fig. 5. At
lower fields, a value close to the ones obtained at ZF is found but a
decreases as a function of LF is observed. Since Eq. (3) is a stretched
exponential, quantitative statement about the absolute values are difficult to
justify. However, the information from the relative field dependence can still
be extracted. In the motional narrowing limit, the relaxation rate is expected
to follow the field dependence of Fourier transform of the dynamical
correlation function, commonly known as the Redfield equation: $\lambda_{\rm
S}(B)=\frac{2\gamma_{\mu}^{2}\Delta^{2}\tau}{1+(\gamma_{\mu}B\tau)^{2}}$. A
fit with the Redfield function does not yield a good representation of the
field-dependence $\lambda_{\rm S}(LF)$ (see Fig. 5 for more details).
Therefore, the data could instead be very well fitted using a more general
power law form: $\lambda_{\rm S}=\frac{1}{1+cB^{\alpha}}$ for which
$\alpha=0.742(2)$ was obtained. It is noted that LF relaxation rate maps the
Fourier transform of the dynamical correlation function, meaning the field
dependence of $\lambda_{\rm S}$ (Fig. 5) corresponds to the spin-spin
correlation function. For a gapped spin liquid, the correlation function is
expected to experience a exponential like decay (i.e. Redfield like)
Schollwöck  _et al._ (2008). On the other hand, for a gapless spin liquid,
$e.g.$ Tomonaga-Luttinger liquid (TLL), the correlation is expected to show a
power law behavior Schollwöck  _et al._ (2008). Based on the fits presented in
Fig. 5, we therefore initially suggest that the QSL ground state in K2V8O16
could be the gapless TLL.

It is worth noticing that the signal from quantum fluctuations originating
from the spin singlet dimers would result in a gaped spin liquid. Naturally,
such fluctuations would also contribute to the LF relaxation rate and will be
included into the $\alpha$ coefficient of the power law (Fig. 5). However, the
obtained $\alpha=0.742(2)$ is a strong deviation from the ideal Redfield
($\alpha=2$), which supports a 1D diffusive picture of the spin fluctuations.
We anticipate that the non-gapped nature (TLL) will be directly confirmed via
neutron scattering at future high-intensity/-resolution neutron spallation
sources.

Figure 5: Relaxation rate ($\lambda_{\rm S}$) as a function of externally
applied LF plotted in logarithmic scale. The solid blue line represents a fit
using a power law: $\lambda_{\rm S}=\frac{1}{1+cB^{\alpha}}$ with
$\alpha=0.742(2)$, whereas the dashed red line is a fit with the Redfield
equation: $\lambda_{\rm
S}(B)=\frac{2\gamma_{\mu}^{2}\Delta^{2}\tau}{1+(\gamma_{\mu}B\tau)^{2}}$,
where $\gamma_{\mu}$ is the muon gyromagnetic ratio, $\Delta$ is the internal
field distribution width and $\tau$ is the spin-spin correlation time.

While there are several types of QSL, let us focus on the TLL, which is a
liquid state present in one dimensional heavily correlated electron systems.
In fact, K2V8O16 may be checking these boxes. DFT+U calculations Kim  _et al._
(2016) show a highly one dimensional character with a flat Fermi surface. It
was proposed that local electron correlation is driving the charge ordering
and orbital ordering of the V t2g bands. Therefore, K2V8O16 is both highly
correlated and 1D. An important consequence of TLL is the prediction of spin-
charge separation. Typical problem with experimental verification of TLL is
the fact that many QSL are usually stabilised under extreme conditions ($i.e.$
high pressure or magnetic field e.g. BaCo2V2O8 Kimura  _et al._ (2008); Faure
_et al._ (2019)). In K2V8O16 however, the QSL/TLL state is stabilized at
ambient condition (zero pressure and magnetic field), although at very low
temperatures. Hence, one would expect a FM or AF ordered states to be
stabilized either at high magnetic field or pressure, at least according to
the phase diagram of the XXZ Hamiltonian Schollwöck  _et al._ (2008).
Verifying these quantities in future experiments would clearly strengthen the
TLL scenario for K2V8O16.

Let us take a step back and compare to other members in the Hollandite family.
K2Ti8O16 is a Pauli paramagnetic metal down to the lowest measured Isobe  _et
al._ (2009) while K2Cr8O16 is a FM insulator down to 2 K Hasegawa  _et al._
(2009). Similarly, Rb2Cr8O16 is a FM metal Sugiyama  _et al._ (2012). While it
is clear that the Hollandite family exhibits a variety of ground states, a QSL
was up until now never reported within the family, even though band structure
calculations Toriyama  _et al._ (2011) of K2Ru8O16 in fact proposed it to
belong to the class of TLL. Hence, this work highlights K2V8O16 as the first
Hollandite to exhibit QSL (possibly TLL) behavior, which originates from the
high-temperature CO formation of isolated 1D spin-$1/2$ Heisenberg chains. It
is well established that CO can play an important role in order to stabilise
various phases. It was shown to be the driver from low to high Spin-State
transition in the AF YBaCo2O5 Vogt  _et al._ (2000) and multiferroicity is
naturally heavily linked to charge ordering van den Brink and Khomskii (2008);
Yamauchi and Picozzi (2010). In fact, detailed CO engineering was proposed in
order to induce desired multiferroic properties He and Jin (2016). Likewise,
CO plays an important part in high temperature superconductivity. CO appears
usually in the vicinity of superconducting state to either co-exist or to
compete with each other Tranquada  _et al._ (1997). Therefore, controlling the
CO are believed to be a key component in order to achieve and fully understand
the paring mechanisms in both the archetypal cuprate superconductor
YBa2Cu3O7-δ Hücker  _et al._ (2014); Comin  _et al._ (2015) as well as others
da Silva Neto  _et al._ (2014); Tabis  _et al._ (2014); Park  _et al._ (2021).
In a similar fashion, K2V8O16 is an example on how the formation of QSL is
stabilized by the CO formation, which effectively reduces the dimensionality
to a true 1D system within the 3D crystallographic structure. To the best of
our knowledge, this is the first clear instance of a CO induced QSL state.

In summary, our $\mu^{+}$SR establish the absence of long- or short-range
magnetic order in K2V8O16 down to $T=100$ mK. The ground state is found to be
dynamic and temperature independent, which matches the properties of a quantum
spin liquid (QSL) forming below $T_{\rm QSL}\approx 2$ K. Field dependent data
confirms the dynamic nature and points towards a Tomonaga-Luttinger liquid
(TLL) phase, which uniquely forms in this stoichiometric compound under zero
field and ambient pressure. The key to such properties is the formation of
charge order (CO) within half of the quasi-one-dimensional chains, which was
confirmed by XRD Komarek  _et al._ (2011). Combined with theoretical
predictions of spin-singlet formation Kim  _et al._ (2016) along with
supporting experimental evidence of a strong decrease in magnetic
susceptibility [Fig. 1(b)]] suggest a mechanism that effectively isolates the
remaining spin chains and thereby stabilizes a QSL. These findings highlights
the high-pressure synthesis route as a platform for discovering emerging
materials and novel physical properties. Finally, we also establish the
hollandites as a new group of model compounds available to study intrinsic QSL
behavior as well as its response to future external tuning via magnetic field,
high pressure and chemical substitution/doping.

