# Quantum Dissipation at Conical Intersections of Quasienergies

Sigmund Kohler  Instituto de Ciencia de Materiales de Madrid, CSIC, E-28049
Madrid, Spain

###### Abstract

We introduce the notion of conical intersections to Floquet theory and work
out consequences of the underlying spatio-temporal symmetries for a driven
two-level system coupled to an ohmic heat bath. We find that on manifolds with
constant quasienergy splitting, the mean energies of the Floquet states vary
significantly. The stationary populations behave in the same way, which
emphasizes the importance of the mean energies for dissipation. Owing to
symmetries, the stationary state may be fully mixed even at zero temperature.
In the limit of large driving frequency, such full mixture is found in the
whole vicinity of the intersection.

Introduction.—Floquet engineering is the construction of effective
Hamiltonians by acting with ac fields or periodic pulses upon a quantum
system. The idea originates in the suppression of tunneling in bistable
systems [1] and tight-binding models [2]. It has been proposed also for many-
particle states in double quantum dots [3], cold atoms [4, 5], and photonics
[6]. Experimental realizations have been achieved with quantum dots [7, 8, 9],
superconducting qubits [10, 11], and optical lattices [12]. These ideas have
been extended to tuning topological properties [13, 14] and artificial gauge
fields [5, 15].

A theoretical tool for studying such systems is Floquet theory with its
quasienergies [16, 17] which emerge in the formal expression for the
propagator and, thus, provide much insight. For example, the suppression of
tunneling can be attributed to exact quasienergy crossings [1, 2], which
require a spatio-temporal symmetry known as generalized parity [18, 19]. When
a small perturbation such as a static detuning breaks this symmetry, the exact
quasienergy crossings become avoided. Then, as will be shown, the quasienergy
surfaces as a function of any original parameter and the detuning form a
double cone with a degeneracy at the tip, a so-called conical or diabolical
intersection. Conical intersections have been investigated mainly in the
rather different context of breakdown of adiabaticity in Born-Oppenheimer
theory [20, 21, 22, 23, 24].

Dissipative transitions between Floquet states lack detailed balance [25].
Therefore, the stationary state generally is not a canonical ensemble, but
must be computed with a quantum master equation [26, 27]. Even for minimal
dissipation, the populations may significantly depend on the operator that
couples the system to the environment [28, 29, 30]. Only in the high-frequency
limit, one may find Floquet-Gibbs states, which are canonical states with
eigenenergies replaced by quasienergies [31, 32]. In many cases, by contrast,
the mean energies are more relevant for the populations [27, 33, 9]. For many
of these results, level crossings play an important role [27, 28, 29, 30, 9,
34]. Therefore, dissipative driven quantum dynamics in the vicinity of conical
intersections is of particular interest.

In this work we analyze the behavior of Floquet states at such intersections
as well as their stationary state which results from a coupling to a heat
bath. It will turn out that many properties can be deduced already from the
behavior of the Floquet states and the system-bath coupling under the symmetry
operation responsible for the emergence of the crossing.

Floquet states at conical intersections.—As a model, we consider the ac-driven
two-level Hamiltonian

$H(t)=\frac{\Delta}{2}\sigma_{x}+\frac{1}{2}[\varepsilon+A\cos(\Omega
t)]\sigma_{z},$ (1)

with tunneling $\Delta$, static detuning $\varepsilon$, driving amplitude $A$
and frequency $\Omega$, while $\sigma_{x,z}$ denotes Pauli matrices.

Owing to time periodicity, the corresponding Schrödinger equation possesses a
complete set of solutions of the form $e^{-iqt}|\phi(t)\rangle$ with
quasienergy $q$ and Floquet state $|\phi(t)\rangle$ which shares the time
periodicity of the driving [16, 17]. Floquet states are eigensolutions of the
operator $\mathcal{H}=H(t)-i\partial_{t}$ (in units with $\hbar=1$).
Quasienergies come in a Brillouin zone structure, i.e., for any
$|\phi(t)\rangle$ and integer $n$, $|\phi^{(n)}(t)\rangle=e^{-in\Omega
t}|\phi(t)\rangle$ is an equivalent Floquet state with quasienergy
$q+n\Omega$. To characterize Floquet states, one may employ their mean
energies $E$ which are the expectation values of $H(t)$ averaged over the
driving period. Equivalent states possess equal mean energy, as is expected
for an observable. For the present purpose, it is convenient to consider
Floquet states as elements of Sambe space which is the direct product of the
underlying Hilbert space and the space of $2\pi/\Omega$-periodic functions
[17]. We denote its elements by a double angle, $|\phi\rangle\\!\rangle$,
where $|\phi(t)\rangle=\langle t|\phi\rangle\\!\rangle$ and
$E=q+\langle\\!\langle\phi|i\partial_{t}|\phi\rangle\\!\rangle$.

Hamiltonian (1) is invariant under time-reversal $t\to-t$ which allows one to
write the Floquet states $|\phi(t)\rangle$ as Fourier series with real
coefficients. Moreover, the chirality $C=i\sigma_{y}K$ with $K$ denoting
complex conjugation maps $\mathcal{H}\to-\mathcal{H}$, which implies that non-
zero quasienergies come in pairs with opposite sign. Without loss of
generality, we will choose the Floquet states from the Brillouin zone
symmetric around zero, $-\Omega/2\leq q<\Omega/2$, and label them such that
$q_{0}\leq 0$, $q_{1}=-q_{0}$ and $E_{1}=-E_{0}$. Notice, however, that this
order is arbitrary and by itself cannot justify designations such as “ground
state”.

Figure 1: (a) Quasienergies $q_{\pm}$ (solid lines) and mean energies
$E_{\pm}$ (dashed) for driving frequency $\Omega=25\Delta$ in the absence of a
static detuning, such that the Floquet states possess definite generalized
parity indicated by colors (green: even, orange: odd). We focus on the
vicinity of the quasienergy crossings marked by grey arrows. (b) The same for
static detuning $\varepsilon=0.03\Delta$, i.e., in the absence of generalized
parity. (c) Quasienergies $q_{0}$ and $q_{1}$ as a function of the detuning
$\varepsilon$ and the driving amplitude $A$. (d) Floquet state with lower
quasienergy, $|\phi_{0}(t)\rangle$, in the vicinity of a degeneracy point
$(\varepsilon,A)=(0,A_{0})$ in terms of the basis states
$|\varphi_{\pm}(t)\rangle$. The underlying color indicates the corresponding
mean energy $E_{0}=E_{-}\cos\vartheta$ [green: $E_{0}>0$, cf. panel (a)].

A $\mathbb{Z}_{2}$ symmetry with direct consequences for the physics discussed
herein is the generalized parity $G=\sigma_{x}\exp(\pi\partial_{t}/\Omega)$
which consists of a unitary transformation with $\sigma_{x}$ in Hilbert space
and a time shift by half a period. It is present for $\varepsilon=0$. Then,
the Floquet states can be classified as even or odd depending on the
respective eigenvalue of $G$ [19, 18]. Quasienergies of states with different
generalized parity may cross to form accidental degeneracies. Also the mean
energies may exhibit exact crossings which, however, need not coincide with
the quasienergy crossings, see Fig. 1(a). Two properties of the generalized
parity will turn out important. First, $\sigma_{x}$ and $i\partial_{t}$ are
even under transformation with $G$, while $\sigma_{z}$ is odd. Second, from
the above definition of the equivalent states in the time domain, it is
obvious that for a Floquet state with generalized parity $\pm 1$, the
equivalent state $|\phi^{(n)}\rangle\\!\rangle$ has generalized parity
$\pm(-1)^{n}$.

The situation changes in the presence of a detuning $\varepsilon\neq 0$. Then
the Floquet states no longer possess generalized parity, and exact quasienergy
crossings turn into avoided crossings, as can be witnessed in Fig. 1(b). This
implies that the quasienergies are degenerate only at isolated points of
parameter space $(\varepsilon,A)$. Therefore, they form accidental symmetry-
allowed conical intersections, as is shown in Fig. 1(c). Henceforth, we focus
on the vicinities of such intersections.

In these regions, we employ degenerate perturbation theory using the Floquet
states $|\varphi_{\pm}\rangle\\!\rangle$ for
$H_{0}(t)\equiv\frac{1}{2}\Delta\sigma_{x}+\frac{1}{2}A_{0}\sigma_{z}\cos(\Omega
t)$ as zeroth order, where $A_{0}$ denotes the driving amplitude at the
respective quasienergy crossing (the notation $\varphi_{\pm}$ is restricted to
$\varepsilon=0$ and $A=A_{0}$). For large $\Omega$, $A_{0}$ can be expressed
with a root of a Bessel function [19], but the present investigation is not
restricted to this limit. Let us emphasize that while the
$|\varphi_{\pm}(t)\rangle$ form a basis of the Hilbert space, the full Sambe
space is spanned by all $|\varphi_{\pm}^{(n)}\rangle\\!\rangle$. By evaluating
the matrix elements of $H_{1}(t)=H(t)-H_{0}(t)$ in the subspace considered,
one finds the time-independent perturbation
$H_{1}^{\prime}=\frac{1}{2}(a\sigma_{x}^{\prime}+b\sigma_{z}^{\prime})$ with

$\displaystyle a={}$
$\displaystyle\frac{A_{0}-A}{A_{0}}\big{(}2E_{-}+\Delta\langle\\!\langle\varphi_{-}|\sigma_{x}|\varphi_{-}\rangle\\!\rangle\big{)}\equiv
r\cos\vartheta,$ (2) $\displaystyle b={}$
$\displaystyle\varepsilon\langle\\!\langle\varphi_{-}|\sigma_{z}|\varphi_{+}\rangle\\!\rangle\equiv
r\sin\vartheta,$ (3)

and $r=\sqrt{a^{2}+b^{2}}$, where the prime refers to the time-dependent basis
$|\varphi_{\pm}(t)\rangle$. To obtain the expression for $a$, we have written
$\sigma_{z}\cos(\Omega t)$ in terms of $H_{0}(t)-i\partial_{t}$ and have used
$E=q+\langle\\!\langle i\partial_{t}\rangle\\!\rangle$ together with the fact
that the quasienergies of the chosen basis states vanish. Diagonalization of
$H_{1}^{\prime}$ provides the perturbed Floquet states

$\displaystyle|\phi_{0}(t)\rangle$
$\displaystyle={}\cos(\vartheta/2)|\varphi_{-}(t)\rangle+\sin(\vartheta/2)|\varphi_{+}(t)\rangle$
(4) $\displaystyle|\phi_{1}(t)\rangle$
$\displaystyle={}-\sin(\vartheta/2)|\varphi_{-}(t)\rangle+\cos(\vartheta/2)|\varphi_{+}(t)\rangle$
(5)

and quasienergies $q_{0,1}=\pm r/2$. Since
$G^{\dagger}i\partial_{t}G=i\partial_{t}$ and hence
$\langle\\!\langle\varphi_{+}|i\partial_{t}|\varphi_{-}\rangle\\!\rangle=0$,
the corresponding mean energies read $E_{0}=E_{-}\cos\vartheta=-E_{1}$. Figure
1(d) visualizes the definition of the angle $\vartheta$ and the result of the
perturbation theory. Interestingly, values of $\vartheta$ that differ by $\pi$
give rise to a duality. Upon the shift $\vartheta\to\vartheta+\pi$,
$|\phi_{0}(t)\rangle$ becomes $\propto|\phi_{1}(t)\rangle$ and vice versa.
Notice however, that while the shape of the Floquet states is interchanged,
their quasienergies remain the same, i.e., $q_{0}<0$ for all values of
$\vartheta$.

