Two faces of Hawking radiation and thin-shell emission: pair-creation vs.
tunneling

Dong-han Yeom$\star$

Department of Physics Education,

Pusan National University, Busan 46241, Republic of Korea

Research Center for Dielectric and Advanced Matter Physics,

Pusan National University, Busan 46241, Republic of Korea

$\star$<EMAIL_ADDRESS>
4th International Conference on Holography,
String Theory and Discrete Approach
Hanoi, Vietnam, 2020

## Abstract

We first revisit Hartle and Hawking’s path integral derivation of Hawking
radiation. In the first point of view, we interpret that a particle-
antiparticle pair is created and the negative energy antiparticle falls into
the black hole. On the other point of view, a particle inside the horizon, or
beyond the Einstein-Rosen bridge, tunnels to outside the horizon, where this
computation requires the analytic continuation of the time. These two faces of
the Hawking radiation process can be extended to not only particles but also
fields. As a concrete example, we study the thin-shell tunneling process; by
introducing the antishell as a negative tension shell, we can give the
consistent interpretation for two pictures, where one is a tunneling from
inside to outside the horizon using instantons, while the other is a shell-
antishell pair-creation. This shows that the Euclidean path integral indeed
carries vast physical implications not only for perturbative, but also for
non-perturbative processes.

###### Contents

  1. 1 Introduction
  2. 2 Hawking radiation as tunneling: revisit Hartle and Hawking
    1. 2.1 Formalism
    2. 2.2 First point of view: particle-antiparticle pair creation
    3. 2.3 Second point of view: tunneling from inside to outside the horizon
  3. 3 Extension to fields and shells
  4. 4 Thin-shell instantons revisited
    1. 4.1 Thermal single-shell: junction equation and causal structures
    2. 4.2 Thermal double-shell: junction equations and causal structures
    3. 4.3 Probability: equivalence to the single-shell interpretation
    4. 4.4 Beyond the thermal shell limit
  5. 5 Conclusion: toward generic complex-valued instantons

##  1 Introduction

The tension between gravity and quantum mechanics is the fundamental but
unresolved problem of modern physics. Although this is a difficult problem,
there is one way to approach the nature of quantum gravity. That is to
consider quantum effects around a given classical metric, where this is known
as semi-classical gravity. This semi-classical approach is not the complete
theory, but it can include some useful and important consequences that should
be implemented by the consistent quantum theory of gravity. One of such
consequences in black hole physics is the emission of particles from the event
horizon, or so-called Hawking radiation [1].

The existence of Hawking radiation was already confirmed by various
approaches. The most famous explanation is to use the Bogoliubov
transformation [1]. Then, due to the redshift of the incoming modes around the
horizon, the expectation value of the number operator of a given mode at
future infinity is non-vanishing, even though the state was chosen to be a
vacuum at the past infinity. This approach includes very essential properties
of Hawking radiation, but there remain questions. For example, what physically
happens at the event horizon?

As a response of the question, one may say that there are several alternative
interpretations. First, in the tunneling picture [2, 3, 4, 5], a particle
tunnels from inside to outside the horizon. Second, in the renormalized
energy-momentum tensor picture [6], a negative energy flux falls into the
black hole. Of course, this must be two faces of the same physics. In fact,
there is a well-known correspondences between antiparticle and particle;
either a particle moves backward in time or an antiparticle moves forward in
time.

In this paper, we first revisit the tunneling picture which was developed by
Hartle and Hawking [2]. The original motivation of this was to explain that a
particle tunnels from inside to outside, or equivalently, an antiparticle
falls in and a particle comes out from the black hole. In order to evaluate
the probability, Hartle and Hawking used the analytic continuation of the time
of the inside the horizon. However, we will observe that this is not the
unique analytic continuation; one can also use the analytic continuation
outside the horizon.

Then, this opens a possibility that the analytic continuation of the time
outside the horizon justifies the use of the Euclidean instantons; indeed,
Hawking radiation can be interpreted as instantons [7]. One strong point of
the instanton is that one can further extend to non-perturbative processes
[8]. In this point of view, will the correspondence between two pictures, the
particle-antiparticle pair-creation vs. the tunneling over the event horizon
or the Einstein-Rosen bridge, still be true even in the non-perturbative
limit?

Although we cannot provide a complete proof about this, our answer is
positive. We will focus on the thin-shell tunneling issue [9] and first
construct a tunneling channel which is very similar to the Hartle-Hawking’s
particle tunneling. From this tunneling instanton, we will reconstruct an
analogous process of the particle-antiparticle pair creation; so to speak, a
shell-antishell pair creation. We will explain that two pictures give
consistent probability interpretations.

This paper is organized as follows. In Sec. 2, we first revisit the particle
tunneling picture of Hartle and Hawking. We can introduce the analytic
continuation of the time of not only inside but also outside the event
horizon. We conclude that a particle-antiparticle pair-creation is indeed
equivalent to a tunneling over the Einstein-Rosen bridge, or the event
horizon. In Sec. 3, we approximately extend this to the field level using the
Euclidean analytic continuation and conclude that if there is a positive
tension shell, its counter part (antishell) can be described by a negative
tension shell, which can be supported by the complexification of the field. In
Sec. 4, as a very natural extension, we consider a shell-antishell pair
creation process, where we can provide two points of view; one is a shell
tunnels from inside to outside, while the other is a negative tension shell
falls into the horizon. Finally, in Sec. 5, we summarize our results and
discuss possible future extensions.