###### Acknowledgements.

This research was supported by the Swedish Research Council (VR) via a Neutron
Project Grant (Dnr. 2016-06955) and the Carl Tryggers Foundation for
Scientific Research (CTS-18:272). E.N. is funded by the Swedish Foundation for
Strategic Research (SSF) within the Swedish national graduate school in
neutron scattering (SwedNess). D.A. acknowledges financial support from the
Romanian UEFISCDI project PN-III-P4-ID-PCCF-2016-0112, Contract Nr. 6/2018.
Y.S. is funded by VR through a Starting Grant (Dnr. 2017-05078) as well as the
Chalmers Area of Advance-Materials Science. J.S. acknowledges support from
Japan Society for the Promotion Science (JSPS) KAKENHI Grants No. JP18H01863
and No. JP20K21149.

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# A Thom Spectrum Model for $C_{2}$-Integral Brown–Gitler Spectra

Guchuan Li and Sarah Petersen and Elizabeth Ellen Tatum

###### Abstract.

A Thom spectrum model for a $C_{2}$-equivariant analogue of integral
Brown–Gitler spectra is established and shown to have a multiplicative
property. The $C_{2}$-equivariant spectra constructed enjoy properties
analogous to classical nonequivariant integral Brown–Gitler spectra and thus
may prove useful for producing splittings of $BP\langle 1\rangle\wedge
BP\langle 1\rangle$ and $bo\wedge bo$ in the $C_{2}$-equivariant setting.

###### Contents

  1. 1 Introduction
  2. 2 Statement of Theorems
  3. 3 Equivariant Preliminaries
  4. 4 A Thom spectrum model for $H\underline{{\mathbb{Z}}}_{2}$
  5. 5 A product fibration
  6. 6 Proof of Main Theorem

##  1\. Introduction

In the 1970’s, Brown and Gitler constructed a family of spectra realizing
certain sub-comodules of the dual Steenrod algebra at the prime $p=2$ [BG73].
Brown–Gitler’s motivation for constructing the original spectra was to study
immersions of manifolds. These spectra were indeed useful for that purpose, in
particular allowing Cohen to prove the immersion conjecture for differentiable
manifolds [Coh82]. They have also been useful for studying maps out of
classifying spaces. For example, they were used by Miller to prove the
Sullivan Conjecture [Mil84].

Brown–Gitler used an obstruction-theoretic approach, first constructing an
algebraic resolution, then constructing a tower of spectra realizing that
resolution, and then taking its inverse limit to obtain the desired spectrum.
In [Mah77], Mahowald suggested an alternative construction using a Thom
spectrum model. In particular, Mahowald posited that the natural filtration of
the space $\Omega^{2}S^{3}$ due to May–Milgram [MM81] produces a filtration of
the Thom spectrum $(\Omega^{2}S^{3})^{\mu}\simeq H{{\mathbb{F}}}_{2}$ by
spectra. Here, $\mu$ is the double loop map $\mu:\Omega^{2}S^{3}\to BO$ of the
classifying map of the Möbius bundle over $S^{1}.$ The resulting filtration
spectra turn out to be the Brown–Gitler spectra of [BG73] (see [BP78, Coh79,
HK99]). In [Coh79], Cohen made this construction precise at all primes.

In [Mah81], Mahowald suggested that integral analogues of Brown–Gitler spectra
should exist: that is, finite spectra realizing an analogous family of sub-
comodules of $H_{*}H\mathbb{Z}$, and proposed a Thom spectrum model. In
[Shi84], Shimamoto gave a construction using the obstruction-theoretic
approach, and also gave more details on Mahowald’s proposed Thom spectrum
model. Kane used an odd primary version of this Thom spectrum construction in
[Kan81]. In [Coh+88], Cohen, Davis, Goerss, and Mahowald made these Thom
spectrum constructions rigorous, in particular by explicitly defining the base
space used in the Thom spectrum construction.

In [GJM86], Goerss–Jones–Mahowald used the obstruction-theoretic approach to
construct Brown–Gitler spectra for $BP\langle 1\rangle$ and $bo$ (i.e. finite
spectra realizing certain sub-comodules of $H_{*}BP\langle 1\rangle$ and
$H_{*}bo$), and Klippenstein extended these arguments to the $BP\langle
2\rangle$ case in [Kli88]. These spectra are collectively known as generalized
Brown–Gitler spectra.

The higher Brown–Gitler spectra have been useful in understanding the smash
products of various ring spectra. In particular, Mahowald used the integral
Brown–Gitler spectra to decompose $bo\wedge bo$ as a sum of finitely generated
$bo$-modules [Mah81]. Kane subsequently produced an analogous decomposition
for $BP\langle 1\rangle$, also using integral Brown–Gitler spectra [Kan81].
These splittings helped make it feasible to use the $bo$\- and $BP\langle
1\rangle$-based Adams spectral sequences for computations and have proved to
be powerful computational tools. For example, Mahowald used the $bo$-based
Adams spectral sequence to prove the telescope conjecture at the prime $2$ and
height $1$ [Mah81].

This paper extends Mahowald’s idea for an integral analogue of Brown–Gitler
spectra to the $C_{2}$-equivariant setting (Theorem 6.1). The
$C_{2}$-equivariant spectra we construct realize an analogous family of sub-
comodules of
$H{\underline{{\mathbb{F}}}_{2}}_{\star}H\underline{{\mathbb{Z}}},$ where $H$
is the Eilenberg-MacLane spectrum associated to the constant $C_{2}$-Mackey
functor $\underline{{\mathbb{F}}}_{2}$ and $H\underline{{\mathbb{Z}}}$ is the
Eilenberg-MacLane spectrum associated to the constant $C_{2}$-Mackey functor
$\underline{{\mathbb{Z}}}$. Our primary motivation for establishing a
construction of these spectra is to enable the study of study
$C_{2}$-equivariant analogues of splittings of $BP\langle 1\rangle\wedge
BP\langle 1\rangle$ and $bo\wedge bo,$ and related $BP\langle 1\rangle$\- and
$bo$\- based spectral sequence computations.

In the $C_{2}$-equivariant setting, the Eilenberg-MacLane spectra
$H\underline{{\mathbb{F}}}_{p},$ $H\underline{{\mathbb{Z}}}_{(2)},$ and
$H\underline{{\mathbb{Z}}}_{2},$ associated to the constant $C_{2}$-Mackey
functors $\underline{{\mathbb{F}}}_{2},$ $\underline{{\mathbb{Z}}}_{(2)},$ and
$\underline{{\mathbb{Z}}}_{2}$ respectively, also arise from Thom spectra. In
[BW18], Behrens–Wilson show there is an equivalence of $C_{2}$-spectra
$(\Omega^{\rho}S^{\rho+1})^{\mu}\simeq H\underline{{\mathbb{F}}}_{2},$ where
$\rho$ is the regular representation of $C_{2}$ and $\mu$ is the $\rho$-loop
map $\mu:\Omega^{\rho}S^{\rho+1}\to B_{C_{2}}O$ of the classifying map of the
Möbius bundle over $S^{1}$ regarded as a $C_{2}$-equivariant virtual bundle of
dimension zero by endowing both $S^{1}$ and the bundle with trivial action.
Note we follow the convention that ${\mathbb{Z}}_{(2)}$ denotes the $2$-local
integers and ${\mathbb{Z}}_{2}$ the $2$-adic integers.

In [HW20], Hahn–Wilson generalize the result of Behrens–Wilson. They show the
$G$-equivariant mod $p$ Eilenberg-MacLane spectrum arises as an equivariant
Thom spectrum for any finite, $p$-power cyclic group $G.$ Hahn–Wilson also
establish a Thom spectrum model for $H\underline{{\mathbb{Z}}}_{(p)},$
building a base space from the $G$-space $\Omega^{\lambda}S^{\lambda+1},$
where $\lambda$ denotes the standard representation of $G$ on the complex
numbers with a generator acting by $e^{2\pi i/p^{n}}.$ They observe that this
space carries the arity filtration from the $E_{\lambda}$-operad and thus one
could define equivariant Brown–Gitler spectra as the spectra coming from this
filtration. This is the perspective we take and extend to the integral setting
in this paper.