A central ingredient of our analysis will be the action of the generalized
parity operator $G$ on the Floquet states. Using
$G|\varphi_{\pm}^{(n)}\rangle\\!\rangle=\pm(-1)^{n}|\varphi_{\pm}^{(n)}\rangle\\!\rangle$,
one obtains

$\displaystyle G|\phi_{0}^{(n)}\rangle\\!\rangle={}$
$\displaystyle(-1)^{n}\left(-|\phi_{0}^{(n)}\rangle\\!\rangle\cos\vartheta+|\phi_{1}^{(n)}\rangle\\!\rangle\sin\vartheta\right),$
(6) $\displaystyle G|\phi_{1}^{(n)}\rangle\\!\rangle={}$
$\displaystyle(-1)^{n}\left(|\phi_{0}^{(n)}\rangle\\!\rangle\sin\vartheta+|\phi_{1}^{(n)}\rangle\\!\rangle\cos\vartheta\right).$
(7)

For $\vartheta=\pm\pi/2$, $G$ maps the one Floquet state to the other, i.e.
$G|\phi_{0}^{(n)}\rangle\\!\rangle\propto|\phi_{1}^{(n)}\rangle\\!\rangle$.

Dissipation and stationary state.—To describe quantum dissipation, we couple
the driven system to a heat bath modeled as harmonic oscillators with the
Hamiltonians [35, 36, 37, 38]
$H_{\text{bath}}=\sum_{\nu}\omega_{\nu}a_{\nu}^{\dagger}a_{\nu}$ and
$H_{\text{int}}=S\xi$, where
$\xi=\sum_{\nu}\lambda_{\nu}(a_{\nu}^{\dagger}+a_{\nu})$ is a collective bath
coordinate and $S$ a system coordinate to be specified later. The influence of
the bath is determined by its spectral density
$J(\omega)=\pi\sum_{\nu}\delta(\omega-\omega_{\nu})$ and the correlation
function $C(t)=\langle\xi(0)\xi(t)\rangle$. At thermal equilibrium, the latter
in frequency space reads $C(\omega)=2J(\omega)n_{\text{th}}(\omega)$ with the
bosonic occupation number $n_{\text{th}}(\omega)=[\exp(\omega/k_{B}T)-1]^{-1}$
and temperature $T$. We assume an ohmic spectral density
$J(\omega)=\pi\alpha\omega/2$ with dimensionless coupling strength $\alpha$
[35].

Focussing on weak dissipation, we employ the Floquet-Bloch-Redfield formalism
of Refs. [39, 29] which provides a master equation in Floquet basis of the
form
$\dot{\rho}_{\alpha\alpha^{\prime}}(t)=\sum_{\beta,\beta^{\prime}}\mathcal{L}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}(t)\rho_{\beta\beta^{\prime}}(t)$.
Generally, the Liouvillian $\mathcal{L}$ is $2\pi/\Omega$-periodic, but within
a rotating-wave approximation can be replaced by its time-average. While this
formulation is suitable for computational purposes, for our analytical
treatment we employ a secular approximation. For $\alpha\lll 1$ the density
operator eventually becomes diagonal [39], such that the stationary state can
be computed from the Pauli-type master equation
$\dot{p}_{\alpha}=\sum_{\beta}(\Gamma_{\alpha\beta}p_{\beta}-\Gamma_{\beta\alpha}p_{\alpha})$
for the Floquet-state populations $p_{\alpha}$ with the golden-rule rates [26,
39, 40]

$\Gamma_{\alpha\beta}=\sum_{k}C(q_{\alpha}-q_{\beta}+k\Omega)|S_{\alpha\beta,k}|^{2},$
(8)

where $S_{\alpha\beta,k}$ denotes the $k$th Fourier component of the time-
dependent transition matrix element
$S_{\alpha\beta}(t)=\langle\phi_{\alpha}(t)|S|\phi_{\beta}(t)\rangle$. In
Sambe formalism, it can be written as
$S_{\alpha\beta,k}=\langle\\!\langle\phi_{\alpha}|S|\phi_{\beta}^{(k)}\rangle\\!\rangle=\langle\\!\langle\phi_{\beta}|S|\phi_{\alpha}^{(-k)}\rangle\\!\rangle^{*}$,
which will allow an elegant formulation of the symmetry properties.

Close to an intersection, the correlation function $C(\omega)$ in Eq. (8) is
evaluated at the frequencies $\omega=\pm r+k\Omega$, where the quasienergy
splitting obeys $r\ll\Omega$. Therefore, $r$ can generally be neglected, such
that the dissipative rates depend only on the basis states
$|\varphi_{\pm}\rangle\\!\rangle$ and the angle $\vartheta$. However, there
exist also exceptional cases in which $S_{\alpha\beta}(t)$ is practically
time-independent, such that in Eq. (8) the term with $k=0$ dominates. Then
$\Gamma_{\alpha\beta}$ is formally given by the corresponding expression for a
time-independent problem, but with the eigenenergies replaced by
quasienergies, which is the constituting feature of a Floquet-Gibbs state [31,
32].

Figure 2: (a) Population $p_{0}$ of the Floquet state with lower quasienergy
in the vicinity of $A=A_{0}\approx 2.4\Omega$ and $\varepsilon=0$ for driving
frequency $\Omega=5\Delta$. The bath with temperature $T=0.01\Delta/k_{B}$ is
coupled with strength $\alpha=0.01$ via $S=\sigma_{x}$. (b) Population $p_{0}$
for $\varepsilon=0$. The dashed line depicts the population of the Floquet
state with even generalized parity, $\phi_{+}$, which is continuous. Inset:
Discontinuity of $p_{0}$ at the intersection as a function of the driving
frequency $\Omega$. The solid line indicates a decay $\propto 1/\Omega$.

The generic behavior.—We start with a bath coupling via $S=\sigma_{x}$ which
represents the generic case in which the Fourier coefficients
$S_{\alpha\beta,k}$ with $k\neq 0$ are significant, such that $r$ can be
neglected. As already discussed, upon $\vartheta\to\vartheta+\pi$ the Floquet
states (4) and (5) are interchanged. Consequently, the transition rates
$\Gamma_{01}$ and $\Gamma_{10}$ and, thus, the populations $p_{0}$ and $p_{1}$
are interchanged as well, i.e., $p_{0}$ at an angle $\vartheta$ must be equal
to $p_{1}=1-p_{0}$ at $\vartheta+\pi$. In short, the stationary populations
exhibit the duality of the Floquet states, such that $p_{0}-1/2$ turns out
anti-symmetric upon point reflection at the cone tip $(0,A_{0})$. This implies
that qualitatively the populations behave like the mean energies, which is
confirmed by the numerical data shown in Fig. 2(a). On a line through the tip,
this may imply a discontinuity which can be attributed to the discontinuities
of the Floquet states (4) and (5). In a continuous basis such as the one
formed by the states with definite generalized parity along the $A$-axis, by
contrast, the populations are continuous as well. This is verified by the
dashed lines in Fig. 2(b).

Since the populations as a function of $\vartheta$ change from $p_{0}<p_{1}$
to the opposite order, we conjecture that for some angle $p_{0}=p_{1}=1/2$.
Figure 2(a) suggests, that this occurs at $\vartheta=\pm\pi/2$. For a proof,
we use $G^{\dagger}\sigma_{x}G=\sigma_{x}$ together with Eqs. (6) and (7). For
$\vartheta=\pm\pi/2$ this provides the relation
$S_{01,k}=\langle\\!\langle\phi_{0}|G^{\dagger}\sigma_{x}G|\phi_{1}^{(k)}\rangle\\!\rangle=(-1)^{k}S_{10,k}$.
Therefore, all terms that contribute to the transition rates are equal such
that $\Gamma_{01}=\Gamma_{10}$ and, thus, $p_{0}=p_{1}$. In other words, the
stationary density operator is proportional to the unit matrix, i.e., it
describes a fully incoherent mixture even at arbitrarily small temperature.

The symmetry properties discussed so far hold for any driving frequency
$\Omega\gtrsim\Delta$. Within the usual high-frequency approximation [19] (for
a summary, see the Appendix), we find a further symmetry relation. It is based
on the transformation with $\sigma_{z}$, which in the Hamiltonian (1) inverts
the sign of the tunnel term, effectively causing $\Delta\to-\Delta$. Since for
large frequencies the quasienergies read
$q_{0,1}=\mp(\Delta/2)J_{0}(A/\Omega)$ [19], $\sigma_{z}|\phi_{0}(t)\rangle$
must be proportional to the Floquet state with opposite quasienergy, i.e.,
$\sigma_{z}|\phi_{0}(t)\rangle\propto|\phi_{1}(t)\rangle$ and vice versa.
Thus, for any value of $\vartheta$, $\sigma_{z}$ acts on the Floquet states
like the generalized parity operator $G$ for the particular value
$\vartheta=\pi/2$. Therefore, we can expect to find in the high-frequency
limit the corresponding consequences in the whole vicinity of the
intersection. Specifically, we find for all $\vartheta$ the fully mixed state
with $p_{0}=p_{1}=1/2$.

It is interesting to see how finite-frequency corrections vanish with
increasing $\Omega$. The inset of Fig. 2(b) reveals that the limit is
approached $\propto 1/\Omega$. Even for $\Omega=15\Delta$, the difference
$\sim 0.1$ is still remarkably large. Thus, while the usual high-frequency
approximation for the quasienergies is reliable for $\Omega\gtrsim 2\Delta$
[19], for dissipation it requires larger frequencies.

Figure 3: The same as Fig. 2 but for a bath coupling via $S=\sigma_{z}$.

Coupling via $\sigma_{z}$ and Floquet-Gibbs state.—A bath coupling via
$S=\sigma_{z}$ anti-commutes with the generalized parity operator such that
$G^{\dagger}\sigma_{z}G=-\sigma_{z}$. Then the symmetry relation of the matrix
element of $S$ acquires a minus sign and reads $S_{01,k}=(-1)^{k+1}S_{10,k}$.
Since the rates $\Gamma_{\alpha\beta}$ depend only on absolute values, the
generic behavior discussed above should be found as well. However, as we can
appreciate in Fig. 3(a), this is the case only very close to the intersection,
while for a slightly smaller amplitude, the population of $\phi_{0}$ becomes
much larger. The reason for this is that there the Fourier component with
$k=0$ dominates. This becomes clear when computing the matrix elements within
the usual high-frequency approximation,
$\langle\phi_{0}(t)|\sigma_{z}|\phi_{1}(t)\rangle=-1$ (see the explicit
expressions in the Appendix), i.e., it is time independent such that
$S_{\alpha\beta,k}$ vanishes unless $k=0$. Therefore, we expect to find a
Floquet-Gibbs state. Indeed the conditions for the emergence of such state
formulated in Refs. [31, 32] are fulfilled, namely the driving frequency is
very large, while the driving and the system-bath coupling commute.

We are also interested in intermediate frequencies for which components with
$k\neq 0$ still contribute. Very close to the intersection, $r$ is
sufficiently small such that the term with $k=0$ may be neglected. Then the
population $p_{0}$ still resembles the generic case, including the
discontinuity at $A_{0}$. When moving away from the intersection, however, $r$
grows such that the contribution of $S_{\alpha\beta,0}$ becomes increasingly
important and one observes a crossover from the generic behavior to a Floquet-
Gibbs state. Data along the $A$-axis in Fig. 3(b) verify that for
$\Omega=5\Delta$, features of the generic behavior are still present. With
increasing frequency, the crossover region shrinks and eventually disappears.

Mixed bath coupling and global picture.—One might suppose that at least for
moderate driving frequency, a bath coupling via any linear combination of
Pauli matrices $S=\sum_{i}c_{i}\sigma_{i}$ would also lead to the generic
behavior depicted in Fig. 2(a). This is however not the case, because the
condition of $S$ being either even or odd under generalized parity no longer
holds. Since $\sigma_{x}$ is even while $\sigma_{y}$ and $\sigma_{z}$ are odd,
the requirement is met only if either $c_{1}=0$ or $c_{2}=c_{3}=0$. For a
visualization of the crossover, see the Appendix.

Figure 4: Floquet state populations for bath coupling via $\sigma_{x}$ (a) and
$\sigma_{z}$ (b), respectively, cf. Figs. 2(a) and 3(a). The driving frequency
$\Omega=1.5\Delta$ is chosen rather low, such that in panel (b) the zone with
deviations from the Floquet-Gibbs state at $\varepsilon=0$ and $A\lesssim
A_{0}$ becomes visible. All other parameters are as in Fig. 2. The rectangles
mark the conical intersections.