##  2 Hawking radiation as tunneling: revisit Hartle and Hawking

In this section, we review the technique introduced by Hartle and Hawking [2]
and add more comments on possible new interpretations.

###  2.1 Formalism

The tunneling amplitude of a particle with energy $\omega$ from
$x=\left(t,\vec{r}\right)$ to
$x^{\prime}=\left(t^{\prime},\vec{r}^{\prime}\right)$ (fixing $t^{\prime}=0$)
is

$\displaystyle\mathcal{S}\left(\vec{r}^{\prime},\vec{r}\right)=\int_{-\infty}^{+\infty}dte^{-i\omega
t}K\left(0,\vec{r}^{\prime};t,\vec{r}\right),$ (1)

where

$\displaystyle
K\left(0,\vec{r}^{\prime};t,\vec{r}\right)=-\frac{i}{4\pi^{2}}\frac{1}{s(x,x^{\prime})\pm
i\epsilon}$ (2)

is the propagator and $s(x,x^{\prime})$ is the square of the geodesic distance
between $x$ and $x^{\prime}$.

Note that there is a propagator dependence. If we choose the Feynman
propagator, then for $x^{\prime}>x$, the pole is shifted to the upper
direction, while if $x^{\prime}<x$, the pole is shifted to the lower
direction. On the other hand, by choosing advanced or retarded propagator, one
can choose the location of the poles by different ways.

Since the propagator only depends on $t^{\prime}-t$ and $s(x^{\prime},x)$ is
the square of the geodesic distance, without loss of generality, we obtain

$\displaystyle
K\left(0,\vec{r}^{\prime};t,\vec{r}\right)=K\left(t,\vec{r};0,\vec{r}^{\prime}\right)=K\left(-t,\vec{r}^{\prime};0,\vec{r}\right).$
(3)

By using the second term, in order to evaluate the amplitude, we interpret
that we fix a point outside the black hole $x^{\prime}$ and integrate over $t$
by sliding the point inside the black hole $x$, which is the same as Hartle
and Hawking. On the other hand, if we use the third term, we interpret that we
fix a point inside the black hole $x$ and integrate over $t$ by sliding the
point outside the black hole $x^{\prime}$. Then we will get the same result
but obtain a different interpretation.

###  2.2 First point of view: particle-antiparticle pair creation

We recall the integration

$\displaystyle\mathcal{S}\left(\vec{r}^{\prime},\vec{r}\right)=\int_{-\infty}^{+\infty}dte^{-i\omega
t}K\left(-t,\vec{r}^{\prime};0,\vec{r}\right).$ (4)

If we redefine $t=-T$, then this integration becomes

$\displaystyle\mathcal{S}\left(\vec{r}^{\prime},\vec{r}\right)=\int_{-\infty}^{+\infty}dTe^{-i(-\omega)T}K\left(0,\vec{r};T,\vec{r}^{\prime}\right).$
(5)

Now we can interpret that, in this new time coordinates $T$, this is
equivalent to see that a particle of negative energy $-\omega$ propagates from
outside to inside the horizon. This means that there is a particle-
antiparticle pair-creation outside the horizon; and the antiparticle with
negative energy falls into the black hole.

###  2.3 Second point of view: tunneling from inside to outside the horizon

Figure 1: Case 1 (upper) and Case 2 (lower). For Case 1 and 2, $x^{\prime}$ is
fixed and $x$ is sliding over time $t$. The green dashed curves show two
possible points which can be connected by null geodesics. Each null geodesic
is denoted by a pole in the complex time plain and the colors (red and blue
dots) corresponds x-marks in the right diagram. For each color, there are more
than one x-mark, because longer geodesics can be obtained by considering
$\theta$ and $\varphi$ angles. The poles are shifted by $\pm i\epsilon$ due to
the Feynman propagator. Figure 2: Case 3. For Case 3, $x$ is fixed and
$x^{\prime}$ is sliding. Note that for Case 1, $x(t-4\pi Mi)$ is the causal
past of $x^{\prime}$, while for Case 3, $x$ is always causal future of
$x^{\prime}$. Hence, in order to have the same analytic structure with Case 1,
we need to introduce the advanced propagator for the poles at
$x^{\prime}(t-4\pi Mi)$.

Now we see the analytic structure of
$K\left(0,\vec{r}^{\prime};t,\vec{r}\right)$ as a function of $t$ by fixing
$\vec{r}$ and $\vec{r}^{\prime}$. The important thing to check is the pole
structure, where the pole happens when $x$ and $x^{\prime}$ can be connected
by null geodesics.

We will see the pole structures for three different cases.

  * – 1.