Non-equivariantly, two conditions characterize Brown–Gitler spectra. First,
these spectra realize certain sub-comodules of the dual Steenrod algebra.
Additionally, they satisfy a surjectivity condition coming from the geometry
involved in Brown–Gitler’s original construction [BG73]. While the Thom
spectrum model does satisfy this additional condition in the non-equivariant
setting, it is not clear, neither from the geometry nor the Thom spectra
filtration models, how to generalize this second condition in the equivariant
setting. We discuss this in Remark 6.2, taking the philosophy proposed by
Hahn–Wilson: one can simply define equivariant Brown–Gitler spectra using
filtrations on the Thom spectra models and see if these spectra are
computationally useful.

Several factors motivate our choice to study $C_{2}$-equivariant, as opposed
to $C_{p}$-equivariant for all primes $p,$ spectra. First, $RO(C_{2})$-graded
$H\underline{{\mathbb{F}}}_{2}$-homology is free over its coefficients in
important examples such as the dual Steenrod algebra
${H\underline{{\mathbb{F}}}_{2}}_{\star}H\underline{{\mathbb{F}}}_{2}$ [HK01]
and ${H\underline{{\mathbb{F}}}_{2}}_{\star}H\underline{{\mathbb{Z}}}$
[Orm11]. This is helpful because integral Brown–Gitler spectra topologically
realize certain sub-comodules of
${H\underline{{\mathbb{F}}}_{2}}_{\star}H\underline{{\mathbb{Z}}}$. In
contrast, $RO(C_{p})$-graded $H\underline{{\mathbb{F}}}_{p}$-homology is often
not free over its coefficients when $p>2.$ In particular, the $C_{p}$-dual
Steenrod algebra is not free [SW22], which suggests the odd primary
$C_{p}$-equivariant story may be more complicated, requiring techniques beyond
those developed in this paper.

Further, when $G=C_{2},$ there is a non-obvious $C_{2}$-equivalence of spaces
$\Omega^{\lambda}S^{\lambda+1}\simeq\Omega^{\rho}S^{\rho+1}$ [HW20]. Thus
there are filtrations of
$\Omega^{\lambda}S^{\lambda+1}\simeq\Omega^{\rho}S^{\rho+1}$ by both the
$E_{\lambda}$ and $E_{\rho}$ operads, leading to at least two definitions of
Brown–Gitler spectra. In this paper, we choose to work with the
$E_{\rho}$-filtration because it is the most computationally accessible.
However, in [Lev22], Levy unified and extended the work of Behrens, Hahn, and
Wilson constructing the Eilenberg–MacLane spectrum
$H\underline{{\mathbb{F}}}_{p}.$ Studying how the $E_{\lambda}$ and $E_{\rho}$
filtrations arise from faithful representations of $p$-groups could unify the
study of these two definitions of Brown–Gitler spectra, and may prove an
interesting direction for future work.

##  2\. Statement of Theorems

Our main result is a $C_{2}$-equivariant analogue of [Coh+88, Theorem
1.5(i),(ii)]. To state this theorem precisely, we recall
${H\underline{{\mathbb{F}}}_{2}}$ has distinguished elements $a\in
H{\underline{{\mathbb{F}}}_{2}}_{\\{-\sigma\\}}$ and $u\in
H{\underline{{\mathbb{F}}}_{2}}_{\\{1-\sigma\\}},$ where $\sigma$ is the one-
dimensional sign representation of $C_{2}$. We define a weight filtration on

${H\underline{{\mathbb{F}}}_{2}}_{\star}H{\underline{{\mathbb{Z}}}}\cong
H{\underline{{\mathbb{F}}}_{2}}_{\star}[\bar{\xi_{1}},\bar{\xi_{2}},\bar{\xi_{3}},\cdots,c(\tau_{1}),c(\tau_{2}),\cdots]/(c(\tau_{i}^{2})=ac(\tau_{i+1})+u\bar{\xi_{i+1}}),$

where $|\tau_{j}|=2^{j}\rho-\sigma,$ $|\bar{\xi}_{i}|=(2^{i}-1)\rho,$ and $c$
denotes the antiautomorphism of the dual Steenrod algebra
${\mathcal{A}}\cong\pi_{\star}H\underline{{\mathbb{F}}}_{2}\wedge
H\underline{{\mathbb{F}}}_{2}$ (the computation of
$H{\underline{{\mathbb{F}}}_{2}}_{\star}H\underline{{\mathbb{Z}}}$ follows
from [Orm11, Theorem 3.8]). The weight filtration is defined by

$\operatorname{wt}(c(\tau_{j}))=\operatorname{wt}(\bar{\xi}_{j})=2^{j},\qquad\operatorname{wt}(xy)=\operatorname{wt}(x)+\operatorname{wt}(y).$

######  Theorem (Theorem 6.1).

For $n>0,$ there is an $H\underline{{\mathbb{F}}}_{2}$-complete spectrum
$B_{1}(n)$ and a map

$B_{1}(n)\xrightarrow[]{g}H{\underline{{\mathbb{Z}}}_{2}}$

such that

  * (i)

$g_{\star}$ sends $H{\underline{{\mathbb{F}}}_{2}}_{\star}(B_{1}(n))$
isomorphically onto the span of monomials of weight $\leq 2n;$

  * (ii)

there are pairings

$B_{1}(m)\wedge B_{1}(n)\to B_{1}(m+n)$

whose homology homomorphism is compatible with the multiplication in
$H{\underline{{\mathbb{F}}}_{2}}_{\star}(H\underline{{\mathbb{Z}}}_{2}).$

The spectra $B_{1}(n)$ are $C_{2}$-equivariant analogues of Cohen, Davis,
Goerss and Mahowald’s integral Brown–Gitler spectra in the sense that they
realize certain sub-comodules of
$H{\underline{{\mathbb{F}}}_{2}}_{\star}H\underline{{\mathbb{Z}}}.$ This
result follows from the more technical Theorem 5.1, which is a
$C_{2}$-equivariant analogue of [Coh+88, Theorem 1.3]. To state Theorem 5.1,
we introduce both an increasing filtration of the space
$\Omega^{\rho}S^{\rho+1}$ and a homotopy fiber sequence.

Work of Rourke-Sanderson [RS00] shows the space $\Omega^{\rho}S^{\rho+1}$
admits an increasing filtration by spaces

(2.1) $\displaystyle F_{n}\Omega^{\rho}S^{\rho+1}\simeq\underset{0\leq
k<n}{\displaystyle{\coprod}}C_{k}(\rho)\underset{\Sigma_{k}}{\times}(S^{1})^{\times
k}/\sim,$

where

$C_{k}(\rho)=\\{m_{1},m_{2},\cdots,m_{k}|m_{i}\neq m_{j}\text{ if }i\neq
j\\}\subset\rho^{k}$

is the configuration space of $k$ ordered points in the $C_{2}$-regular
representation $\rho.$ Note when $k=0,$ this gives the base point $x_{0}.$ If
$x_{n}=x_{0},$ the relation $\sim$ identifies

$(m_{1},\cdots,m_{n};x_{1},\cdots,x_{n})\sim(m_{1},\cdots,m_{n-1};x_{1},\cdots,x_{n-1}).$

Let $X_{2}$ denote the Bousfield localization of $X$ with respect to
$H{\underline{{\mathbb{F}}}_{2}}$ and let
${\mathcal{F}}_{n}=(F_{n}\Omega^{\rho}S^{\rho+1})_{2}.$ Then there are product
maps