While we have worked out the scenario in the vicinities of conical
intersections, it remains to integrate this into a global picture. Therefore,
we show in Fig. 4 the population in a much larger parameter range. The stripes
with equal population (white) indicate that the choice of $S$ influences the
global behavior significantly. For a coupling via $\sigma_{x}$, the stripes
are merely bent, but nevertheless extend over a rather large range. For
$\sigma_{z}$, by contrast, the switching between the generic behavior and the
Floquet-Gibbs state brings about a further line with $p_{0}=1/2$. Details are
beyond the present symmetry analysis.

Conclusions.—We have worked out the properties of Floquet states in the
vicinity of conical intersections of quasienergies with a focus on
dissipation. It turned out that many features of the Floquet states as well as
their stationary populations follow already from symmetry relations. The
analytical achievements are based on a rotating-wave approximation which, as
our numerical results verify, holds even for relatively small driving
frequencies of the order of the tunnel matrix element. In particular, they are
beyond the common high-frequency approximation with Bessel functions. While
the quasienergies of the Floquet states form a double cone, their mean
energies exhibit a strong parameter dependence on manifolds with constant
quasienergy. In a generic scenario, the populations qualitatively behave like
the mean energies, which underlines the role of the latter. Remarkably, a
consequence of the symmetry properties is that the stationary state may be
fully mixed even at zero temperature. In the high-frequency limit, this
property extends to the whole double cone. This represents an analogy to the
breakdown of adiabaticity at conical intersections. Further insight may
require an investigation of the temperature dependence of the populations and
stronger system-bath coupling.

###### Acknowledgements.

This work was supported by the Spanish Ministry of Science, Innovation, and
Universities (Grant No. PID2020-117787GB-I00), and by the CSIC Research
Platform on Quantum Technologies PTI-001.

Figure 5: Difference of the numerically computed quasienergy and the
analytical approximation in Eq. (10) as a function of the driving amplitude.

Figure 6: Population of the Floquet state $\phi_{0}$ for system-bath coupling
via the operator $S$ defined in Eq. (12) with the angle $\theta$ given on the
top of each panel. The driving frequency is $\Omega=8\Delta$.

##  Appendix A High-frequency approximation

For completeness and reference, we provide the well-known expressions for the
Floquet states in the limit of large driving frequencies [19],

$|\phi_{\pm}(t)\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}\pm e^{-iF(t)}\\\
e^{iF(t)}\end{pmatrix},$ (9)

with $F(t)=(A/2\Omega)\sin(\Omega t)$ and their quasienergies

$q_{\pm}=\pm\frac{\Delta}{2}J_{0}(A/\Omega)$ (10)

with $J_{0}$ the zeroth-order Bessel function of the first kind. Thus, a
position $A_{0}$ of a conical intersection requires that $A_{0}/\Omega$
matches a root of $J_{0}$.

For a derivation of Eqs. (9) and (10) [19], one uses the Hamiltonian in the
absence of the tunnel term as zeroth order,
$H_{0}(t)=(A/2)\sigma_{z}\cos(\Omega t)$, where the corresponding propagator
reads $U_{0}(t)=\exp[-i(A/2\Omega)\sigma_{x}\sin(\Omega t)]$. Transformation
of the remaining contribution, $H_{1}=(\Delta/2)\sigma_{z}$ with $U_{0}$ and
subsequent time-average, i.e. a rotating-wave approximation, yields
$H_{1}^{\prime}=(\Delta/2)J_{0}(A/\Omega)\sigma_{x}$. Its eigenvalues are the
quasienergies, while the states in Eq. (9) are the eigenstates of $H^{\prime}$
transformed back to the Schrödinger picture. Figure 5 shows that already for
$\Omega\gtrsim 2\Delta$, the approximation in Eq. (10) deviates from the exact
value by only a few percent of the tunnel matrix element $\Delta$.

A particular feature of the Floquet states in Eq. (9) is that the transition
matrix element $\langle\phi_{+}(t)|\sigma_{z}|\phi_{-}(t)\rangle=-1$ is time-
independent. For $\vartheta=0$, we have $\phi_{0}=\phi_{-}$ and
$\phi_{1}=\phi_{+}$ (and vice versa for $\vartheta=\pi$) such that only the
Fourier component with $k=0$ contributes to the dissipative rates
$\Gamma_{\alpha\beta}$. For sufficiently large driving frequency, this finally
leads to the emergence of a Floquet-Gibbs state [31, 32]. For intermediate
$\Omega$, however, components with $k\neq 0$ still play a role such that
sufficiently close to $A_{0}$, the generic behavior is restored.

Within the present high-frequency approximation, the Floquet states possess a
further symmetry, namely a hidden chirality given by a transformation with
$\sigma_{z}$. Obviously, in the Hamiltonian in Eq. (1) of the main text, the
terms with the detunings $\varepsilon$ and $A\cos(\Omega t)$ are invariant
under transformation with $\sigma_{z}$. By contrast, the tunnel term changes
sign, effectively causing $\Delta\to-\Delta$. Thus, already from Eq. (10) it
becomes clear that $\sigma_{z}|\phi_{0}(t)\rangle$ and $|\phi_{1}(t)\rangle$
can differ at most by a phase factor. For the present choice of the phases of
the Floquet states in Eq. (9), one finds

$\sigma_{z}|\phi_{\pm}(t)\rangle=-|\phi_{\mp}(t)\rangle,$ (11)

which implies that for large driving frequencies, the Floquet states possess a
hidden chirality.

##  Appendix B Data for mixed system-bath coupling

As discussed in the main text, a mixed system-bath coupling via a linear
combination of $\sigma_{x}$ and $\sigma_{z}$ does not fulfill the symmetry
properties required for the generic behavior, because these two operators
transform differently under generalized parity. To see how the transition
between the two symmetric cases occurs, we consider a bath coupling via

$S=\sigma_{x}\sin\theta+\sigma_{z}\cos\theta.$ (12)

Figure 6 shows the population $p_{0}$ for various values of the coupling angle
$\theta$. In consistency with Refs. [29, 30], it shows that the coupling via
$\sigma_{z}$ is rather sensitive to any small admixture of $\sigma_{x}$.
Already for a rather small value of $\theta$, the behavior changes
qualitatively as the vertical symmetry axis has been lost completely. With
increasing angle $\theta$, the line with equal population $p_{0}=p_{1}=1/2$
turns smoothly into the behavior for pure $\sigma_{x}$ coupling.

## References

  * Grossmann  _et al._ [1991] F. Grossmann, T. Dittrich, P. Jung, and P. Hänggi, Coherent destruction of tunneling, Phys. Rev. Lett. 67, 516 (1991).
  * Holthaus [1992] M. Holthaus, Collapse of minibands in far-infrared irradiated superlattices, Phys. Rev. Lett. 69, 351 (1992).
  * Creffield and Platero [2002] C. E. Creffield and G. Platero, ac-driven localization in a two-electron quantum dot molecule, Phys. Rev. B 65, 113304 (2002).
  * Gong  _et al._ [2009] J. Gong, L. Morales-Molina, and P. Hänggi, Many-body coherent destruction of tunneling, Phys. Rev. Lett. 103, 133002 (2009).
  * Eckardt [2017] A. Eckardt, Colloquium: Atomic quantum gases in periodically driven optical lattices, Rev. Mod. Phys. 89, 011004 (2017).
  * Longhi [2011] S. Longhi, Many-body coherent destruction of tunneling in photonic lattices, Phys. Rev. A 83, 034102 (2011).
  * Stehlik  _et al._ [2012] J. Stehlik, Y. Dovzhenko, J. R. Petta, J. R. Johansson, F. Nori, H. Lu, and A. C. Gossard, Landau-Zener-Stückelberg interferometry of a single electron charge qubit, Phys. Rev. B 86, 121303(R) (2012).
  * Forster  _et al._ [2014] F. Forster, G. Petersen, S. Manus, P. Hänggi, D. Schuh, W. Wegscheider, S. Kohler, and S. Ludwig, Characterization of qubit dephasing by Landau-Zener-Stückelberg-Majorana interferometry, Phys. Rev. Lett. 112, 116803 (2014).
  * Chen  _et al._ [2021] M.-B. Chen, B.-C. Wang, S. Kohler, Y. Kang, T. Lin, S.-S. Gu, H.-O. Li, G.-C. Guo, X. Hu, H.-W. Jiang, G. Cao, and G.-P. Guo, Floquet state depletion in ac-driven circuit QED, Phys. Rev. B 103, 205428 (2021).
  * Sillanpää  _et al._ [2006] M. Sillanpää, T. Lehtinen, A. Paila, Y. Makhlin, and P. Hakonen, Continuous-time monitoring of Landau-Zener interference in a Cooper-pair box, Phys. Rev. Lett. 96, 187002 (2006).
  * Berns  _et al._ [2008] D. M. Berns, M. S. Rudner, S. O. Valenzuela, K. K. Berggren, W. D. Oliver, L. S. Levitov, and T. P. Orlando, Amplitude spectroscopy of a solid-state artificial atom, Nature (London) 455, 51 (2008).
  * Lignier  _et al._ [2007] H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O. Morsch, and E. Arimondo, Dynamical control of matter-wave tunneling in periodic potentials, Phys. Rev. Lett. 99, 220403 (2007).
  * Gómez-León and Platero [2013] A. Gómez-León and G. Platero, Floquet-Bloch theory and topology in periodically driven lattices, Phys. Rev. Lett. 110, 200403 (2013).
  * Atala  _et al._ [2013] M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, Direct measurement of the Zak phase in topological Bloch bands, Nat. Phys. 9, 795 (2013).
  * Aidelsburger  _et al._ [2018] M. Aidelsburger, S. Nascimbène, and N. Goldman, Artiﬁcial gauge ﬁelds in materials and engineered systems, C. R. Physique 19, 394 (2018).
  * Shirley [1965] J. H. Shirley, Solution of the Schrödinger equation with a Hamiltonian periodic in time, Phys. Rev. 138, B979 (1965).
  * Sambe [1973] H. Sambe, Steady states and quasienergies of a quantum-mechanical system in an oscillating field, Phys. Rev. A 7, 2203 (1973).
  * Peres [1991] A. Peres, Dynamical quasidegeneracies and quantum tunneling, Phys. Rev. Lett. 67, 158 (1991).
  * Großmann and Hänggi [1992] F. Großmann and P. Hänggi, Localization in a driven two-level dynamics, Europhys. Lett. 18, 571 (1992).
  * Yarkony [1996] D. R. Yarkony, Diabolical conical intersections, Rev. Mod. Phys. 68, 985 (1996).
  * Domcke  _et al._ [2011] W. Domcke, D. R. Yarkony, and H. Köppel, eds.,  _Conical Intersections_ , Advanced Series in Physical Chemistry, Vol. 17 (World Scientific, Singapore, 2011).
  * Larson  _et al._ [2020] J. Larson, E. Sjöqvist, and P. Öhberg,  _Conical Intersections in Physics: An Introduction to Synthetic Gauge Theories_ , 1st ed., Lecture Notes in Physics, Vol. 965 (Springer, Heidelberg, 2020).
  * Chen  _et al._ [2016] L. Chen, M. F. Gelin, V. Y. Chernyak, W. Domcke, and Y. Zhao, Dissipative dynamics at conical intersections: Simulations with the hierarchy equations of motion method, Faraday Discuss. 194, 61 (2016).
  * Duan and Thorwart [2016] H.-G. Duan and M. Thorwart, Quantum mechanical wave packet dynamics at a conical intersection with strong vibrational dissipation, J. Phys. Chem. Lett. 7, 382 (2016).
  * Kohler  _et al._ [2005] S. Kohler, J. Lehmann, and P. Hänggi, Driven transport on the nanoscale, Phys. Rep. 406, 379 (2005).
  * Blümel  _et al._ [1991] R. Blümel, A. Buchleitner, R. Graham, L. Sirko, U. Smilansky, and H. Walther, Dynamical localisation in the microwave interaction of Rydberg atoms: The influence of noise, Phys. Rev. A 44, 4521 (1991).
  * Kohler  _et al._ [1998] S. Kohler, R. Utermann, P. Hänggi, and T. Dittrich, Coherent and incoherent chaotic tunneling near singlet-doublet crossings, Phys. Rev. E 58, 7219 (1998).
  * Ferrón  _et al._ [2012] A. Ferrón, D. Domínguez, and M. J. Sánchez, Tailoring population inversion in Landau-Zener-Stückelberg interferometry of flux qubits, Phys. Rev. Lett. 109, 237005 (2012).
  * Blattmann  _et al._ [2015] R. Blattmann, P. Hänggi, and S. Kohler, Qubit interference at avoided crossings: The role of driving shape and bath coupling, Phys. Rev. A 91, 042109 (2015).
  * Engelhardt  _et al._ [2019] G. Engelhardt, G. Platero, and J. Cao, Discontinuities in driven spin-boson systems due to coherent destruction of tunneling: Breakdown of the Floquet-Gibbs distribution, Phys. Rev. Lett. 123, 120602 (2019).
  * Shirai  _et al._ [2015] T. Shirai, T. Mori, and S. Miyashita, Condition for emergence of the Floquet-Gibbs state in periodically driven open systems, Phys. Rev. E 91, 030101(R) (2015).
  * Shirai  _et al._ [2016] T. Shirai, J. Thingna, T. Mori, S. Denisov, P. Hänggi, and S. Miyashita, Effective Floquet-Gibbs states for dissipative quantum systems, New J. Phys. 18, 053008 (2016).
  * Ketzmerick and Wustmann [2010] R. Ketzmerick and W. Wustmann, Statistical mechanics of Floquet systems with regular and chaotic states, Phys. Rev. E 82, 021114 (2010).
  * Ivakhnenko  _et al._ [2023] O. V. Ivakhnenko, S. N. Shevchenko, and F. Nori, Nonadiabatic Landau-Zener-Stückelberg-Majorana transitions, dynamics, and interference, Phys. Rep. 995, 1 (2023).
  * Leggett  _et al._ [1987] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Dynamics of the dissipative two-state system, Rev. Mod. Phys. 59, 1 (1987).
  * Hänggi  _et al._ [1990] P. Hänggi, P. Talkner, and M. Borkovec, Reaction-rate theory: Fifty years after Kramers, Rev. Mod. Phys. 62, 251 (1990).
  * Weiss [2012] U. Weiss,  _Quantum Dissipative Systems_ , 4th ed. (World Scientific, Singapore, 2012).
  * Landi  _et al._ [2022] G. T. Landi, D. Poletti, and G. Schaller, Nonequilibrium boundary-driven quantum systems: Models, methods, and properties, Rev. Mod. Phys. 94, 045006 (2022).
  * Kohler  _et al._ [1997] S. Kohler, T. Dittrich, and P. Hänggi, Floquet-Markovian description of the parametrically driven, dissipative harmonic quantum oscillator, Phys. Rev. E 55, 300 (1997).
  * Grifoni and Hänggi [1998] M. Grifoni and P. Hänggi, Driven quantum tunneling, Phys. Rep. 304, 229 (1998).