Choice of $K\left(t,\vec{r};0,\vec{r}^{\prime}\right)$, while $\vec{r}$ is
inside the black hole and $\vec{r}^{\prime}$ is outside the black hole (upper
of Fig. 1): Following Hartle-Hawking, there appears poles in $t$ (inside the
black hole region) as well as $t-4\pi Mi$ (inside the white hole region)
lines. There are two kinds of poles, where one is $x^{\prime}>x$ (at
$\mathrm{Im}t=0$) and the other one is $x^{\prime}<x$ (at $\mathrm{Im}t=-4\pi
Mi$). Hence, by introducing the Feynman propagator, one can shift the poles to
the upper direction for the former and the lower direction for the latter.
Then, due to the analyticity, one can shift the time integration from $t$ to
$t-4\pi Mi$.

  * – 2.

Choice of $K\left(t,\vec{r};0,\vec{r}^{\prime}\right)$, while both of
$\vec{r}$ and $\vec{r}^{\prime}$ are outside the black hole (lower of Fig. 1):
Again, as described by Hartle-Hawking, there appears poles only in the $t$
line. There are two kinds of poles, where one is $x^{\prime}>x$ and the other
one is $x^{\prime}<x$. Hence, by introducing the Feynman propagator, the half
of the poles is shifted to the upper direction and the other half of the poles
is shifted to the lower direction. Then, in the end, there is no good way to
analytically continue to the Euclidean time.

  * – 3.

Choice of $K\left(-t,\vec{r}^{\prime};0,\vec{r}\right)$, while $\vec{r}$ is
inside the black hole and $\vec{r}^{\prime}$ is outside the black hole (Fig.
2): Now $\vec{r}$ can be connected via null geodesics to $\vec{r}^{\prime}$ by
two ways: one is in the $t$ side (right side of the Penrose diagram), while
the other one is in the $t-4\pi Mi$ side (left side of the Penrose diagram).
Note that both of them satisfy $x^{\prime}>x$. Therefore, in order to obtain
the same analytic structure, we need to introduce the retarded propagator for
the former poles and the advanced propagator for the latter poles.

Figure 3: Comparison with Case 1 (left) and Case 3 (right). The contour 2 of
left should be identified to the contour 2-1 and 2-2 of right. This requires
the advanced propagator for 2-1.

Therefore, the choice of the propagator for Case 3 means that we choose such a
propagator in order to make the contour integration proportional to the
absorption probability (Fig. 3). In terms of the particle trajectory, for Case
1 (left), one can approximately understand that the tunneling rate is
calculated by two parts, where one is over the Euclidean manifold (trajectory
1, inside the horizon) and the other is from inside the white hole to the
future infinity (trajectory 2), while the latter is the same as the absorption
probability. The same process is described by Case 3 (right), and hence, the
absorption probability is described by the trajectories 2-1 and 2-2. This
means that in order to interpret this contribution as the absorption
probability, the causality of 2-1 should be regarded as opposite to the usual
way.

Finally, for Case 1 and Case 3, due to the analytic structure, we can shift
the time integration from $t$ to $t-4\pi Mi$. Then we obtain the following
integration

$\displaystyle\mathcal{S}\left(\vec{r}^{\prime},\vec{r}\right)=e^{-4\pi
M\omega}\int_{-\infty}^{+\infty}dte^{-i\omega
t}K\left(0,\vec{r}^{\prime};t-4\pi Mi,\vec{r}\right).$ (6)

After squaring both sides, we obtain

$\displaystyle(\mathrm{emission\;probability})=e^{-8\pi
M\omega}\times(\mathrm{absorption\;probability}),$ (7)

that means the Hawking temperature is $T=1/8\pi M$.

##  3 Extension to fields and shells

Now we understand that two points of view are equivalent, in other words, the
in-coming negative energy particle is equivalent to a positive energy particle
that comes out through the Einstein-Rosen bridge. This equivalence can be
generalized to field level descriptions if we interpret a field as a bunch of
particles.

Let us consider a propagator between an initial hypersurface defined on past
infinity $(h^{\mathrm{in}}_{ab},\phi^{\mathrm{in}})$ to a final hypersurface
defined on the future infinity $(h^{\mathrm{out}}_{ab},\phi^{\mathrm{out}})$:

$\displaystyle\Psi\left[h^{\mathrm{out}}_{ab},\phi^{\mathrm{out}};h^{\mathrm{in}}_{ab},\phi^{\mathrm{in}}\right]=\int\mathcal{D}g_{\mu\nu}\mathcal{D}\phi\;\;e^{iS[g_{\mu\nu},\phi]},$
(8)

where we sum over all metric $g_{\mu\nu}$ and matter field $\phi$ those
connect from $(h^{\mathrm{in}}_{ab},\phi^{\mathrm{in}})$ to
$(h^{\mathrm{out}}_{ab},\phi^{\mathrm{out}})$. For the ground state, we may
introduce the Euclidean analytic continuation and introduce the Euclidean path
integral [10] such as

$\displaystyle\Psi_{0}\left[h^{\mathrm{out}}_{ab},\phi^{\mathrm{out}};h^{\mathrm{in}}_{ab},\phi^{\mathrm{in}}\right]=\int\mathcal{D}g_{\mu\nu}\mathcal{D}\phi\;\;e^{-S_{\mathrm{E}}[g_{\mu\nu},\phi]}.$
(9)

This Euclidean path integral can be further approximated by two assumptions
[7]:

  * –

First, we can assume a kind of symmetry and restrict to the mini-superspace,
e.g., the spherical symmetry.