${\mathcal{F}}_{m}\times{\mathcal{F}}_{n}\overset{\mu}{\rightarrow}{\mathcal{F}}_{m+n}$

induced by the corresponding maps for the filtration spaces and the fact that
localization preserves finite products. Define $A_{n}$ by the homotopy fiber
sequence

(2.2) $\displaystyle A_{n}\to{\mathcal{F}}_{2n+1}\to S^{1}_{2}$

where the second map is the localization of the composite

$F_{2n+1}\Omega^{\rho}S^{\rho+1}\to\Omega^{\rho}S^{\rho+1}\to S^{1}.$

######  Theorem (Theorem 5.1).

The fiber sequence (2.2) is equivalent to a product fibration. Indeed, there
is a $C_{2}$-equivariant map
$A_{n}\overset{\phi}{\rightarrow}{\mathcal{F}}_{2n}$ and a commutative diagram
of fibrations

${A_{n}}$${A_{n}}$${S_{2}^{1}\times
A_{n}}$${S_{2}^{1}\times{\mathcal{F}}_{2n}={\mathcal{F}}_{1}\times{\mathcal{F}}_{2n}}$${{\mathcal{F}}_{2n+1}}$${S_{2}^{1}}$${S_{2}^{1}}$$\scriptstyle{1\times\phi}$$\scriptstyle{p_{1}}$$\scriptstyle{m}$

which is an equivalence on total spaces and on fibers.

Our argument for deducing Theorem 6.1 from Theorem 5.1 follows that of
Cohen–Davis–Goerss–Mahowald and thus relies on a Thom spectrum construction of
$H\underline{{\mathbb{Z}}}_{2}$ with a $2$-complete base space (Theorem 4.1).

######  Remark 2.3.

Hahn–Wilson [HW20] establish a construction of
$H\underline{{\mathbb{Z}}}_{(2)}$ as a $C_{2}$-Thom spectrum. However, they
observe this construction does not yield $H\underline{{\mathbb{Z}}}_{(p)}$ as
a $C_{2}$-spectrum for other primes $p,$ and thus does not readily generalize
to a full integral construction.

In Section 4, we extend Hahn–Wilson’s arguments to construct
$H\underline{{\mathbb{Z}}}_{2}$ as a $C_{2}$-equivariant Thom spectrum with a
$2$-complete base space, resulting in Theorem 4.1, an equivariant analogue of
[CMT81, Theorem 1], which was originally proposed by Mahowald in the
nonequivariant setting.

To state Theorem 4.1, we require the following maps. Consider the $\rho$-loops
of the unit map $S^{\rho+1}\to K({\mathbb{Z}},\rho+1).$ Composing with the
adjoint to $-1\in\pi_{0}^{C_{2}}(S_{2}^{0})$ we get a map

$\Omega^{\rho}S^{\rho+1}\to\Omega^{\rho}K(\underline{{\mathbb{Z}}},\rho+1)\to
BGL_{1}(S_{2}^{0}).$

Note $\Omega^{\rho}K(\underline{{\mathbb{Z}}},\rho+1)\simeq S^{1}$ with
trivial $C_{2}$-action.

Let $\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle$ denote the homotopy fiber of
the map $\Omega^{\rho}S^{\rho+1}\to S^{1}$ and consider the composition

$\mu:\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle\to\Omega^{\rho}S^{\rho+1}\to
S^{1}\to BGL_{1}(S_{2}^{0}).$

######  Theorem (4.1).

There is an equivalence of $C_{2}$-spectra

$\displaystyle(\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle_{2})^{\mu}\to
H\underline{{\mathbb{Z}}}_{2}.$

### Acknowledgments

The authors would like to thank to Agnès Beaudry, Tobias Barthel, Mark
Behrens, David Chan, Bert Guillou, Paul Goerss, and Jeremy Hahn for
enlightening conversations. The first and second authors would also like to
thank the Max Planck Institute for Mathematics in Bonn for its hospitality and
financial support. The third author thanks the Hausdorff Institute in Bonn for
its hospitality and financial support, as well as the Knut and Alice
Wallenberg Foundation for financial support. All three authors are grateful to
the Hausdorff Institute in Bonn for its hospitality during the Spectral
Methods in Algebra, Geometry, and Topology trimester program in Fall of 2022,
funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy- EXC-2047/1-390685813. This
material is based upon work supported by the National Science Foundation under
Grant No. DMS 2135884.

###  2.1. Structure of Argument

We extend the arguments of Cohen–Davis–Goerss–Mahowald to the
$C_{2}$-equivariant setting. To make this extension clear, we first recall
Cohen–Davis–Goerss–Mahowald’s nonequivariant argument, then outline the
structure of our equivariant extension.

In [Coh+88], Cohen–Davis–Goerss–Mahowald make a Thom spectrum model for
integral Brown–Gitler spectra precise by explicitly defining the base space.
Specifically, they consider the space $\Omega^{2}S^{3}\langle 3\rangle,$ the
double loop space of the homotopy fiber of the unit map $S^{3}\to
K({\mathbb{Z}},3),$ and show that after localization with respect to mod $p$
homology, a weight filtration on the homology of $\Omega^{2}S^{3}\langle
3\rangle$ is in fact induced by an actual filtration of the space
$\Omega^{2}S^{3}\langle 3\rangle.$ The filtration pieces of the space
$\Omega^{2}S^{3}\langle 3\rangle_{2}$ are the base spaces in the Thom spectrum
model for integral Brown–Gitler spectra.

To extend Cohen–Davis–Goerss–Mahowald’s nonequivariant argument to the
$C_{2}$-equivariant setting, we localize with respect to
$H\underline{{\mathbb{F}}}_{2}$-homology, and then show that a weight
filtration on the homology of $\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle$ is
in fact induced by an actual filtration of the space
$\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle.$ This requires a model of
$H\underline{{\mathbb{Z}}}_{2}$ as a Thom spectrum with $2$-complete base
space $\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle_{2}.$

We provide equivariant preliminaries in Section 3. We construct
$H\underline{{\mathbb{Z}}}_{2}$ as a $C_{2}$-equivariant Thom spectrum with
$2$-complete base space (Theorem 4.1) in Section 4. In Section 5, we extend
Cohen–Davis–Goerss–Mahowald’s technical argument making the filtration of the
base space rigorous. This culminates in the proof of Theorem 5.1, from which
our main result concerning a Thom spectrum model for $C_{2}$-integral
Brown–Gitler spectra (Theorem 6.1) immediately follows.

##  3\. Equivariant Preliminaries

As in [BW18, Section 4], we will make use of an identification of the
$C_{2}$-fixed points of the space $\Omega^{\rho}S^{\rho+1}.$ Consider the
cofiber sequence

${C_{2}}_{+}\to S^{0}\hookrightarrow S^{\sigma}.$

Mapping out of this cofiber sequence gives a fiber sequence

$\Omega N^{\times}\Omega S^{\rho+1}\to\Omega^{\rho}S^{\rho+1}\to\Omega
S^{\rho+1}\xrightarrow{\Delta}N^{\times}\Omega S^{\rho+1},$

where
$N^{\times}X:=\text{Map}(C_{2},X)=X\underset{\begin{subarray}{c}\curvearrowbotleftright\\\
C_{2}\end{subarray}}{\times}X$ is the norm with respect to Cartesian product
(i.e. the coinduced space). On taking fixed points, we get a fiber sequence

$\Omega^{2}S^{3}\xrightarrow[]{}(\Omega^{\rho}S^{\rho+1})^{C_{2}}\to\Omega
S^{2}\xrightarrow[]{\text{null}}\Omega S^{3}.$