# Rethinking Quality of Experience for Metaverse Services: A Consumer-based
Economics Perspective

Hongyang Du, Bohao Ma, Dusit Niyato, , Jiawen Kang, Zehui Xiong, and Zhaohui
Yang H. Du and D. Niyato are with the School of Computer Science and
Engineering, Nanyang Technological University, Singapore (e-mail:
<EMAIL_ADDRESS>dniyato@ntu.edu.sg).B. Ma is with the Shool of Civil
and Environmental Engineering, Nanyang Technological University, Singapore
(e-mail: bohao001@e.ntu.edu.sg).J. Kang is with the School of Automation,
Guangdong University of Technology, China (e-mail: kavinkang@gdut.edu.cn).Z.
Xiong is with the Pillar of Information Systems Technology and Design,
Singapore University of Technology and Design, Singapore (e-mail:
zehui_xiong@sutd.edu.sg).Z. Yang is with the College of Information Science
and Electronic Engineering, Zhejiang University, China (e-mail:
yang_zhaohui@zju.edu.cn).

###### Abstract

The Metaverse is considered to be one prototype of the next-generation
Internet, which contains people’s expectations for the future world. However,
the academic discussion of the Metaverse still mainly focused on the technical
system design, and few research studied Metaverse challenges from the
perspective of consumers, i.e., Metaverse users. One difficulty is that the
analysis from the consumer’s perspective requires interdisciplinary
theoretical framework and quantifiable Quality of Experience (QoE)
measurements. In this article, pioneering from consumers’ point of view, we
explore an interaction between Metaverse system design and consumer behaviors.
Specifically, we rethink QoE and propose an framework that encompasses both
the Metaverse service providers (MSPs) and consumer considerations. From the
macro perspective, we introduce a joint optimization scheme that
simultaneously considers the Metaverse system design, consumers’ utility, and
profitability of the MSPs. From the micro perspective, we advocate the
Willingness-to-Pay (WTP) as an easy-to-implement measurement for future
Metaverse system studies. To illustrate the usability of the proposed
integrated framework, a use case, i.e., virtual traveling, is presented. We
show that our framework can benefit the MSPs in offering competitive and
economical service design to consumers while maximizing the profit.

###### Index Terms:

Metaverse, quality of experience, consumer behavior, interdisciplinary
research

##  I Introduction

Supported by the development of communication technologies, the Metaverse is
an emerging concept considered to be the next generation of the Internet.
Instead of accessing the Internet through mobile phones and computers, users
can completely immerse themselves in the virtual world of the Metaverse
through interactive technologies such as Virtual Reality (VR) and Augmented
Reality (AR). In addition, Metaverse can provide support for the construction
of the real world. For example, complex experiments that are difficult to be
performed in the real world can be simulated in the Metaverse, and then the
results are fed back to the real world for decision-making. These desirable
features have drawn the attention of giant multinational companies globally.
For instance, Facebook, which recently changed its name to Meta, announced
ambitious the plan to invest $\$10$ billion in Metaverse development.
Microsoft acquired company Activision Blizzard, Inc. for $\$70$ billion to
support its Metaverse strategy.

To exploit the full potential of the Metaverse, several system design
indicators have been proposed and widely adopted, e.g., high-reliability
feedback, high-definition virtual object perception, and feeling of immersion.
Both the industry and academia have made numerous efforts to improve the
performance based on the proposed indicators. For example, the development of
computational graphic software such as Unreal Engine has made it possible to
render scenes that are close to places of interest in the real world. In
addition, advances in hardware, such as head-mounted devices (HMDs), can
provide users with high-quality virtual scene perception over long periods of
time to create an immersive experience in the Metaverse. Meanwhile, it is
foreseeable that Extended Reality (XR) devices in the near future can further
improve the quality of immersion. Massive data transmission can be efficiently
supported by advances in communication technologies, e.g., massive multi-input
multi-output (MIMO) technology, reconfigurable intelligent surfaces, mmWave,
and semantic communication techniques. Beyond fifth-generation (B5G) and the
sixth-generation (6G) wireless communication network can provide a huge number
of Metaverse devices with real-time, ubiquitous, seamless, and ultra-reliable
communications [1].

Besides the technical development, a conceptual shift required by the
Metaverse is the adoption of a user-centric perspective throughout the entire
design process. This shift can also be observed in the new generation
Internet, e.g., Web3. Unlike “read based” Web1 and “read and write based”
Web2, Web3 is based on “read, write, and own”. Although the success of the
Metaverse critically depends on active user participation, in the current
development of the Metaverse and related technologies, consumers as the vital
success factor, have received little attention. Attempts by Metaverse service
providers (MSPs) to enhance user-centric design are facing many difficulties.
The reason is that, unlike the objective and directly measurable technical
indicators, the interaction between system design with consumer behaviors is
rather complex, as shown in Fig. 1 (Part I). Thus, it is unclear how the
consumer perspective can be incorporated into the Metaverse system design
process. Several studies attempted to propose Quality-of-Experience (QoE)
measurements for the user-centric design in the Metaverse [2]. Yet the
proposed indicators are considered to be highly simplified and may not capture
the complex interaction between consumer behavior and system designs.
Considering that the Business-to-Customer (B2C) model will play an important
role in the promotion of Metaverse, behavior analysis from consumer
perspectives is essential for the MSPs.

With all above motivation, in this article, we propose an interdisciplinary
framework that qualitatively synergies the technical and consumer perspectives
from both macro-level (system-level), and micro-level (individual-level).
Specifically, the contributions of this article can be summarized as follows.

  * •

We explore the causal relationship between Metaverse system design and
consumer behaviors with the support of consumer economic research. A Stimulus-
Organism-Response (S-O-R) behavioral framework is established.

  * •

We propose the implementation of econometric discrete choice model (DCM) as
the macro-level and a Willingness-to-Pay (WTP) measure as the micro-level to
connect the system design with consumer behaviors.

  * •

Taking a specific Metaverse service, i.e., virtual traveling, as a case study,
we demonstrate how to use our proposed framework to jointly consider the
Metaverse system design, consumers’ utility and profitability of the MSPs.

Figure 1: Structure of this article.

The structure of the article is shown in Fig. 1. We discuss the existing
system design research focusing on the consumer-centric concept, and then
propose our integrated Metaverse design framework. Using a case study, we
demonstrate the applicability of the framework. Finally, we discuss possible
future research directions and conclude this article.

##  II Research Related to Consumer Behaviors

In this section, we first introduce the analysis methods of consumer behaviors
in existing Metaverse research with their limitations. Then, we present how
user behavior is generally considered in the design of products and services.

###  II-A Metaverse Services

The progress of multimedia technologies, e.g., VR and AR, and digital economy
technologies, e.g., blockchain, has promoted the emergence of Metaverse
services. Although many Metaverse services are still in their infancy, we can
observe that the services have shown a diversified evolution direction. For
example, MSPs can provide virtual educational, traveling, game, and
teleconferencing services, as shown in Fig. 1 (Part III). To provide formal
guidance for the Metaverse system, quality of service (QoS) and QoE are widely
used as utility functions. Specifically, the QoS attempts to measure
objectively system performance metrics, i.e., data transmission rate and bit
error probability. QoE is a subjective measure from the perspective of the
user [2].

However, it is not realistic to consider QoS or QoE as an objective function
in the design of Metaverse systems directly and blindly. Although these two
indicators are positively correlated to the profit of MSPs in most cases, the
relationship between the profit and the utility, i.e., QoS or QoE, is actually
complicated. When a MSP invests more resources in the system design to improve
QoE and QoS, the MSP needs to raise prices to ensure its profitability [3].
However, different users have different sensitivities to the rising service
prices, i.e., users are heterogenous instead of homogeneous. Moreover,
although the increase in QoE can bring the increase in the loyalty of existing
users and the increase in the number of new users, the specific relationships
are difficult to quantify. Thus, most of the current literature ignores
complex user behavior responses by various assumptions and simplifications. To
fill this research gap, we aim to rethink QoE measurement and design a
comprehensive utility function analysis framework for MSP by considering
consumer behavior in depth in the Metaverse system design.

###  II-B Analysis Methods in Product and Service Design

Figure 2: Illustration of consumer behaviors in product/service design.

In the literature on product line design and hospitality management, the
importance and value of incorporating consumer behaviors have long been noted.
As shown in Fig. 2, in the early years, research was limited to single-
attribute design, namely pricing optimization. In addition, consumers are
assumed to be homogeneous. Yet, significant progress has been made in recent
years, as a result of both advancements in mathematical modeling and
improvement in computational powers. In a product line design case, the
authors in [4] formulate a pricing problem with consumers’ heterogeneity, and
apply a latent-class multinomial formulation. The authors assume that
consumers hold different views towards product attributes. Such heterogeneity
can be described with tailored models for each consumer. For example, because
consumers’ sensitivity to prices may vary, the whole population can be
segmented into high-income, medium-income and low-income groups. Within each
group, specific price sensitivity is applied. The latent-class-based model was
applied to a real-life consumer electronic pricing optimization case, and
significant performance improvement of the manufacturer was observed. From
another perspective, the authors in [5] studied an integrated product design
problem whereby a firm jointly determines multiple product attributes, e.g.,
price, quality and service duration. It is shown that such product
differentiation can both benefit the consumers and improve the profitability
of the firm.