  * –

Second, we can approximate the path-integral by using the steepest-descent
approximation, so-called sum-over instantons:

$\displaystyle\Psi_{0}\left[h^{\mathrm{out}}_{ab},\phi^{\mathrm{out}};h^{\mathrm{in}}_{ab},\phi^{\mathrm{in}}\right]\simeq\sum_{\mathrm{on-
shell}}e^{-S_{\mathrm{E}}^{\mathrm{on-shell}}[g_{\mu\nu},\phi]}.$ (10)

One interesting point is that because of the analytic continuation, every
fields should be complexified [11]. On the other hand, we need to impose the
reality condition for the initial hypersurface as well as the final
hypersurface. This means that for the intermediate geometry between the
initial and the final surfaces, it is allowed to introduce imaginary valued
fields, as long as the imaginary field value does not reach the future
infinity.

In this paper, we will restrict the complexification of only matter fields for
simplicity. Then what are the effective roles of the imaginary part,
especially, of the scalar field? Let us focus on the following points. For a
(classical) scalar field, for a given time slice, we can use the Fourier
transform and split each different momentum modes.

The scalar field of each mode will be proportional to $\phi_{\omega}\propto
a(\omega)e^{-i\omega t}$. If the internal proper time $t$ of the scalar field
is following the backward direction, then it is equivalent to change
$\omega\rightarrow-\omega$; the energy becomes negative. Or, equivalently, if
we introduce overall imaginary factor $i=\sqrt{-1}$ to the scalar field, i.e.,
$\phi_{\omega}\rightarrow i\phi_{\omega}$, then it is also equivalent to
consider a negative kinetic energy (hence, an effective ghost field [12]),
because the effective number of each mode becomes negative
$|a(\omega)|^{2}\rightarrow-|a(\omega)|^{2}$. To summarize, for a given mode,
the follows are equivalent:

  * –

A real scalar field with positive energy moves backward in time,

  * –

A real scalar field with negative energy moves forward in time,

  * –

An imaginary scalar field with positive energy moves forward in time.

Because of these relations, it is justifiable to introduce a negative tension
shell (as long as it does not reach to past or future infinity), that is
motivated from the complexified scalar field. For simplicity, we interpret
that these imaginary (or negative tension) shell as a positive tension shell
that moves the opposite way of the coordinate time.

##  4 Thin-shell instantons revisited

In this section, we apply for the results of the previous sections. First, for
the Hawking radiation, we have two points of view for the same phenomenon: (1)
a particle-antiparticle pair-creation and (2) a particle tunneling from inside
to outside the horizon, or a particle tunnels through the Einstein-Rosen
bridge, where the latter can be described by Euclidean approaches. Second, if
we extend this to the field level, then the notion of the antiparticle can be
well matched to the analytically continued imaginary part of the field. In the
thin-shell case, the imaginary field will effectively give a negative tension,
because its kinetic term is negative.

Now we have the following question. Will the thin-shell tunneling process
(which is usually described by the Euclidean instantons [8, 13]) be
equivalently described by the shell-antishell (or, positive-tension-shell-
negative-tension-shell) pair-creation process? Will these two pictures be
consistent? In this section, we will answer for these questions.

###  4.1 Thermal single-shell: junction equation and causal structures

We first follow rather a canonical approach: a tunneling process of a positive
tension shell. We consider a spacetime with the spherical symmetric metric
ansatz

$\displaystyle
ds_{\pm}^{2}=-f_{\pm}(R)dT^{2}+\frac{1}{f_{\pm}(R)}dR^{2}+R^{2}d\Omega^{2},$
(11)

where we prepare a thin-shell that locates at $r$: outside the shell is $r<R$
(denoted by $+$) and inside the shell is $R<r$ (denoted by $-$). The thin-
shell will satisfy the metric

$\displaystyle ds^{2}=-dt^{2}+r^{2}(t)d\Omega^{2}.$ (12)

We impose the metric ansatz for outside and inside the shell

$\displaystyle f_{\pm}(R)=1-\frac{2M_{\pm}}{R}-\frac{R^{2}}{\ell_{\pm}^{2}}.$
(13)

Here, $M_{+}$ and $M_{-}$ are the mass parameters of each region and

$\displaystyle\ell^{2}_{\pm}=\frac{3}{8\pi V_{\pm}}$ (14)

is the parameter due to the vacuum energy $V_{\pm}$. We assume $M_{+}>M_{-}$
(for the opposite case, see [14]).

The equation of motion of the thin-shell is determined by the junction
equation [15]:

$\displaystyle\epsilon_{-}\sqrt{\dot{r}^{2}+f_{-}(r)}-\epsilon_{+}\sqrt{\dot{r}^{2}+f_{+}(r)}=4\pi
r\sigma.$ (15)

Here, $\epsilon_{\pm}=\pm 1$, where these $\epsilon$ parameters denotes the
outward normal direction of the shell. $\sigma$ is the tension parameter and
we choose positive value so that this satisfies the null energy condition.