In particular, there is an equivalence

(3.1) $\displaystyle(\Omega^{\rho}S^{\rho+1})^{C_{2}}\simeq\Omega
S^{2}\times\Omega^{2}S^{3}.$

Behrens–Wilson [BW18] also established an additive isomorphism

(3.2) $\displaystyle
H{\underline{{\mathbb{F}}}_{2}}_{\star}\Omega^{\rho}S^{\rho+1}\cong
H{\underline{{\mathbb{F}}}_{2}}_{\star}\otimes E[t_{0},t_{1},\cdots]\otimes
P[e_{1},e_{2},\cdots]$

where

$\displaystyle|t_{i}|$ $\displaystyle=2^{i}\rho-\sigma$ $\displaystyle|e_{i}|$
$\displaystyle=(2^{i}-1)\rho.$

We define a weight on the monomials in
$H{\underline{{\mathbb{F}}}_{2}}_{\star}(\Omega^{\rho}S^{\rho+1})$ by

$\operatorname{wt}(t_{j})=\operatorname{wt}(e_{j})=2^{j},\qquad\operatorname{wt}(ab)=\operatorname{wt}(a)+\operatorname{wt}(b).$

and recall the space $\Omega^{\rho}S^{\rho+1}$ admits an increasing filtration
by spaces

$F_{n}\Omega^{\rho}S^{\rho+1}\simeq\underset{0\leq
k<n}{\displaystyle{\coprod}}C_{k}(\rho)\underset{\Sigma_{k}}{\times}(S^{1})^{\times
k}/\sim,$

where the relation is defined in equation (2.1). This filtration is such that
$H{\underline{{\mathbb{F}}}_{2}}_{\star}(F_{n}\Omega^{\rho}S^{\rho+1})$ is the
span of monomials of weight $\leq n.$

######  Remark 3.3.

To verify the claim that
$H{\underline{{\mathbb{F}}}_{2}}_{\star}(F_{n}\Omega^{\rho}S^{\rho+1})$ is the
span of monomials of weight $\leq n,$ observe that there is a
$C_{2}$-equivariant Snaith splitting [Lew+86, Chapter VII Theorem 5.7] and
inclusions

${\Sigma_{+}^{\infty}\Omega^{\rho}S^{\rho+1}\simeq\bigvee\mathcal{E}_{n}^{+}\wedge_{\Sigma_{n}}(S^{1})^{\wedge
n}}$${\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\Sigma_{+}^{\infty}\Omega^{\rho}S^{\rho+1}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Sigma_{+}^{\infty}\mathcal{E}_{n}\underset{\Sigma_{n}}{\times}(S^{1})^{\times
n}}$

where $\mathcal{E}_{\bullet}$ is the little $2$-disks operad in the
$C_{2}$-regular representation $\rho.$ We will consider the cases where
$n=2,3,4.$ The remaining cases are similar. Note that $\mathcal{E}_{n}\simeq
C_{n}(\rho).$ The $\rho$ loops space structure means we have a diagram

${\mathcal{E}_{2}\underset{\Sigma_{2}}{\times}(\Omega^{\rho}S^{\rho+1}\times\Omega^{\rho}S^{\rho+1})}$${\Omega^{\rho}S^{\rho+1}}$${S^{1}\underset{\Sigma_{2}}{\times}(\Omega^{\rho}S^{\rho+1}\times\Omega^{\rho}S^{\rho+1})}$$\scriptstyle{\simeq}$

Using the notation and Dyer–Lashof operations defined in [BW18], the map

$H{\underline{{\mathbb{F}}}_{2}}_{\star}S^{1}\underset{\Sigma_{2}}{\times}\Omega^{\rho}S^{\rho+1}\times\Omega^{\rho}S^{\rho+1}\to
H{\underline{{\mathbb{F}}}_{2}}_{\star}\Omega^{\rho}S^{\rho+1}$

sends

$\displaystyle\iota\otimes t_{0}\otimes t_{0}$ $\displaystyle\mapsto
Q^{\rho}t_{0}=t_{1}$ $\displaystyle*\otimes t_{0}\otimes t_{0}$
$\displaystyle\mapsto n(x_{1})=e_{1}$ $\displaystyle*\otimes*\otimes t_{0}$
$\displaystyle\mapsto t_{0}.$

For degree reasons, we pick up no additional generators when $n=3.$ To
complete the statement for $n=4,$ consider the diagram

${\mathcal{E}_{2}\underset{\Sigma_{2}}{\times}(\mathcal{E}_{2}\underset{\Sigma_{2}}{\times})^{\times
2}(\Omega^{\rho}S^{\rho+1})^{\times
2}}$${\mathcal{E}_{2}\underset{\Sigma_{2}}{\times}(\Omega^{\rho}S^{\rho+1})^{\times
2}}$${\Omega^{\rho}S^{\rho+1}}$${\mathcal{E}_{2}\times(\mathcal{E}_{2}^{\times
2}\underset{\Sigma_{2}\int\Sigma_{2}}{\times}{(\Omega^{\rho}S^{\rho+1})}^{\times
4})}$${\mathcal{E}_{4}\underset{\Sigma_{4}}{\times}(\Omega^{\rho}S^{\rho+1})^{\times
4}.}$$\scriptstyle{\Sigma_{2}\int\Sigma_{2}\subset\Sigma_{4}}$

Recall $\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle$ denotes the homotopy
fiber of the map $\Omega^{\rho}S^{\rho+1}\xrightarrow{r}S^{1},$ the
$\rho$-loops of the unit map $S^{\rho+1}\to
K(\underline{{\mathbb{Z}}},\rho+1),$ so

$\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle\to\Omega^{\rho}S^{\rho+1}\xrightarrow{r}S^{1}$

is a fiber sequence. The fibration splits if there is an epimorphism

$\pi_{1}^{C_{2}}\Omega^{\rho}S^{\rho+1}\to\pi_{1}^{C_{2}}S^{1}.$

######  Lemma 3.4.

There is an epimorphism
$\pi_{1}^{C_{2}}\Omega^{\rho}S^{\rho+1}\to\pi_{1}^{C_{2}}S^{1}.$ Hence,
$\Omega^{\rho}S^{\rho+1}\simeq
S^{1}\times\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle$ and
$H{\underline{{\mathbb{F}}}_{2}}_{\star}(\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle)\subseteq
H{\underline{{\mathbb{F}}}_{2}}_{\star}(\Omega^{\rho}S^{\rho+1})$ is the span
of monomials of weight divisible by 2.

###### Proof.

Stably,
$\underline{\pi}_{1}(\Omega^{\rho}S^{\rho+1})\cong\underline{\pi}_{\rho+1}S^{\rho+1}$
is the Mackey functor

${{\mathbb{Z}}^{2}}$${{\mathbb{Z}}.{}}$$\scriptstyle{(1\,\,2)}$

The induced map
$\underline{\pi}_{1}(r):\underline{\pi}^{s}_{1}\Omega^{\rho}S^{\rho+1}\to\underline{\pi}_{1}K({\mathbb{Z}},1)$
must be an epimorphism since the diagram of Mackey functors

${{\mathbb{Z}}^{2}}$${{\mathbb{Z}}}$${{\mathbb{Z}}}$${{\mathbb{Z}}}$$\scriptstyle{(1\,\,2)}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

commutes. Evaluation at the $C_{2}/C_{2}$ level gives an epimorphism

$\pi_{1}^{C_{2}}\Omega^{\rho}S^{\rho+1}\to\pi_{1}^{C_{2}}S^{1}.$

∎

###  3.1. Thom spectra of stable spherical fibrations

We describe a Lewis–May Thom spectrum functor for $p$-complete stable
spherical fibrations $\mu:X\to BF_{p}$ (see [BCS10, §3.4], for instance, for a
$p$-local description). In the language of structured ring spectra, $BF_{p}$
can be identified with $BGL_{1}({\mathbb{S}}_{p}),$ the classifying space of
the units of the $p$-complete sphere spectrum ${\mathbb{S}}_{p}$, and a
spectrum can be viewed as an ${\mathbb{S}}_{p}$-module. This language appears
in Lemma 3.5 and Theorem 4.1.