One observation is that the most notable success in the above analysis is made
by the multinomial formulation based on the Random Utility Model [6]. Such a
formulation supports design decisions by quantifying the impacts of product
and service attributes on consumers’ behaviors. More details on applying the
Random Utility Model in Metaverse system design are given in Section III-B.

##  III An Integrated System Design Framework

Figure 3: Stimulus-Organism-Response (S-O-R) behavioral framework.

In this section, we propose an integrated framework that synergizes the
response of consumers and Metaverse system design. We first introduce the
design-to-behavior causal chain, and then connect the system design with
consumer behaviors.

###  III-A The Design-to-Behavior Causal Chain

In recent years, a rich body of research has been conducted to explore the
latent psychological mechanisms that systematically explain how technological
attributes impact consumer behaviors. Existing research can generally be
organized into the S-O-R behavioral framework [7], as shown in Fig. 3 (Part
I). Technological attributes, i.e., the stimulus (S), triggers certain
cognitive or affective shifts within consumers (O), which consequently leads
to behavioral or attitudinal responses (R). For example, if the MSP invests
funds to improve the comfort of VR hardware and increase the types of services
in the virtual world (System Design), users can directly feel a smooth
interactive experience and enjoy informative services (S). The users will then
consider that these Metaverse services are useful (O), resulting in brand
loyalty to the MSP and willingness to promote the services (R).

Specifically, in the context of VR/AR, as the core access to Metaverse,
stimulus constructs can be grouped into three clusters. The first cluster
includes the media attributes, such as the perceived interactivity, the
perceived vividness, and the perceived personalization [8, 3] to the virtual
tourism attraction. The second cluster covers the media quality, including the
richness of information and system response time [8]. Finally, the third
cluster in the stimulus concerns the ergonomic design of equipment [9].
Despite being the antecedent of consumer behaviors, the stimulus serves as
direct outcomes of system designs. For example, the resource allocation
decision can directly impact the media quality perceived by consumers. Such a
dual property explicitly bridges Metaverse system design considerations with
consumer behaviors, which well supports applications of the interdisciplinary
framework.

Receiving the stimulus, consumers start to form psychological evaluations of
the system technologies, which is called organism. The evaluation can be
specified as cognitive attitudes such as expected utilitarian value from
technologies and affective attitudes including enjoyment [8]. Meanwhile, the
authors in [10] suggest that privacy concerns and worries about loss of
control over information access may arise, which can hinder the adoption of
cutting-edge technologies. Finally, the psychological organism results in
behavioral responses including purchase, Word-of-Mouth recommendations to
attractions, and attitudinal responses such as brand loyalty to MSPs [8].

###  III-B Connect System Design with Consumer Behaviors

Although the S-O-R causal chains can connect the Metaverse system designs with
consumer behaviors, the establishment of a quantitative relationship is a non-
trivial task. Researchers have to solve explicitly the following two
questions:

  * Q1)

How can the psychological states described in Section III-A be reliably
measured?

  * Q2)

How can we connect consumers’ psychological states with attributes that can be
directly applied to Metaverse system design processes?

####  III-B1 Structural Equation Modelling

To address (Q1), the Structural Equation Modelling (SEM) can be used, which
covers a family of causal inference methods [11]. The SEM allows systematical
and reliable measurements of the latent constructs, i.e., the psychological
states when users participate in the Metaverse services. By considering that
the latent constructs are the causes of observable measurements, e.g. users’
answers to a Liker-scale survey questions designed by MSP, the latent
constructs can be inferred. To ensure the reliability of the inference,
several observable measurements should be simultaneously applied for one
latent construct. In addition, the level of variance commonality across the
observable measurements can be examined to validate the assumption that the
selected observable measurements are “caused” by the same latent construct.

Figure 4: Detailed illustration of Metaverse system-consumer integration.

The detailed progress discussed above can be illustrated in Fig. 4 (Part III).
For example, to infer the latent constructs such as user enjoyment in the
Metaverse, researchers can collect several observable measurements including
biometric data, such as eye concentration. The variance of a measurement can
be decomposed into several factors, e.g., measurement error, variance specific
to this measurement, and the common variance caused by latent constructs. With
the common variance, the causal parameters between observable measurements and
latent constructs can be derived. Consequently, latent constructs can be
reliably inferred.

####  III-B2 Econometric Discrete Choice Modelling

Although SEM can provide quantitative evidence for latent causal
relationships, such scale-based numerical parameters can hardly be interpreted
meaningfully for Metaverse system design. Hence, it is vital to explicitly
connect latent constructs with practical attributes, for instance, the
expected revenue. As illustrated in Fig. 4 (Part II), the DCM can provide a
paradigm to link psychological constructs with observable attributes (Q2). The
DCM is based on the Random Utility Model (RUM), which considers the Metaverse
users choose the option that maximizes their utility. The utility is composed
of a systematic portion, which is calculated linearly with identified
attributes, and a random portion [6]. Specifically, different latent
constructs, e.g., service enjoyment and usefulness, have different weights on
different users. We call the weight as the parameter to the corresponding
latent construct. Then, the systematic utility can be obtained by adding
latent constructs weighted by corresponding parameters, taking into account
the price and its weight. The utility function can finally be expressed as the
addition of the systematic utility and a random variable. Furthermore, through
integration over the probability density function of the random portion, the
probability of consumers choosing a certain option can be analytically
expressed as functions of systematic utility in binomial/multinomial
formulations, as shown in Fig. 4 (Part II-C). On the user population level,
such probability can also be interpreted as the market potential for a
product. With observations of consumers’ actual choices, e.g., to use
Metaverse service or not, the structural parameters of systematic utility can
be estimated through methods such as maximum likelihood and methods of the
moment. Managerial implications can be readily obtained with the estimated
structural parameters.

####  III-B3 Insights

First, with the estimated structural parameters, researchers can readily
derive the sensitivities of Metaverse consumers’ purchase probability/market
potential with respect to various attributes. For example, to understand
consumers’ responses to price adjustment, the MSPs can take the derivative of
purchase probability with respect to price. Here the price could be the
Metaverse service subscription fee. The resulting numerical values can provide
insights into the potential market shrinking because of the increase of unit
price.

Second, the DCM framework acknowledges the MSPs about consumers’ trade-offs
between different pairs of attributes. One important economic indicator is the
ratio between the price parameter and any other parameter, a.k.a, the
willingness to pay (WTP), as shown in Fig. 4 (Part II-B). In this way, DCM
allows the monetization of psychological shifts and directly connects
improvement in consumers’ satisfaction with system designs through the WTP
measurement [12]. For example, the MSP can quantitatively determine whether to
invest more resources to improve a system design indicator, by calculating the
ratio of the price increase due to more resource input and the latent
construct increase brought by the improvement of system designs. If the ratio,
i.e., WTP, is calculated to be greater than 1, this decision of the MSP is
reasonable.

Compared to the existing QoS or QoE measurements, the DCM framework provides a
macro-level path for implementing consumer-centric approaches. The MSPs can
explicitly model the expected revenue as the product of Metaverse service
purchase probability and price, e.g., service subscription fee. Then, the
revenue model can be readily embedded into the objective function for the
Metaverse system design. In addition, the consumers’ purchase probability is
modeled as a function of Metaverse service attributes, which are directly
determined by service costs. For instance, the probability of users purchasing
Metaverse services can be formulated as a function of VR media quality, which
is directly associated with the MSPs’ available bandwidth and modeling
ability. Compared to the existing system design method which optimizes the
technical indicators only, the integrated framework leads to the overall
system optimality, bringing more profits to MSP.

###  III-C Heterogeneity in Metaverse Users

In real-world Metaverse applications, it is significant to consider the
interpersonal and contextual differences among users, as known as
heterogeneity. In the case of Metaverse, for instance, attitudes toward
privacy can vary significantly across different socio-demographic groups. For
example, the authors in [13] conclude that individuals who are male, younger,
earn higher incomes, and hold higher education qualifications are more
sensitive to their information privacy. Hence, from the WTP perspective, it
implies such consumer group needs to be compensated more for the collection of
personal data. For example, if the MSP plans to analyze the user’s attention
by collecting the eye movements of users to design a personalized resource
allocation scheme [3], it is necessary to provide more compensation for such
privacy-sensitive users or consider alternative schemes. Note that the
heterogeneity is not only attributed to interpersonal differences. The usage
contexts may also differentiate consumers’ behaviors. For example, in a
working context such as Metaverse conference, informativeness is the dominant
factor for technology adoption. While in a virtual traveling context, the
authenticity of Metaverse users experience prevails [14].

Hence, at the design stage, a homogeneous consumer measurement may not suffice
to deal with real-life Metaverse system design complexities. One
straightforward solution is the incorporation of latent constructs into the
DCM that is discussed in Section III-B. As a result, the interpersonal
heterogeneity among Metaverse users arising from attitudinal differences can
be captured [4]. In Section III-B, we treat the structural parameters in the
DCM to be deterministic. To capture the interpersonal heterogeneity,
econometrician has developed more advanced DCMs by treating the structural
parameters as random variables. Such models include the Mixed Multinomial
Logit model (MMNL), which assumes structural parameters to be continuously
distributed, and the Latent Class Model (LCM), which assumes a discrete
distribution [6]. These methods have great potential to be applied in the
Metaverse system design as an alternative to the DCM in our proposed
framework.

##  IV A Metaverse Service Design Case

In this section, we consider a Metaverse service design case, and show how the
proposed integrated framework can facilitate consumer-centric decision
makings.

###  IV-A Design Case Introduction

Here we consider the virtual traveling as the Metaverse service case.
Specifically, users access the Metaverse with devices such as VR and AR. Then,
users are free to choose the scenes of the virtual traveling, such as a
digital copy of an attraction or a completely virtual place. The MSP needs to
ensure multiple system technical indicators such as [15]

  * •

Accurate Sensing Ability: To create a digital copy of the real world, the MSP
needs to sense the real world through various sensors. The inauthenticity
caused by inaccurate sensing leads to a degraded service experience.

  * •

High Downlink Data Transmission Rate: To send a large amount of virtual object
data to users, the MSP needs to ensure a high downlink data rate. Untimely
data transmission causes slow loading of Metaverse scenes.

  * •

Low Latency: To provide a realistic travel experience in real time, the MSP
needs to ensure low latency. High latency creates jitter in visual effects.

  * •

Hardware Design: To avoid problems such as heat and excess weight of wearable
devices, the MSP needs to invest in hardware design. Uncomfortable hardware
design leads to adverse reactions such as motion sickness.

###  IV-B Psychological Causal Chain for Virtual Travel

To adopt the S-O-R causal framework described in Section III-A, the MSP should
first formulate the mechanism behind system design and consumers’ reactions.
In the virtual traveling setting, perceived authenticity of the travel
experience is regarded as a significant stimulus (S). It is defined as the
perceptions of the extent to which the experiences are real, original and
exceptional. Then, the perceived authenticity leads to consumers’ enjoyment
and emotional involvement in the Metaverse as the psychological organism (O).
Consequently, consumers’ purchase intention can be induced [14] as the
response (R) to the organism. As discussed in Section III-B, the SEM allows
the estimation of causal parameters along the S-O-R framework. For example,
with a scale increase in the perceived authenticity, the MSP can estimate the
magnitude of increase in consumers’ enjoyment, emotional involvement in the
Metaverse, and the purchase intention to the virtual goods that may be
involved in virtual traveling. Furthermore, to establish connection between
system design with the psychological process, the dual property of stimuli
constructs shall be addressed as described in Section III-A. The MSP can adopt
a regression method to estimate the impacts of technical indicators on the
level of perceived authenticity in the virtual traveling. Consequently, a
complete technical-to-response causal chain can be established.

###  IV-C Consumer-Centric System Design Optimization

As noted in Section III-B, though the SEM method enhances MSP’s understanding
on Metaverse consumers’ behaviors, results are hardly to be applied directly
to system design process such as virtual attraction creation. The reason is
that all parameters estimated under the SEM method are “scale” based with no
direct practical explanation. Thus, we explicitly describe a consumer-centric
service design optimization progress with the application of the DCM part in
our proposed integrated framework. We first start with a formulation with
technical indicators being directly applied in the DCM equation. Although such
an approach omits the psychological mechanism behind consumers’ behaviors, it
allows the joint optimization in a more straightforward and easy-to-implement
manner. Following that, we incorporate the latent constructs, e.g., the
enjoyment users obtain from the virtual traveling and the perceived usefulness
of the service, into the Metaverse system design progress.