After simple calculations, we can reduce the junction equation by a simpler
formula [16]:

$\displaystyle\dot{r}^{2}+V(r)$ $\displaystyle=$ $\displaystyle 0,$ (16)
$\displaystyle V(r)$ $\displaystyle=$ $\displaystyle
f_{+}(r)-\frac{\left(f_{-}(r)-f_{+}(r)-16\pi^{2}\sigma^{2}r^{2}\right)^{2}}{64\pi^{2}\sigma^{2}r^{2}}.$
(17)

Now we consider the nucleation of thin-shells. Such a nucleation can be
explained by an instanton. For simplicity, we first consider the case of the
thermal excitations [17, 18], where this requires the conditions
$V(r_{s})=V^{\prime}(r_{s})=0$ for a given $r_{s}$; later, we will extend to a
generic instanton. The detailed condition for the thermal excitation condition
is described in the Appendix of [19].

After we classify the classical trajectories, we determine signs of $\epsilon$
parameters by comparing extrinsic curvatures:

$\displaystyle\beta_{\pm}(r)\equiv\frac{f_{-}(r)-f_{+}(r)\mp
16\pi^{2}\sigma^{2}r^{2}}{8\pi\sigma
r}=\epsilon_{\pm}\sqrt{\dot{r}^{2}+f_{\pm}(r)}.$ (18)

If we choose $\ell_{+}=\infty$, $\ell_{-}^{-2}<0$, and
$-\ell_{-}^{-2}-16\pi^{2}\sigma^{2}>0$, then the only allowed solution is
$\epsilon_{\pm}=+1$ for all regions.

In the Euclidean signatures, the shell is in a stable local minimum and will
maintain a constant radius. Therefore, the causal structure is easy to
explain. On the other hand, it is on a unstable local maximum in the
Lorentzian signatures. Therefore, after a nucleation, it is reasonable to
think that the shell either collapses or expands [18].

###  4.2 Thermal double-shell: junction equations and causal structures

Now we ask whether the same process can be described by a shell-antishell
process. In order to do this, we need to introduce not only a positive tension
shell, but also a negative tension shell. Due to the symmetry, it is not
difficult to extend the thin-shell formalism.

We consider spacetime with the metric,

$\displaystyle
ds_{+,0,-}^{2}=-f_{+,0,-}(R)dT^{2}+\frac{1}{f_{+,0,-}(R)}dR^{2}+R^{2}d\Omega^{2}.$
(19)

We prepare two thin-shells where one is the outer shell $r_{2}$ and the other
is the inner shell $r_{1}$ ($r_{1}\leq r_{2}$): outside the outer shell is
$r_{2}<R$ (denoted by $+$), inside the inner shell is $R<r_{1}$ (denoted by
$-$), and the intermediate region is $r_{1}<R<r_{2}$ (denoted by $0$).

Thin-shells will follow the metric

$\displaystyle ds^{2}=-dt^{2}+r_{1,2}^{2}(t)d\Omega^{2}.$ (20)

We impose the metric ansatz for outside and inside the shell

$\displaystyle
f_{+,0,-}(R)=1-\frac{2M_{+,0,-}}{R}-\frac{R^{2}}{\ell_{+,0,-}^{2}}.$ (21)

The equations of motion of thin-shells are as follows:

$\displaystyle\epsilon^{(2)}_{0}\sqrt{\dot{r}_{2}^{2}+f_{0}(r_{2})}-\epsilon^{(2)}_{+}\sqrt{\dot{r}_{2}^{2}+f_{+}(r_{2})}$
$\displaystyle=$ $\displaystyle 4\pi r_{2}\sigma_{2},$ (22)
$\displaystyle\epsilon^{(1)}_{-}\sqrt{\dot{r}_{1}^{2}+f_{-}(r_{1})}-\epsilon^{(1)}_{0}\sqrt{\dot{r}_{1}^{2}+f_{0}(r_{1})}$
$\displaystyle=$ $\displaystyle 4\pi r_{1}\sigma_{1}.$ (23)

Here, $\epsilon^{(1,2)}_{+,0,-}=\pm 1$ to denote the outward normal
directions. $\sigma_{1,2}$ are tension parameters.

If we assume $M_{+}=M_{-}=M$, $\sigma_{2}=-\sigma_{1}=\sigma$,
$\ell_{+}=\ell_{-}=\ell$, and $\Delta_{\pm}\equiv M\pm M_{0}$, then

$\displaystyle\dot{r}_{1,2}^{2}+V_{1,2}(r_{1,2})=0,$ (24)

where

$V_{2}(r_{2})=1-\left(\frac{\rho_{+}^{2}}{64\pi^{2}\sigma^{2}}+\frac{\ell_{0}^{-2}+\ell_{+}^{-2}}{2}+4\pi^{2}\sigma^{2}\right)r_{2}^{2}-\left(1+\frac{\rho_{+}}{16\pi^{2}\sigma^{2}}\frac{\Delta_{-}}{\Delta_{+}}\right)\frac{\Delta_{+}}{r_{2}}-\frac{\Delta_{-}^{2}}{16\pi^{2}\sigma^{2}r_{2}^{4}},$
(25)
$V_{1}(r_{1})=1-\left(\frac{\rho_{+}^{2}}{64\pi^{2}\sigma^{2}}+\frac{\ell_{+}^{-2}+\ell_{0}^{-2}}{2}+4\pi^{2}\sigma^{2}\right)r_{1}^{2}-\left(1+\frac{\rho_{+}}{16\pi^{2}\sigma^{2}}\frac{\Delta_{-}}{\Delta_{+}}\right)\frac{\Delta_{+}}{r_{1}}-\frac{\Delta_{-}^{2}}{16\pi^{2}\sigma^{2}r_{1}^{4}},$
(26)

and hence the effective potential coincide.