Let $V$ be a finite dimensional real $C_{2}$-representation and $F_{p}(V)$ be
the topological monoid of basepoint-preserving $C_{2}$-equivariant homotopy
equivalences of $S^{V}_{p},$ the $p$-completion of the one-point
compactification of $V.$ There is an associated quasi-fibration

$EF_{p}(V)\to BF_{p}(V)$

with fiber $S^{V}_{p}.$ Write $BF_{p}$ for the colimit of the spaces
$BF_{p}(V)$ where the colimit is taken over a diagram with one object for each
finite dimensional real $C_{2}$-representation and one arrow $V\to W$ if and
only if $V\subset W.$

Given a map $f:X\to BF_{p},$ let $X(V)$ be the closed subset $f^{-1}BF_{p}(V)$
and

$T(f)(V):=f^{*}EF_{p}(V)/X$

be the Thom space of the induced map $f_{V}:X(V)\to BF(V).$ Here
$f^{*}EF_{p}(V)$ is the pullback of $EF_{p}(V)$ and $X$ is viewed as a
subspace via the induced section.

This defines an object $T(f)$ in the category of $C_{2}$-orthogonal
prespectra. Composition with the spectrification functor gives a Thom spectrum
functor

$T_{{\mathbb{S}}_{p}}:\mathcal{U}/BF_{p}\to{\mathbb{S}}_{p}$

defined on the category of $\mathcal{U}$ of $C_{2}$-spaces over the
classifying space $BF_{p}$ for $p$-complete stable spherical fibrations with
values in $p$-complete spectra ${\mathbb{S}}_{p}.$

###  3.2. A Thom spectrum model for $\mathbf{H\underline{{\mathbb{F}}}_{2}}$

Consider the $\rho$-loops of the unit map $S^{\rho+1}\to
K({\mathbb{Z}},\rho+1).$ Let $\mu$ denote composition with the adjoint to
$-1\in\pi_{0}^{C_{2}}(S_{2}^{0}):$

$\mu:\Omega^{\rho}S^{\rho+1}\to\Omega^{\rho}K(\underline{{\mathbb{Z}}},\rho+1)\to
BGL_{1}(S_{2}^{0}).$

######  Lemma 3.5.

The Thom class $(\Omega^{\rho}S^{\rho+1})^{\mu}\to
H\underline{{\mathbb{F}}}_{2}$ is an equivalence of $C_{2}$-spectra.

This follows from [Lev22, Proof of Theorem A].

##  4\. A Thom spectrum model for $H\underline{{\mathbb{Z}}}_{2}$

Arguing as in Hahn–Wilson [HW20, Theorem 9.1], which uses arguments of
Antolín-Camarena-Barthel [AB19, § 5.2], we prove

######  Theorem 4.1.

There is an equivalence of $C_{2}$-spectra

$\displaystyle(\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle_{2})^{\mu}\to
H\underline{{\mathbb{Z}}}_{2}.$

###### Proof.

We first show the Thom class

(4.1) $\displaystyle(\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle)^{\mu}\to
H\underline{{\mathbb{Z}}}_{2}.$

is an equivalance of $C_{2}$-spectra. Decomposing $S^{1}$ into a $0$-cell and
a $1$-cell and trivializing the fiber on each cell produces a decomposition of
the Thom spectrum $(\Omega^{\rho}S^{\rho+1})^{\mu}$ as a cofiber

$(\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle)^{\mu}\xrightarrow{x}(\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle)^{\mu}\to(\Omega^{\rho}S^{\rho+1})^{\mu}\simeq
H\underline{{\mathbb{F}}}_{2}.$

Each of these Thom spectra come from bundles classified by
$\mathbb{A}_{2}$-maps, which is enough to ensure the map $x$ induces a map

$\underline{\pi}_{*}(\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle)^{\mu}\to\underline{\pi}_{*}(\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle)^{\mu}$

of modules over
$\underline{\pi}_{0}(\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle)^{\mu}.$ In
particular, on homotopy the map corresponds to multiplication by some element
$x\in\underline{\pi}_{0}(\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle)^{\mu}.$

Taking loops of $\mu:\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle\to
BGL_{1}(S_{2}^{0}),$ we obtain a map

$\Omega\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle\to GL_{1}(S_{2}^{0}).$

By the universal property of $GL_{1}(S_{2}^{0}),$ we equivalently have an
$E_{1}$-ring map

$f:\Sigma_{+}^{\infty}\Omega\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle\to
S_{2}^{0}.$

Similarly, there is also a map
$\epsilon:\Sigma_{+}^{\infty}\Omega\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle\to
S^{0}_{2}$ coming from the trivial map
$\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle\to GL_{1}(S_{2}^{0}).$

The construction of Thom spectra as bar constructions [And+14, Definition 4.1]
then implies
$\underline{\pi}_{0}(\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle)^{\mu}\cong\text{coker}\underline{\pi}_{0}(f-\epsilon).$

Thus $\underline{\pi}_{0}(\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle)^{\mu}$
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modulo classes in the image of $f-\epsilon.$ This also fits into a short exact
sequence

$\underline{\pi}_{0}\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle)^{\mu}\xrightarrow{x}\underline{\pi}_{0}\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle)^{\mu}\to\underline{{\mathbb{F}}}_{2}.$

From the $C_{2}/e$ spot we deduce $x=2$ and from the short exact sequence and
Mackey functor structure deduce that

$\underline{\pi_{0}}(\Omega^{\rho}S^{\rho}+1\langle\rho+1\rangle)^{\mu}=\underline{{\mathbb{Z}}}_{2}.$

We have checked that the Thom class (4.1) induces an isomorphism on
$\underline{\pi_{0}}$. To show that it is an equivalence of $C_{2}$-spectra,
it remains to check that this map induces isomorphism in $\underline{\pi}_{V}$
for $V\neq 0$, that is,
$\underline{\pi}_{V}((\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle)^{\mu})$ is
trivial.

The underlying level follows from the classical nonequivariant result.

The genuine fixed point level follows from Nakayama’s lemma once one argues
that the genuine fixed point spectra have finitely generated homotopy groups
in each degree. The isotropy separation reduces us to the corresponding
statement on geometric fixed points.

By the Thom isomorphism,

$H_{*}[\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle)^{\mu}]^{\Phi C_{2}}\cong
H_{*}((\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle)^{C_{2}}).$

Recall from Lemma 3.4 that $\Omega^{\rho}S^{\rho+1}\simeq
S^{1}\times\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle$, and from Equation
(3.1) that $\Omega^{2}S^{3}\times\Omega
S^{2}\simeq(\Omega^{\rho}S^{\rho+1})^{C_{2}}$. It follows that

$\Omega^{2}S^{3}\times\Omega
S^{2}\simeq(\Omega^{\rho}S^{\rho+1})^{C_{2}}\simeq(S^{1}\times\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle)^{C_{2}}\simeq
S^{1}\times(\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle)^{C_{2}}.$

Thus $H_{*}(\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle)^{C_{2}}\subseteq
H_{*}(\Omega^{2}S^{3}\times\Omega S^{2})$ has finitely generated homology
groups in each degree. Hence, it also has finitely generated homotopy groups
in each degree and using Nakayama’s lemma finishes the proof of the genuine
fixed level.