####  IV-C1 Direct Applications of DCM with Technical Indicators

As illustrated in Section III-B, the measurement of latent constructs can be a
non-trivial task, which requires to collect additional panel surveys or
biometric data. In cases when such data is unavailable, MSPs can directly
incorporate the relevant technical indicators as we discussed in Section IV-A
into the DCM-based utility formulation as shown in Fig. 4. The systematic
utility can be expressed as a linear function with respect to all technical
indicators, with a set of structural parameters to quantify the impacts of
each indicator on Metaverse consumers’ choices. Specifically, these structural
parameters can be estimated by recording consumers’ choices of virtual
traveling adoption under different technical designs.

With the estimated structural parameters, a joint optimization problem can be
formulated to maximize the product of the purchase probability and the
Metaverse service subscription fee, i.e., the price, as shown in Fig. 4 (Part
II-C). Consequently, the optimal set of product attributes, including
technical indicators and pricing, can be determined. An example is shown in
Fig. 5, after obtaining the parameters of the latent constructs of Metaverse
users (Here we show the results of the simulation), the MSPs can analyze the
shifts in consumer utility, purchase probability, and revenue under different
service subscription fee. The best pricing strategy can then be determined.

Figure 5: An illustration of shifts in consumer utility, purchase probability
and revenue under different Metaverse service subscription fee.

Furthermore, our proposed framework can also support MSP’s service
differentiation scheme. For example, the MSP intends to establish a premium
virtual traveling service with higher data transmission rate. By substituting
the new services’ technical details into the DCM, the MSP can estimate the
potential market size of the new service offers. Meanwhile, through
investigating the ratio of the structural parameter of data transmission rate
to that of service price, the MSP can derive consumers’ willingness to pay for
incremental data transmission rates for a smoother virtual traveling
experience. Such WTP measurement can well-support the pricing scheme for the
premium services.

####  IV-C2 A Full Application with Latent Constructs

Although the direct incorporation of technical indicators can suffice to
provide meaningful practical implementations, the incorporation of latent
constructs can bring several additional benefits. First, from the statistical
perspective, it has been shown that such incorporation can improve the
forecast accuracy of the DCM. Also, the variances of estimations can be
reduced, and the biases can sometimes be eliminated [12]. Second, by opening
up the “black-box” relationship between technical indicators and Metaverse
consumer behaviors, the explainability and transparency of consumer modeling
can be improved. For example, when only technical indicators are incorporated,
the MSP may find that the low bit error probability leads to greater purchase
probability yet limited consumer-related insights can be drawn. In contrast,
if the latent constructs are adopted, the aforementioned relationship can be
explained with the “bit error probability $\to$ authenticity in virtual
traveling service $\to$ purchase probability” causal chain. In a scenario when
the authenticity is not a priority, the MSP can readily de-prioritize the bit
error probability indicator as well.

##  V Future Research Directions

###  V-A Advertising Strategy of Metaverse Companies

The advertising can be designed to attract new Metaverse users and encourage
existing users to make multiple purchases, for increasing the sale of
Metaverse services or digital products. Moreover, advertising strategies can
be used in the allocation scheme of network resources, i.e., bandwidth,
required by users to access the Metaverse. Specifically, users could be
incentivized to buy more network resources to increase the Metaverse access
rate. Considering the total budget of a MSP is limited, the allocation of the
budget between advertising and network resources needs to be optimized. The
competition between multiple Metaverse companies is also worth studying.

###  V-B Social Network Analysis

As noted in Section IV, the successful implementation of the consumer-centric
design depends on the knowledge to consumers, which requires experiments among
consumers. However, the experiment design is complex. For example, to estimate
DCM structural parameters, the MSP may invite users to experience pre-defined
Metaverse environments and record their choices. The environments should be
defined in a way that much information can be revealed. If in one experiment,
all technical indicators are set to the highest standard, consumers are likely
to adopt the service and little information can be obtained from such a
experiment design. To facilitate this process, interdisciplinary
collaborations with social scientists may be pursued.

###  V-C Dataset Study for Metaverse

Because the research of Metaverse is still at an early stage, there is not
enough datasets to meet the needs of various system design schemes. Therefore,
an important research direction of Metaverse is the establishment and analysis
of datasets. For example, datasets related to virtual scenes can help us to
analyze users’ behaviors to design real-time resource allocation schemes to
improve QoE, and datasets of consumer responses to various strategies of MSPs
can help MSPs formulate suitable market solutions.

##  VI Conclusion

We have proposed an integrated research framework that encompasses both
consumer behaviors and system design considerations for the MSPs, which
facilitates the consumer-centric Metaverse ecology development. Under the
proposed framework, the S-O-R causal chain was introduced to bridge
systematically technical indicators with consumers’ responses. By rethinking
QoE measurement and analyzing the underlying causal chain, a micro-level WTP
measurement and a macro-level DCM-based optimization formulation were
introduced. Compared to the current Metaverse system design solutions that
only focus on QoE, our proposed framework systematically describes the
relationships between technical design, consumers’ utility and profitability
for the MSPs. It allows a joint optimization with more transparent and
explainable process. Meanwhile, a virtual traveling design case is introduced
to operationalize the proposed framework. This article also aims to advocate
the wider adoption of interdisciplinary perspectives for the study of
increasingly diversified and decentralized next generation Internet. Efforts
shall be made to synergies knowledge from the social science track into the
futurist system design.

## References

  * [1] H. Tataria, M. Shafi, A. F. Molisch, M. Dohler, H. Sjöland, and F. Tufvesson, “6G wireless systems: Vision, requirements, challenges, insights, and opportunities,” _Proc. IEEE_ , vol. 109, no. 7, pp. 1166–1199, Jul. 2021.
  * [2] Y. Wang, P. Li, L. Jiao, Z. Su, N. Cheng, X. S. Shen, and P. Zhang, “A data-driven architecture for personalized QoE management in 5G wireless networks,” _IEEE Wireless Commun._ , vol. 24, no. 1, pp. 102–110, Jan. 2016\.
  * [3] H. Du, J. Liu, D. Niyato, J. Kang, Z. Xiong, J. Zhang, and D. I. Kim, “Attention-aware resource allocation and QoE analysis for metaverse xURLLC services,” _arXiv preprint arXiv:2208.05438_ , 2022.
  * [4] H. Li, S. Webster, N. Mason, and K. Kempf, “Product-line pricing under discrete mixed multinomial logit demand: Winner-2017 M&SOM practice-based research competition,” _Manuf. Serv. Oper. Manag._ , vol. 21, no. 1, pp. 14–28, Jan. 2019.
  * [5] R. Wang, C. Ke, and S. Cui, “Product price, quality, and service decisions under consumer choice models,” _Manuf. Serv. Oper. Manag._ , vol. 24, no. 1, pp. 430–447, Jan. 2022.
  * [6] K. E. Train, _Discrete choice methods with simulation_. Cambridge University Press, 2009.
  * [7] R. R. Ahmed, J. A. Qureshi, F. Arshad, Z. A. Channar, V. Parmar, N. Z. M. Salleh _et al._ , “The social media break-up: Psycho-behavioral measures and implications,” _IEEE Access_ , vol. 10, pp. 58 116–58 135, 2022.
  * [8] H. Kumar, “Augmented reality in online retailing: A systematic review and research agenda,” _Int. J. Retail. Distrib. Manag._ , Sept. 2021.
  * [9] L. Tsao, L. Li, and L. Ma, “Human work and status evaluation based on wearable sensors in human factors and ergonomics: A review,” _IEEE Trans. Hum. Mach. Syst._ , vol. 49, no. 1, pp. 72–84, Jan. 2018.
  * [10] Y. Feng and Q. Xie, “Privacy concerns, perceived intrusiveness, and privacy controls: An analysis of virtual try-on apps,” _J. Interact. Mark._ , vol. 19, no. 1, pp. 43–57, 2019.
  * [11] D. Pal, S. Funilkul, V. Vanijja, and B. Papasratorn, “Analyzing the elderly users’ adoption of smart-home services,” _IEEE Access_ , vol. 6, pp. 51 238–51 252, 2018.
  * [12] A. Vij and J. L. Walker, “How, when and why integrated choice and latent variable models are latently useful,” _Transp. Res. Part B Meth._ , vol. 90, pp. 192–217, 2016.
  * [13] H. Lee, S. F. Wong, J. Oh, and Y. Chang, “Information privacy concerns and demographic characteristics: Data from a korean media panel survey,” _Gov. Inf. Q._ , vol. 36, no. 2, pp. 294–303, Feb. 2019.
  * [14] M. J. Kim, C.-K. Lee, and T. Jung, “Exploring consumer behavior in virtual reality tourism using an extended stimulus-organism-response model,” _J. Travel Res._ , vol. 59, no. 1, pp. 69–89, Jan. 2020.
  * [15] D. Van Huynh, S. R. Khosravirad, A. Masaracchia, O. A. Dobre, and T. Q. Duong, “Edge intelligence-based ultra-reliable and low-latency communications for digital twin-enabled metaverse,” _IEEE Wireless Commun. Lett._ , to appear, 2022.


# Spherical fibrations over certain complexes

Wen Shen  Department of Mathematics, Capital Normal University, Beijing,
P.R.China<EMAIL_ADDRESS>
###### Abstract.

This paper presents any spherical fibration over certain CW complexes is
stably fibre homotopy equivalent to a $\mathrm{TOP}$-linear spherical bundle.
Moreover, if the CW complex is a Poincar$\mathrm{\acute{e}}$ complex, then it
is of the homotopy type of a topological manifold.

###### 2020 Mathematics Subject Classification:

Primary 57N99, 57P10, 57Q50, 57R22

##  1\. Introduction

In this paper, $H_{\ast}(-)$ and $H^{\ast}(-)$ denote integer coefficient
homology and cohomology respectively. $\mathbb{Z}/p$ is integers mod $p$,
$\mathbb{Z}_{(p)}$ is integers localized at $p$, where $p$ is a prime.

$X$ is a Poincar$\mathrm{\acute{e}}$ complex of dimension $m$, if there is an
element $[X]\in H_{m}(X)$ such that $[X]\cap:H^{q}(X)\to H_{m-q}(X)$ is an
isomorphism for all $q$. An oriented spherical fibration $\xi:E_{0}\to X$ with
fibre $F$ homotopy equivalent to $S^{k-1}$, is called a Spivak normal
fibration for $X$ if there is $\alpha\in\pi_{m+k}(T(\xi))$ of degree $1$, i.e.
such that $h(\alpha)\cap U_{\xi}=[X]$, where $T(\xi)=X\cup_{\pi}C(E_{0})$,
$U_{\xi}\in H^{k}(T(\xi))$ is the Thom class; $h:\pi_{\ast}\to H_{\ast}$ the
Hurewicz homomorphism. The Spivak normal fibration is the analog of the stable
normal bundle to the embedding of a manifold in the sphere. In [12], Spivak
showed that if $\pi_{1}(X)=0$ then a Poincar$\mathrm{\acute{e}}$ complex X has
a Spivak normal fibration, unique up to stable fibre homotopy equivalence.

For stable spherical fibrations, there exist a classifying space
$\mathrm{BG}$, and natural maps
$\mathrm{BO}\to\mathrm{BPL}\to\mathrm{BTOP}\to\mathrm{BG}$ (see [7] [11]). Let
$f:Y\to\mathrm{BG}$ be a classifying map of a stable spherical fibration $\xi$
over $Y$. $\xi$ is stably fibre homotopy equivalent to an $\mathrm{O}$,
$\mathrm{PL}$ or $\mathrm{TOP}$-linear spherical bundle if $f$ has a lift to
$\mathrm{BO}$, $\mathrm{BPL}$ or $\mathrm{BTOP}$.