After we classify the classical trajectories, we determine signs of $\epsilon$
parameters by comparing extrinsic curvatures:

$\displaystyle\beta^{(2)}_{+}(r_{2})$ $\displaystyle\equiv$
$\displaystyle\frac{f_{0}(r_{2})-f_{+}(r_{2})-16\pi^{2}\sigma_{2}^{2}r_{2}^{2}}{8\pi\sigma_{2}r_{2}}=\epsilon^{(2)}_{+}\sqrt{\dot{r}_{2}^{2}+f_{+}(r_{2})},$
(27) $\displaystyle\beta^{(2)}_{0}(r_{2})$ $\displaystyle\equiv$
$\displaystyle\frac{f_{0}(r_{2})-f_{+}(r_{2})+16\pi^{2}\sigma_{2}^{2}r_{2}^{2}}{8\pi\sigma_{2}r_{2}}=\epsilon^{(2)}_{0}\sqrt{\dot{r}_{2}^{2}+f_{0}(r_{2})},$
(28) $\displaystyle\beta^{(1)}_{0}(r_{1})$ $\displaystyle\equiv$
$\displaystyle\frac{f_{-}(r_{1})-f_{0}(r_{1})-16\pi^{2}\sigma_{1}^{2}r_{1}^{2}}{8\pi\sigma_{1}r_{1}}=\epsilon^{(1)}_{0}\sqrt{\dot{r}_{1}^{2}+f_{0}(r_{1})},$
(29) $\displaystyle\beta^{(1)}_{-}(r_{1})$ $\displaystyle\equiv$
$\displaystyle\frac{f_{-}(r_{1})-f_{0}(r_{1})+16\pi^{2}\sigma_{1}^{2}r_{1}^{2}}{8\pi\sigma_{1}r_{1}}=\epsilon^{(1)}_{-}\sqrt{\dot{r}_{1}^{2}+f_{-}(r_{1})}.$
(30)

Let us define $M_{+}=M_{-}=M$ and $\sigma_{2}=-\sigma_{1}=\sigma$, so that the
outer shell is a real shell and the inner shell is an imaginary shell. We
assume that $\ell=\infty$, $\ell_{0}^{-2}<0$, and
$|4\pi\sigma\ell_{0}|^{-2}>1$. Then, the equation of motion for each shell
will follow the same potential that was discussed in the previous section. In
addition, since the tension of the inner shell is negative, the signs of the
extrinsic curvatures should be always positive for both of inner and outer
shells. Therefore, from the point of crossing radius $r_{s}$, the allowed
solutions are only two types those are located right patch of the Penrose
diagram: branching and emerging solutions.

We consider an Euclidean analytic continuation when the effective potential
satisfies the matching condition $V^{\prime}(r_{s})=V(r_{s})=0$. In this
limit, if we fold two shells at $r=r_{s}$, then $\dot{r}=0$; in Euclidean
signatures this is a stable local minimum. Also, since one positive and one
negative shells are folded, effectively the Euclidean Schwarzschild manifold
with mass $M$ is the entire Euclidean solution.

After analytic continuation to the Lorentzian time, the order of $+$ part and
$-$ part are changed for the left side of the causal patch. So, for the right
part, we can continue to a trivial branching shell solution. On the other
hand, for the left part, the directions of outward normal vectors are flipped.
For these flipped outward normal vectors, the only possible corresponding
solution is the rotated result of the emerging shells.

###  4.3 Probability: equivalence to the single-shell interpretation

For the thermal instantons, the velocity of the shell is zero on the Euclidean
manifold and maintains a constant radius $r_{0}$. Since there is no dynamics
of the shell on the Euclidean manifold, the only contribution of the Euclidean
action comes from the regularization of the cusp of the horizon. We can
calculate the probability of a single shell based on the Euclidean path
integral formalism [13], where the result is $P\sim e^{-2B}$ with

$\displaystyle
2B=2\left[S_{\mathrm{E}}(\mathrm{solution})-S_{\mathrm{E}}(\mathrm{background})\right]=-\frac{\Delta\mathcal{A}}{4},$
(31)

where $\mathcal{A}$ is the area of the horizon. For the evaporating case,
$\Delta\mathcal{A}<0$ and hence $2B$ is positive definite.

On the other hand, if we interpret the shell-antishell pair-creation process
and regard that the negative energy falls into the black hole, the areal
entropy is then changed while the energy is conserved. Then after a sufficient
Lorentzian time, we compare two solutions, where one is the pure Schwarzschild
solution with mass $M_{+}$ and the other is the Schwarzschild-AdS solution
with mass $M_{-}$ and the cosmological constant $V_{-}$. We can evaluate the
Euclidean action difference between the two solutions, because the shell
dynamics will not contribute to the probability during the Lorentzian time. As
a result, the action difference between two solutions will be the same as the
areal entropy difference. As a simple check, we can evaluate following the
thermodynamic way. We can estimate the Helmholtz free energy difference as
follows [18]:

$\displaystyle 2B=\frac{\Delta\mathcal{F}}{T}=-\frac{\Delta\mathcal{A}}{4},$
(32)

where $\mathcal{F}=E-ST$ is the Helmholtz free energy, $E$ is the energy,
$S=\mathcal{A}/4$ is the areal entropy, and the last equation is obtained
since the energy is conserved ($\Delta E=0$).