The completion map

$\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle\to\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle_{2}$

induces an isomorphism on homology and using this one can show that the map

$(\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle)^{\mu}\to(\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle_{2})^{\mu}$

is a 2-complete equivalence. On the other hand, the left hand side is
automatically $2$-complete, being an (equivariant) homotopy colimit of
$2$-complete spectra by construction. ∎

##  5\. A product fibration

Recall $X_{2}$ denotes the Bousfield localization of $X$ with respect to
$H{\underline{{\mathbb{F}}}_{2}},$ we have set

${\mathcal{F}}_{n}=(F_{n}\Omega^{\rho}S^{\rho+1})_{2},$

and there are product maps

${\mathcal{F}}_{m}\times{\mathcal{F}}_{n}\overset{\mu}{\rightarrow}{\mathcal{F}}_{m+n}.$

Further $A_{n}$ is defined by the homotopy fiber sequence

(5.1) $\displaystyle A_{n}\to{\mathcal{F}}_{2n+1}\to S^{1}_{2}$

where the second map is the localization of the composite

$F_{2n+1}\Omega^{\rho}S^{\rho+1}\to\Omega^{\rho}S^{\rho+1}\to S^{1}.$

We prove the $C_{2}$-equivariant analogue of [Coh+88, Theorem 1.3].

######  Theorem 5.1.

The fiber sequence (2.2) is equivalent to a product fibration. Indeed, there
is a $C_{2}$-equivariant map
$A_{n}\overset{\phi}{\rightarrow}{\mathcal{F}}_{2n}$ and a commutative diagram
of fibrations

${A_{n}}$${A_{n}}$${S_{2}^{1}\times
A_{n}}$${S_{2}^{1}\times{\mathcal{F}}_{2n}={\mathcal{F}}_{1}\times{\mathcal{F}}_{2n}}$${{\mathcal{F}}_{2n+1}}$${S_{2}^{1}}$${S_{2}^{1}}$$\scriptstyle{1\times\phi}$$\scriptstyle{p_{1}}$$\scriptstyle{m}$

which is an equivalence on total spaces and on fibers.

Our proof of Theorem 5.1 is a $C_{2}$-equivariant analogue of [Coh+88, §2.
Proof of Theorem 1.3]. Following [Coh+88], we first show that the inclusion

$F_{m-1}\Omega^{\rho}S^{\rho+1}\to F_{m}\Omega^{\rho}S^{\rho+1}$

may be considered as a $C_{2}$-equivariant inclusion into a mapping cone. For
a pointed $C_{2}\times\Sigma_{m}$-space $X,$ we define

$M_{m}(X)=C_{m}(\rho)\underset{\Sigma_{m}}{\times}X/C_{m}(\rho)\underset{\Sigma_{m}}{\times}*.$

Let $I$ denote the unit interval, $I^{m}$ the $m$-dimensional unit interval,
and $\partial I^{m}$ the boundary of $I^{m},$ all with trivial $C_{2}$-action.
Note that $I/\partial I\simeq S^{1}$ and $\partial I^{m}\simeq S^{m-1}.$

######  Lemma 5.2.

Let $F_{m}=F_{m}\Omega^{\rho}S^{\rho+1}.$ There is a cofibration sequence

$M_{m}(\partial I^{m})\xrightarrow[]{c}F_{m-1}\to F_{m}.$

###### Proof.

Let $T^{m}(S^{1})$ denote the wedge with trivial $C_{2}$-action consisting of
points in the $m$-fold Cartesian product having at least one component the
basepoint. Viewing $I^{m}$ as the cone of the natural map $\partial I^{m}\to
T^{m}(I/\partial I)$ yields a $C_{2}\times\Sigma_{m}$-equivariant cofibration

$\partial I^{m}\to T^{m}(I/\partial I)\hookrightarrow(I/\partial I)^{\times
m},$

with trivial $C_{2}$-action and hence a cofibration

(5.3) $\displaystyle M_{m}(\partial
I^{m})\xrightarrow[]{k}M_{m}(T^{m}(I/\partial I))\to M_{m}((I/\partial
I)^{\times m}.$

Recall from [RS00] that

$F_{m}=\left(\bigcup_{k\leq m}M_{k}((I/\partial I)^{\times k})\right)/\sim.$

The map $c$ of Lemma 5.2 is the composite

$M_{m}(\partial I^{m})\to M_{m}(T^{m}(I/\partial I))\to F_{m-1},$

where the second map uses the equivalence relation from (2.1) to ignore the
basepoint in at least one component. The required homeomorphism from the
mapping cone of $c$ to $F_{m}$ is a quotient of the homeomorphism in (5.3)
from the mapping cone of $k$ to $M_{m}((I/\partial I)^{\times m}.$ ∎

We now return to the proof of Theorem 5.1. Assume by induction that the
theorem has been proved for $n-1$ so ${\mathcal{F}}_{2(n-1)+1}\simeq
S_{2}^{1}\times A_{n-1}.$ Note that
${\mathcal{F}}_{2(n-1)+1}\to{\mathcal{F}}_{2n-1}$ is an equivalence.
Localizing Lemma 5.2 yields a map

(5.4) $\displaystyle M_{2n}(\partial I^{2n})_{2}\to{\mathcal{F}}_{2n-1}\simeq
S_{2}^{1}\times A_{n-1}.$

######  Lemma 5.5.

The cohomology $H^{1}_{C_{2}}(M_{2n}(\partial
I^{2n})_{2};{\underline{{\mathbb{Z}}}_{p}}\,)=0,$ so the map (5.4) is of the
form $*\times h$, for some map $h:M_{2n}(\partial I^{2n})_{2}\rightarrow
A_{n-1}$.

###### Proof.

We will show $H^{1}(M_{2n}(\partial I^{2n})^{C_{2}};{{\mathbb{Z}}})=0.$ Since
$H^{1}(M_{2n}(\partial I^{2n})^{e};{{\mathbb{Z}}})$ is observed to be zero in
[Coh+88], this will imply $H^{1}_{C_{2}}(M_{2n}(\partial
I^{2n});{\underline{{\mathbb{Z}}}})=0$ by [BW18, Lemma 2.8], and upon
localizing, $H^{1}_{C_{2}}(M_{2n}(\partial
I^{2n})_{2};{\underline{{\mathbb{Z}}}_{p}}\,)=0.$

To compute $H^{1}(M_{2n}(\partial I^{2n})^{C_{2}};{{\mathbb{Z}}}),$ we first
identify $M_{2n}(\partial I^{2n})^{C_{2}}.$ Since $M_{2n}(\partial I^{2n})$ is
a quotient space, it can be constructed as the colimit

$C_{2n}(\rho)\times_{\Sigma_{2n}}*\hookrightarrow
C_{2n}(\rho)\times_{\Sigma_{2n}}S^{2n-1}\to M_{2n}(S^{2n-1}).$

Further, since taking fixed points commutes with filtered colimits for finite
groups, there is an equivalence

$\textstyle{(C_{2n}(\rho)\times_{\Sigma_{2n}}*)^{C_{2}}}$$\textstyle{\hookrightarrow}$$\textstyle{(C_{2n}(\rho)\times_{\Sigma_{2n}}S^{2n-1})^{C_{2}}}$$\textstyle{\to}$$\textstyle{M_{2n}(S^{2n-1})^{C_{2}}}$

$\,\simeq$

$\,\simeq$

$\,\simeq$

$\textstyle{C_{2n}({\mathbb{R}})\times_{\Sigma_{2n}}*}$$\textstyle{\hookrightarrow}$$\textstyle{C_{2n}({\mathbb{R}})\times_{\Sigma_{2n}}S^{2n-1}}$$\textstyle{\to}$$\textstyle{M_{2n}(S^{2n-1})^{C_{2}}}$