The Poincar$\mathrm{\acute{e}}$ complex is a weaker structure than the well-
known structure on manifold. In differential topology and surgery theory, one
studies the obstructions to lifting the structure of
Poincar$\mathrm{\acute{e}}$ complex to the strong type found on a manifold.
Novikov and Browder presented the following theorem independently.

######  Theorem 1.1.

[2] [10] Let $X$ be a $1$-connected Poincar$\mathrm{\acute{e}}$ complex of
dimension $m\geq 5$, $\eta$ an oriented linear $k$-vector bundle over $X$,
$k>m+1$, $\alpha\in\pi_{m+k}(T(\eta))$ of degree $1$ where $T(\eta)$ is the
Thom space of $\eta$, (i.e. the sphere bundle of $\eta$ is stably fibre
homotopy equivalent to the Spivak normal fibration of $X$). If

  * (1)

$m$ is odd, or

  * (2)

$m=4k$ and
$\mathrm{Index}X=(L_{k}(p_{1}(\eta^{-1}),\cdots,p_{k}(\eta^{-1})))[X]$,

then there is a homotopy equivalence $f:M\to X$, $M$ a smooth $m$-manifold,
such that $\nu=f^{\ast}(\eta)$, $\nu=$ normal bundle of $M\subset S^{m+k}$.

Moreover, the above theorem can be extended to the piecewise and topological
cases by [3] and [6].

######  Theorem 1.2.

[1] Let $X$ be a $1$-connected Poincar$\mathrm{\acute{e}}$ complex of
dimension $m\geq 5$. Then $X$ is of the homotopy type of a PL or TOP manifold
if and only if its Spivak normal fibration is stably fibre homotopy equivalent
to a PL or TOP-linear spherical bundle.

Due to Theorem 1.1 and 1.2, the problem of making a general spherical
fibration into a linear bundle (up to fibre homotopy equivalence) is central
to the theory of surgery.

In this paper, we consider which spherical fibration over a certain CW complex
is stably fibre homotopy equivalent to a TOP-linear spherical bundle. Any
considered space is with a base point, and any map preserves the base points.

Let $\mathrm{G/TOP}$ be the homotopy fibre of $\mathrm{BTOP}\to\mathrm{BG}$.
Let $X$ be a 1-connected CW complex. Any spherical fibration over $X$ has a
classifying map $X\to\mathrm{BG}$ that admits a lift to the 2-fold cover
$\mathrm{BSG}$ of $\mathrm{BG}$. Let $f$ denote the lift. Since
$\mathrm{G/TOP}$ is an infinite loop space [9], there is a fibration sequence

$\mathrm{G/TOP}\to\mathrm{BSTOP}\to\mathrm{BSG}\stackrel{{\scriptstyle\gamma}}{{\to}}\mathrm{B(G/TOP)}$

where $\mathrm{BSTOP}$ is the 2-fold cover of $\mathrm{BTOP}$.

The following theorem is obvious by the classic results for
$\pi_{\ast}(\mathrm{G/TOP})$.

######  Theorem 1.3.

Let $X$ be a finite $1$-connected CW complex satisfying that $H^{4n+1}(X)=0$
and $H^{4n+3}(X;\mathbb{Z}/2)=0$ for any $n\geq 0$. Then any spherical
fibration over $X$ is stably fibre homotopy equivalent to a TOP-linear
spherical bundle.

In the following theorem, $H_{n}(X)$ permits to be nontrivial for $n=1$ or $3$
mod $4$, but additional conditions need to be satisfied.

######  Theorem 1.4.

Let $X$ be a finite $1$-connected CW complex satisfying that $H_{2n}(X)$ is
torsion free for any $n\geq 1$. If the classifying map $f:X\to\mathrm{BSG}$ of
a spherical fibration over $X$ satisfies that

$(\gamma\circ f)_{\ast}:H_{4n+1}(X)\to H_{4n+1}(\mathrm{B(G/TOP)})$ (1.1)
$(\gamma\circ f)_{\ast}:H_{4n-1}(X;\mathbb{Z}/2)\to
H_{4n-1}(\mathrm{B(G/TOP)};\mathbb{Z}/2)$ (1.2)

are trivial for any $n\geq 1$, then the fibration is stably fibre homotopy
equivalent to a TOP-linear spherical bundle.

######  Example 1.5.

Let $f:S^{4k+1}\to\mathrm{BSG}$ be the classifying map of any spherical
fibration over $S^{4k+1}$. By comparing the homotopy groups of $\mathrm{BSG}$
and $\mathrm{G/TOP}$, $f$ has a lift to $\mathrm{BSTOP}$.

From the above example, we see that some conditions in Theorem 1.4 are
excessive. Applying fibration theory and local homotopy theory, we have a more
precise theorem as follows.

######  Theorem 1.6.

Let $X$ be a finite $1$-connected CW complex satisfying that $H_{2n}(X)$ is
torsion free for $n\geq 1$. If the classifying map $f:X\to\mathrm{BSG}$ of a
spherical fibration over $X$ satisfies that

$(\gamma\circ f)_{\ast}:H_{2^{n+1}-1}(X;\mathbb{Z}/2)\to
H_{2^{n+1}-1}(\mathrm{B(G/TOP)};\mathbb{Z}/2)$

is trivial for any $n\geq 1$, then the fibration is stably fibre homotopy
equivalent to a TOP-linear spherical bundle.

By Theorem 1.2 and 1.6, we get

######  Corollary 1.7.

If a finite $1$-connected Poincar$\mathrm{\acute{e}}$ complex $X$ of dimension
$m\geq 5$ satisfies that (1) $H_{2n}(X)$ is torsion free for $n\geq 1$.

  * (2)

$H_{2^{n+1}-1}(X;\mathbb{Z}/2)=0$ for $n\geq 1$.

Then $X$ is of the homotopy type of a topological manifold.

We first prove Theorem 1.4 in Section 2. The computations for the homomorphism
$(\gamma\circ f)_{\ast}$ as in Theorem 1.4 are divided into two cases, shown
in Section 3 and 4. The proof of Theorem 1.6 follows from the computations.

##  2\. The trivial homomorphism of homology

Let $X$ be a finite 1-connected CW complex. Any spherical fibre bundle over
$X$ has a classifying map $f:X\to\mathrm{BSG}$. Let $X_{n}$ be the
$n$-dimensional skelton of $X$, $f_{n}:X_{n}\to\mathrm{BSG}$ be the
restriction of $f$ on $X_{n}$. In this section, we assume that $X$ satisfies

$H_{2k}(X)$ is torsion free for any $k\geq 1$ (2.1)

Recall the fibration sequence

$\mathrm{G/TOP}\to\mathrm{BSTOP}\to\mathrm{BSG}\stackrel{{\scriptstyle\gamma}}{{\to}}\mathrm{B(G/TOP)}$

######  Lemma 2.1.

For any space $Y$, any map $f:Y\to\mathrm{BSG}$ has a lift to $\mathrm{BSTOP}$
iff $\gamma\circ f$ is homotopic to zero.

###### Proof.

Since $\mathrm{G/TOP}$ is a H-space, $\gamma$ is a classifying map of the
fibration $\mathrm{G/TOP}\to\mathrm{BSTOP}\to\mathrm{BSG}$. By the given map
$f$, we have an induced fibration over $Y$. Moreover, the above fibrations are
principal fibrations [5]. So there is an action $\mu:\mathrm{G/TOP}\times E\to
E$.

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{BSTOP}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathrm{BSG}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{\mathrm{B(G/TOP)}}$

If $\gamma\circ f$ is homotopic to zero, then $E$ is homotopy equivalent to
$\mathrm{G/TOP}\times Y$. Clearly, $f$ has a lift to $\mathrm{BSTOP}$.

Note that the lifts of $f$ are in one-to-one correspondence with the cross-
sections of the fibration induced by $f$. If $f$ has a lift to
$\mathrm{BSTOP}$, then there is a cross-section $g:Y\to E$. Thus
$T:\mathrm{G/TOP}\times Y\to E$ given by $T(x,y)=\mu(x,g(y))$ defines a
homotopy equivalence, which means that the fibration induced by $f$ is
trivial. Hence $\gamma\circ f$ is homotopic to zero. ∎

By (2.1) and simplest cell structure [14], $X_{2}=\vee_{i\in I^{2}}S_{i}^{2}$
where $I^{2}$ is a finite set or empty set. Clearly,
$f_{2}:X_{2}\to\mathrm{BSG}$ has a lift to $\mathrm{BSTOP}$.

Assume $f_{n}:X_{n}\to\mathrm{BSG}$ has a lift to $\mathrm{BSTOP}$, then
$\gamma\circ f_{n}$ is homotopic to zero. Thus we have the following
commutative diagram up to homotopy where $f\circ\iota=f_{n+1}$, $\phi$ denotes
the $2$-localization (see [8] [13]).

$\textstyle{X_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathrm{BSG}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{X_{n+1}/X_{n}=\vee_{i\in
I^{n}}S_{i}^{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\iota}}$$\textstyle{\mathrm{B(G/TOP)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\mathrm{B(G/TOP)}[2]}$

By [9],
$\mathrm{B(G/TOP)}[2]\simeq\Pi_{k=1}^{\infty}\mathrm{K}(\mathbb{Z}_{(2)},4k+1)\times\mathrm{K}(\mathbb{Z}/2,4k-1).$

######  Lemma 2.2.

A map $S^{j}\to\mathrm{B(G/TOP)}$ is nontrivial iff its induced homomorphism
of $\mathbb{Z}$-coefficient homology is nontrivial for $j=1\mod 4$; iff its
induced homomorphism of $\mathbb{Z}/2$-coefficient homology is nontrivial for
$j=3\mod 4$.

###### Proof.

The lemma follows by applying the Hurewicz homomorphism on the composition of
the map and $\phi:\mathrm{B(G/TOP)}\to\mathrm{B(G/TOP)}[2]$. ∎

If $n$ is odd, $\bar{\iota}$ is trivial. Thus $f_{n+1}$ has a lift to
$\mathrm{BSTOP}$. Next, we discuss the case for $n=2k$.

Note that $H_{2k}(X)$ is torsion free. By simplest cell structure,

$H_{2k+1}(X_{2k+1})\cong H_{2k+1}(X_{2k+1}/X_{2k})=H_{2k+1}(\vee S^{2k+1})$
(2.2)

If the homomorphisms (1.1) and (1.2) are trivial, $\bar{\iota}$ is homotopic
to zero by Lemma 2.2. Thus $f_{n+1}$ has a lift to $\mathrm{BSTOP}$.

By induction, $f$ has a lift to $\mathrm{BSTOP}$.

##  3\. The obstructions in $H^{4n-1}(X;\pi_{4n-2}(\mathrm{G/TOP}))$

In this section, $X$ is a finite 1-connected CW complex and satisfies the
torsion-free condition (2.1). $f$ is a classifying map of a spherical
fibration over $X$ and satisfies that

$(\gamma\circ f)_{\ast}:H_{2^{j+1}-1}(X;\mathbb{Z}/2)\to
H_{2^{j+1}-1}(\mathrm{B(G/TOP)};\mathbb{Z}/2)$ (3.1)

is trivial for any $j\geq 1$.

Since localization preserves fibrations in the category of 1-connected spaces
[13], we have

$\textstyle{\mathrm{G/TOP}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{\mathrm{G/TOP}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{\mathrm{G/TOP}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{G/TOP}[2]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{BSTOP}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathrm{BSG}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{\mathrm{B(G/TOP)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{B(G/TOP)}[2]}$

Let
$\mathbb{K}=\Pi_{k=1}^{\infty}\mathrm{K}(\mathbb{Z}_{(2)},4k)\times\mathrm{K}(\mathbb{Z}/2,4k-2)$.
It is well known that

$\mathrm{G/TOP}[2]\simeq\mathbb{K}.$

######  Lemma 3.1.

$\varphi^{\ast}:H^{\ast}(E;\mathbb{Z}/2)\to
H^{\ast}(\mathrm{G/TOP};\mathbb{Z}/2)$ is surjective.