Therefore, we obtain the consistent and equivalent probabilities from two
pictures: from the thin-shell tunneling picture, the probability is decided by
the Euclidean action integral, while the shell-antishell pair-creation
picture, the probability can be interpreted by the thermodynamic way.

###  4.4 Beyond the thermal shell limit

If one goes beyond the thermal shell, the shell will have dynamics in the
Euclidean domain. Usually, one can interpret this process such that this is a
tunneling from a small radius (say, $r_{1}$) to a larger radius (say,
$r_{2}$). Or, alternatively, one can interpret that two shells are created at
the same time, where one is left side and the other is right side of the
Einstein-Rosen bridge. The difference of two interpretations is whether there
is a contribution of the areal entropy dependence, where the former has no
such a contribution while the latter must have the term from the
regularization of the cusp of the Euclidean manifold.

Now what happens if we apply this for the double-shell model? We will
interpret that the negative tension shell will collapse from the smaller
radius, while the positive tension shell will expand from the larger radius in
the Lorentzian domain. Then, in the Euclidean domain, two shells must meet
each other. Note that the Euclidean action contribution becomes [13]

$\displaystyle
2B=\frac{\mathcal{A}_{\mathrm{i}}-\mathcal{A}_{\mathrm{f}}}{4}+2\int_{r_{1}}^{r_{2}}drr\left|\cos^{-1}\left(\frac{f_{+}+f_{-}-16\pi^{2}\sigma^{2}r^{2}}{2\sqrt{f_{+}f_{-}}}\right)\right|,$
(33)

where the last term of the right hand side is due to the dynamics of the
shell. Note that this is symmetric up to the change of $f_{\pm}\rightarrow
f_{\mp}$ and $\sigma\rightarrow-\sigma$. Therefore, if the two shells meet at
$r=r_{c}$, then the positive tension shell integration from $r_{1}$ to $r_{2}$
is the same as the negative tension shell integration from $r_{1}$ to $r_{c}$
plus the positive tension shell integration from $r_{c}$ to $r_{2}$. If we
interpret the first term using the thermodynamic way (as the negative tension
shell collapses, the area decreases and hence the entropy changes), then we
will eventually obtain the same probabilistic interpretation [20].

One potential assumption is that there is no singular contribution when two
shells meet each other. This is beyond the scope of the thin-shell
approximation. We leave this detailed clarification for a future research
topic.

##  5 Conclusion: toward generic complex-valued instantons

In this paper, we first revisited the particle tunneling from a black hole
using the Hartle-Hawking’s tunneling picture. One can interpret this process
as a pair-creation of particle and antiparticle, where the antiparticle moves
from outside to inside the horizon by carrying negative energy. On the other
hand, in order to evaluate the tunneling probability, one needs to introduce
the Euclidean analytic continuation of the time; original Hartle and Hawking
introduced the analytic continuation inside the horizon, but equivalently, we
can introduce the analytic continuation outside the horizon. Therefore, we can
interpret that the pair-creation is equivalent to a particle tunneling from
inside to outside the horizon, where initially the particle is located beyond
the Einstein-Rosen bridge and connected by the Euclidean manifold.

We can ask whether this picture can be extended to more generic contexts. Due
to the analytic continuation of the time, in principle, all fields can be
analytically continued to complex-valued functions. The imaginary part of the
matter field will have a negative kinetic energy, and hence, it can be
interpreted by a flux of negative energy antiparticles. If this is
accumulated, in the thin-shell approximation, one may describe the antishell
as a negative tension shell.

In this paper, we further extended this idea. We could describe the dynamics
of the shell by using the junction equation. In addition, we can evaluate the
tunneling probability following the Euclidean path integral approach. The same
thing can be done by introducing two shells, where one has the positive
tension (shell) and the other has the negative tension (antishell). We showed
that we obtain the same probability from the two interpretations, not only for
thermal shells, but also for generic processes.

Therefore, the correspondence between the pair-creation and the instanton
tunneling look very natural and fundamental, not only for the perturbative
level (Hawking radiation) but also for the non-perturbative level. This
strongly indicate that the Euclidean path integral indeed include very
important and essential nature of the quantum gravity. The construction of a
shell-antishell is not difficult to imagine at once we have a matter field.
One benefit of the shell-antishell interpretation is that the understanding of
the causal structure is rather simpler, because we do not need to assume the
geometry beyond the Einstein-Rosen bridge. This will allow to understand the
physical causal structure of an evaporating black hole with non-perturbative
or non-adiabatic processes, where this will do a very important role in order
to understand the information loss problem, for example, to show the
possibility of a naked black hole firewall [21] or to make the Einstein-Rosen
bridge traversable [22]. In addition, the contributions from the trivial
geometries without a singularity might be dominated in the late time [23].
This will shed some lights to the answer for the information loss paradox.