Thus $M_{2n}(\partial I^{2n})^{C_{2}}\simeq
C_{2n}({\mathbb{R}})\times_{\Sigma_{2n}}S^{2n-1}/C_{2n}({\mathbb{R}})\times_{\Sigma_{2n}}*.$

The freeness of the action of $\Sigma_{2n}$ on $C_{2n}(\mathbb{R})$ implies

$C_{2n}(\mathbb{R})\times_{\Sigma_{2n}}S^{2n-1}\simeq\big{(}C_{2n}(\mathbb{R})/\Sigma_{2n}\big{)}\times
S^{2n-1}.$

Since $S^{2n-1}$ is $2n-1$-connective, we can conclude that
$H^{1}(C_{2n}(\mathbb{R})\times_{\Sigma_{2n}}S^{2n-1})=0$, and so
$H^{1}(M_{2n}(\partial I^{2n})^{C_{2}};\mathbb{Z})=0$. ∎

We return to the proof of Theorem 5.1. Let $Y$ denote the mapping cone of $h.$
The map of cofibrations

${M_{2n}(\partial
I^{2n})_{2}}$${A_{n-1}}$${Y}$${{\mathcal{F}}_{2n}/{\mathcal{F}}_{2n-1}}$${M_{2n}(\partial
I^{2n})_{2}}$${{\mathcal{F}}_{2n-1}}$${{\mathcal{F}}_{2n}}$${{\mathcal{F}}_{2n}/{\mathcal{F}}_{2n-1}}$$\scriptstyle{\phi}$

shows that $H{\underline{{\mathbb{F}}}_{2}}_{\star}Y\to
H{\underline{{\mathbb{F}}}_{2}}_{\star}{\mathcal{F}}_{2n}$ is injective with
image spanned by monomials of weight divisible by $2.$ There is a map of
fibrations

${Y}$${A_{n}}$${S_{2}^{1}\times
Y}$${S_{2}^{1}\times{\mathcal{F}}_{2n}={\mathcal{F}}_{1}\times{\mathcal{F}}_{2n}}$${{\mathcal{F}}_{2n+1}}$${S_{2}^{1}}$${S_{2}^{1}.}$$\scriptstyle{1\times\phi}$$\scriptstyle{p_{1}}$$\scriptstyle{\mu}$

The map of total spaces induces an isomorphism in
$H\underline{{\mathbb{F}}}_{2}$-homology, and since ${\mathcal{F}}_{2n+1}$ is
$H{\underline{{\mathbb{F}}}_{2}}_{\star}$-local, this map is an
$H{\underline{{\mathbb{F}}}_{2}}_{\star}$-localization. Hence, there is an
equivalence of fibrations

${Y_{p}}$${A_{n}}$${S_{2}^{1}\times
Y_{p}}$${{\mathcal{F}}_{2n+1}}$${S_{2}^{1}}$${S_{2}^{1},}$

extending the induction and completing the proof of Theorem 5.1.

##  6\. Proof of Main Theorem

Now we can use Theorems 4.1 and 5.1 to define $B_{1}(n),$ a
$C_{2}$-equivariant analogue of the $n$th integral Brown-Gilter spectrum at
the prime $2$ as a Thom spectrum. Let $\tilde{A_{n}}$ denote the homotopy
fiber of the composite

$F_{2n+1}\Omega^{\rho}S^{\rho+1}\to\Omega^{\rho}S^{\rho+1}\to S^{1}$

so the $H\underline{{\mathbb{F}}}_{2}$-localization of $\tilde{A_{n}}$ is
$A_{n}.$ Using the $H\underline{{\mathbb{F}}}_{2}$-localization of the
commutative diagram

(6.1)
${\tilde{A_{n}}}$${F_{2n+1}}$${\Omega^{\rho}S^{\rho+1}\langle\rho+1\rangle}$${\Omega^{\rho}S^{\rho+1},}$

we define $B_{1}(n)$ to be the Thom spectrum $(A_{n})^{{\mu}}.$

Taking the Thom spectrum of the composites

$A_{m}\times
A_{n}\xrightarrow[]{\phi_{m}\times\phi_{n}}{\mathcal{F}}_{2m}\times{\mathcal{F}}_{2n}\to{\mathcal{F}}_{2m+2n}\to{\mathcal{F}}_{2n+2m+1}\to
A_{m+n},$

where the last map splits the equivalence of Theorem 5.1, yields pairings

$B_{1}(m)\wedge B_{1}(n)\to B_{1}(m+n).$

Theorem 5.1 implies the Thom map ${(i_{n})}^{{\mu}}:B_{1}(n)\to
H\underline{{\mathbb{Z}}}_{2}$ induces a monomorphism in homology. Recall

${H\underline{{\mathbb{F}}}_{2}}_{\star}H{\underline{{\mathbb{Z}}}_{2}}\cong
H{\underline{{\mathbb{F}}}_{2}}_{\star}[\bar{\xi_{1}},\bar{\xi_{2}},\bar{\xi_{3}},\cdots,c(\tau_{1}),c(\tau_{2}),\cdots]/(c(\tau_{i}^{2})=ac(\tau_{i+1})+uc(\xi_{i+1}))$

where $|c(\tau_{j})|=2^{j}\rho-\sigma,$ $|\bar{\xi}_{i}|=(2^{i}-1)\rho,$ and
$c$ denotes the antiautomorphism of the dual Steenrod algebra
${\mathcal{A}}\cong\pi_{\star}H\underline{{\mathbb{F}}}_{2}\wedge
H\underline{{\mathbb{F}}}_{2}$ [Orm11]. Also recall that

$\operatorname{wt}(\bar{\tau}_{j})=\operatorname{wt}(\bar{\xi}_{j})=2^{j},\qquad\operatorname{wt}(xy)=\operatorname{wt}(x)+\operatorname{wt}(y),$

and that all monomials have weight divisible by $2.$

Then our $C_{2}$-equivariant analogue of [Coh+88, Theorem 1.5(i),(ii)] follows
immediately from Theorem 5.1:

######  Theorem 6.1.

For $n>0,$ there is an $H\underline{{\mathbb{F}}}_{2}$-complete spectrum
$B_{1}(n)$ and a map

$B_{1}(n)\xrightarrow[]{g}H{\underline{{\mathbb{Z}}}_{2}}$

such that

  * (i)

$g_{\star}$ sends $H{\underline{{\mathbb{F}}}_{2}}_{\star}(B_{1}(n))$
isomorphically onto the span of monomials of weight $\leq 2n;$

  * (ii)

there are pairings

$B_{1}(m)\wedge B_{1}(n)\to B_{1}(m+n)$

whose homology homomorphism is compatible with the multiplication in
$H{\underline{{\mathbb{F}}}_{2}}_{\star}(H\underline{{\mathbb{Z}}}_{2}).$

######  Remark 6.2.

It is not currently known if a $C_{2}$-equivariant analogue of [Coh+88,
Theorem 1.5(iii)] holds or if there should be some other criterion determining
(integral) Brown–Gitler spectra in the $C_{2}$-equivariant case. In the non-
equivariant case, [Coh+88, Theorem 1.5(iii)] states that for any CW-complex
$X,$

$g_{*}:B_{1}(n)_{i}(X)\to H_{i}(X;{\mathbb{Z}}_{2})$

is surjective if $i\leq 2p(n+1)-1.$ This condition originated in the geometry
of imersions of manifolds and would be interesting to study further in the
$C_{2}$-equivariant setting.

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