###### Proof.

There is a map $\mathrm{G/TOP}\to\mathbb{K}$ which induces an isomorphism of
Hopf algebras over the Steenrod algebra

$H^{\ast}(\mathbb{K};\mathbb{Z}/2)\to H^{\ast}(\mathrm{G/TOP};\mathbb{Z}/2).$

Let ${\ell}_{2k}$ be the image of the fundamental class

$\iota_{2k}\in H^{2k}(\mathrm{K}(\mathbb{Z}/2,2k);\mathbb{Z}/2)$ or
$H^{2k}(\mathrm{K}(\mathbb{Z}_{(2)},2k);\mathbb{Z}/2)$

Applying [[4] Theorem 4.3, 4.4] and the Serre spectral sequence for

$\mathrm{SG}\to\mathrm{G/TOP}\stackrel{{\scriptstyle q}}{{\to}}\mathrm{BSTOP}$

we have the following statements

${\ell}_{4k-2}$ is in the image of $q^{\ast}$ iff $k\neq 2^{j}$ for any $j\geq
0$;

${\ell}_{4k}$ is in the image of $q^{\ast}$ if $k\neq 2^{j}$ for any $j\geq
0$;

${\ell}_{2^{j+1}}+\ell_{2}^{2^{j}}$ is in the image of $q^{\ast}$ for $j\geq
1$.

By the following commutative diagram

$\textstyle{\mathrm{G/TOP}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{\mathrm{G/TOP}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{BSTOP}}$

the above statements also hold for the homomorphism $\varphi^{\ast}$. Next, we
consider the remaining cases.

For ${\ell}_{2^{j+1}-2}$ (any $j\geq 1$), in the Serre spectral sequence

$E_{2}^{p,q}=H^{p}(\mathrm{B(G/TOP)};H^{q}(\mathrm{G/TOP};\mathbb{Z}/2))\Longrightarrow
H^{p+q}(\ast;\mathbb{Z}/2)$ (3.2) ${\ell}_{2^{j+1}-2}\in
E_{2^{j+1}-1}^{0,2^{j+1}-2}$,
$d_{2^{j+1}-1}({\ell}_{2^{j+1}-2})={l}_{2^{j+1}-1}$

where ${l}_{2^{j+1}-1}\in H^{2^{j+1}-1}(\mathrm{B(G/TOP)};\mathbb{Z}/2)$
corresponds to the fundamental class of
$H^{2^{j+1}-1}(\mathrm{K}(\mathbb{Z}/2,2^{j+1}-1);\mathbb{Z}/2).$ Thus, in the
Serre spectral sequence

$\mathbb{E}_{2}^{p,q}=H^{p}(X;H^{q}(\mathrm{G/TOP};\mathbb{Z}/2))\Longrightarrow
H^{p+q}(E;\mathbb{Z}/2)$ (3.3)

we first have ${\ell}_{2^{j+1}-2}\in\mathbb{E}_{2^{j+1}-1}^{0,2^{j+1}-2}$.
Since the homomorphism (3.1) is trivial, $(\gamma\circ
f)^{\ast}:H^{2^{j+1}-1}(\mathrm{B(G/TOP)};\mathbb{Z}/2)\to
H^{2^{j+1}-1}(X;\mathbb{Z}/2)$ is trivial. Then, in the Serre spectral
sequence (3.3),

$\mathrm{d}_{2^{j+1}-1}({\ell}_{2^{j+1}-2})=0$

${\ell}_{2^{j+1}-2}$ survives to $\mathbb{E}_{\infty}$, which means that
${\ell}_{2^{j+1}-2}$ is in the image of $\varphi^{\ast}$. Moreover,
$\ell_{2}^{2^{j}}$ is in the image of $\varphi^{\ast}$. By statement (3),
$\ell_{2^{j+1}}$ is also in the image of $\varphi^{\ast}$ for $j\geq 1$.

Recall that $H^{\ast}(\mathrm{K}(\mathbb{Z}_{(2)},2k);\mathbb{Z}/2)$ and
$H^{\ast}(\mathrm{K}(\mathbb{Z}/{2},2k);\mathbb{Z}/2)$ are polynomial rings
with generators ${\ell}_{2k}$ and $\mathrm{Sq}^{\mathrm{I}}{\ell}_{2k}$, where
$\mathrm{I}=(i_{1},i_{2},\cdots,i_{r})$ is the admissible sequence. Therefore,
the desired epimorphism follows from the module homomorphism $\varphi^{\ast}$
over the Steenrod algebra. ∎

Let $n\neq 2^{j}$ for any $j\geq 0$. Assume that
$f_{4n-2}:X_{4n-2}\to\mathrm{BSG}$ has a lift to $\mathrm{BSTOP}$. Then we
consider the map $\gamma\circ f_{4n-1}:X_{4n-1}\to\mathrm{B(G/TOP)}$. For
convenience, let $X_{4n-1}=X_{4n-2}\cup e^{4n-1}$. As in last section,
$\gamma\circ f_{4n-1}$ is trivial if the map
$\bar{\iota}:S^{4n-1}\to\mathrm{B(G/TOP)}$ is trivial.

By Lemma 2.2, if $\bar{\iota}$ is nontrivial, the following homomorphism

$H_{4n-1}(S^{4n-1})\to H_{4n-1}(\mathrm{B(G/TOP)})\to
H_{4n-1}(\mathrm{B(G/TOP)}[2])$

is nontrivial. Recall equation (2.2) $H_{4n-1}(X_{4n-1})\cong
H_{4n-1}(S^{4n-1})$. Consider $\mathbb{Z}/2$-coefficient cohomology. By the
following homotopy commutative diagram, we have the equation (3.4)

$\textstyle{X_{4n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathrm{BSG}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{X_{4n-1}/X_{4n-2}=S^{4n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\iota}}$$\textstyle{\mathrm{B(G/TOP)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{B(G/TOP)}[2]}$
$\epsilon=(\gamma\circ f)^{\ast}(l_{4n-1})$ (3.4)

where $l_{4n-1}\in H^{4n-1}(\mathrm{B(G/TOP)};\mathbb{Z}/2)$ corresponds to
the fundamental class of
$H^{4n-1}(\mathrm{K}(\mathbb{Z}/2,4n-1);\mathbb{Z}/2)$, $0\neq\epsilon\in
H^{4n-1}(X;\mathbb{Z}/2)$.

In the Serre spectral sequence (3.2), $d_{4n-1}({\ell}_{4n-2})=l_{4n-1}$. By
equation (3.4) and the naturality of spectral sequences, $\epsilon$ is killed
by some differential in the Serre spectral sequence (3.3). However, by Lemma
3.1, $\epsilon$ survives to $\mathbb{E}_{\infty}$ in the Serre spectral
sequence (3.3), which makes a contradiction. Hence, $\bar{\iota}$ is trivial,
i.e. $f_{4n-1}$ has a lift to $\mathrm{BSTOP}$.

##  4\. The obstructions in $H^{4n+1}(X;\pi_{4n}(\mathrm{G/TOP}))$

In this section, $X$ is also a finite 1-connected CW complex and satisfies the
torsion-free condition (2.1). Assume that $f_{4n}:X_{4n}\to\mathrm{BSG}$ has a
lift to $\mathrm{BSTOP}$ for some $n\geq 1$. Then we consider the map
$\gamma\circ f_{4n+1}:X_{4n+1}\to\mathrm{B(G/TOP)}$. For convenience, let
$X_{4n+1}=X_{4n}\cup e^{4n+1}$. Similarly, $\gamma\circ f_{4n+1}$ is trivial
if the map $\bar{\iota}:S^{4n+1}\to\mathrm{B(G/TOP)}$ is trivial.

######  Lemma 4.1.

For a nontrivial map $\delta:S^{4n+1}\to\mathrm{B(G/TOP)}$, its image
$\mathrm{H}(\delta)$ is equal to $sl_{4n+1}+R\in H_{4n+1}(\mathrm{B(G/TOP)})$
under the Hurewicz homomorphism $\mathrm{H}$, where $0\neq s\in\mathbb{Z}$,
$l_{4n+1}$ is a free generator, $R$ denotes other components. Moreover, under
the homomorphism

$H_{\ast}(\mathrm{B(G/TOP)})\to H_{\ast}(\mathrm{B(G/TOP)}[2])\to
H_{\ast}(\mathrm{K}(\mathbb{Z}_{(2)},4n+1))$

the image of $R$ is zero, the image of $l_{4n+1}$ is nontrivial.

###### Proof.

The lemma follows by applying the Hurewicz homomorphism on the map
$S^{4n+1}\to\mathrm{B(G/TOP)}\to\mathrm{B(G/TOP)}[2]\to\mathrm{K}(\mathbb{Z}_{(2)},4n+1)$.
∎

Consider $\mathbb{Q}$-coefficient homology. As the arguments in last section,
if $\bar{\iota}$ is nontrivial, there is an element $\epsilon\in
H_{4n+1}(\mathrm{BSG};\mathbb{Q})$ such that

$\gamma_{\ast}(\epsilon)=sl_{4n+1}+R.$

In particular, $0\neq s\ell_{4n+1}+R\in
H_{4n+1}(\mathrm{B(G/TOP)};\mathbb{Q})$.

Since $\mathrm{BSG}$ have finite homotopy groups, the map

$q:\mathrm{G/TOP}\to\mathrm{BSTOP}$

defines a rational equivalence, i.e.

$q_{\ast}:H_{\ast}(\mathrm{G/TOP};\mathbb{Q})\to
H_{\ast}(\mathrm{BSTOP};\mathbb{Q})$

is an isomorphism. Furthermore, $H_{k}(\mathrm{BSG};\mathbb{Q})=0$ for any
$k\geq 1$, which makes a contradiction. So $\bar{\iota}$ is trivial, then
$\gamma\circ f_{4n+1}$ is trivial.

######  Data Availability.

Data sharing not applicable to this article as no data sets were generated or
analyzed during the current study.

######  Declaration.

The author states that there is no conflict of interest.

## References

  * [1] Browder, W. Manifolds and homotopy theory. Manifolds-Amsterdam 1970 (Proc. Nuffic Summer School), pp. 17-35.
  * [2] Browder, W. Surgery on simply-connected manifolds. Ergeb. Math. Grenzgeb., Band 65. Springer-Verlag, New York-Heidelberg, 1972.
  * [3] Browder, W; Hirsch, M. W. Surgery on piecewise linear manifolds and applications. Bull. Amer. Math. Soc. 72 (1966), 959-964.
  * [4] Brumfiel, G.; Madsen, I.; Milgram, R. J. PL characteristic classes and cobordism. Bull. Amer. Math. Soc. 77 (1971), 1025-1030.
  * [5] Dold, A.; Lashof, R. Principal quasi-fibrations and fibre homotopy equivalence of bundles. Illinois J. Math. 3 (1959), 285-305.
  * [6] Kirby, R. C.; Siebenmann, L. C. On the triangulation of manifolds and the Hauptvermutung. Bull. Amer. Math. Soc. 75 (1969), 742-749.
  * [7] Kirby, R. C.; Siebenmann, L. C. Foundational Essays on Topological Manifolds, Smoothings and Triangulations. Ann. of Math. Studies 88, Princeton Univ. Press, Princeton, NJ, 1977.
  * [8] May, J. P.; Ponto, K. More concise algebraic topology. Localization, completion, and model categories. Chicago Lectures in Math. University of Chicago Press, Chicago, IL, 2012.
  * [9] Madsen, I.; Milgram, R. J. The classifying spaces for surgery and cobordism of manifolds. Ann. of Math. Stud., No. 92. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1979.
  * [10] Novikov, S. P. Homotopy equivalent smooth manifolds I. AMS Translations 48 (1965), 271-396.
  * [11] Rudyak, Y. B. On Thom Spectra, Orientability, and Cobordism, Springer Mongraphs in Mathematics,Corrected 2nd printing, Springer, 2008.
  * [12] Spivak, M. Spaces satisfying Poincar$\mathrm{\acute{e}}$ duality. Topology 6 (1967), 77-101.
  * [13] Sullivan, D. P. Geometric topology: localization, periodicity and Galois symmetry. $K$-Monogr. Math., 8. Springer, Dordrecht, 2005.
  * [14] Wall, C. T. C. Finiteness conditions for CW-complexes. Ann. of Math. (2) 81 (1965), 56-69.