Of course, there are several issues that must be clarified. For example, we
relied on the thin-shell approximation, but we need to check whether the
correspondence can be consistent even beyond the thin-shell approximation, or
not. Also, if two shells are collided, can there be any new effects? We leave
these topics for future investigations.

## Acknowledgment

The author would like to thank Pisin Chen and Misao Sasaki. DY was supported
by the National Research Foundation of Korea (Grant No.: 2018R1D1A1B07049126).

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# Retractive spaces and Bousfield-Kan completions

Zeshen Gu  and  John E. Harper  Department of Mathematics, The Ohio State
University, 231 West 18th Ave, Columbus, OH 43210, USA<EMAIL_ADDRESS>Department of Mathematics, The Ohio State University, Newark, 1179 University
Dr, Newark, OH 43055, USA<EMAIL_ADDRESS>
###### Abstract.

In this short paper we apply some recent techniques developed by Schonsheck,
and subsequently Carr-Harper, in the context of operadic algebras in
spectra—on convergence of Bousfield-Kan completions and comparisons with
convergence of the Taylor tower of the identity functor in Goodwillie’s
functor calculus—to the setting of retractive spaces: this arises when working
with spaces centered away from the one-point space. Interestingly, in the
retractive spaces context, the comparison results are stronger in terms of
convergence outside of functor calculus’ notion of “radius of (strong)
convergence” for analytic functors. In particular, we give a new proof (and
generalization to retractive spaces) of the Arone-Kankaanrinta result for
convergence of the Taylor tower of the identity functor to various Bousfield-
Kan completions; it’s notable that no use is made of Snaith splittings—rather,
we make extensive use of the kinds of homotopical estimates that appear in
earlier work of Dundas and Dundas-Goodwillie-McCarthy.

##  1\. Introduction

This paper is written simplicially so that “space” means “simplicial set”
unless otherwise noted; see [7, 17]. In particular, we refer to the category
of pointed simplicial sets ${\mathsf{S}_{*}}$ as pointed spaces; this is
equipped with the usual homotopy theory ([7, 17]). Our basic assumption is
that $Z$ is a 0-connected fibrant pointed space. Denote by
${\mathsf{S}_{*}^{Z}}{\ \cong\
}{\mathrm{id}}_{Z}\downarrow({\mathsf{S}_{*}}\downarrow Z)$ the factorization
category ([2, 2.1], [20, 4.9]) of the identity map on $Z$, called the category
of retractive pointed spaces over $Z$, equipped with the homotopy theory ([2,
2.1], [20, 4.9]) inherited from ${\mathsf{S}_{*}}$; in particular, it has the
structure of a simplicial cofibrantly generated model structure ([21], [29])
with an action of ${\mathsf{S}_{*}}$ ([22, 4.2]). The setting of retractive
spaces naturally arises in Goodwillie’s homotopy functor calculus [18, 19]
when working with Taylor towers centered away from the one-point space; see
also [24]. When working with Bousfield-Kan completions, we make extensive use
of the kinds of homotopical resolutions studied in [5]. We say that a
retractive space $X$ over $Z$ is _$k$ -connected relative to $Z$_ (or
$k$-connected (rel. $Z$)) if the structure map $Z\rightarrow X$ is
$k$-connected.

Here are our main results. In the special case when $Z=*$ (the one-point
space), Theorem 1.1 is proved by Bousfield and Hopkins [6] (for $r\geq 1$) and
Carlsson [8] (for $r=\infty$) for any 0-connected nilpotent space $X$, and
subsequently in [4] for any 1-connected space $X$ (using different arguments
closely related to [3, 11, 13, 14]). Our result in Theorem 1.2, generalizes
this to any 0-connected (rel. $Z$) retractive space $F$ over $Z$, provided
that, furthermore, $F$ fits into an appropriate homotopy pullback square. Our
technical approach is motivated by the work in [26], and the subsequent
development in [9], for operadic algebras in spectra (where the estimates are
different). We make extensive use of (the retractive version of) the
homotopical estimates worked out in [4]; these are the kinds of homotopical
estimates that appear in earlier work of Dundas [13] and Dundas-Goodwillie-
McCarthy [14].

In the following theorems, ${\tilde{\Sigma}}_{Z}^{r}$ (resp.
${\tilde{\Omega}}_{Z}^{r}$) denotes the derived $r$-fold suspension ([16, 25])
(resp. derived $r$-fold loops ([16, 25])) in ${\mathsf{S}_{*}^{Z}}$, and
${\tilde{\Sigma}}_{Z}^{\infty}$ (resp. ${\tilde{\Omega}}_{Z}^{\infty}$)
denotes derived stabilization ([23]) on ${\mathsf{S}_{*}^{Z}}$ (resp. derived
0-th object functor ([23]) on Hovey spectra
${\mathsf{Sp}^{\mathbb{N}}}({\mathsf{S}_{*}^{Z}})$ on ${\mathsf{S}_{*}^{Z}}$).

######  Theorem 1.1.

Assume that $Z$ is a 0-connected pointed space. Let $X$ be a retractive
pointed space over $Z$. If $X$ is $1$-connected (rel. $Z$), then the
coaugmentations

$\displaystyle X{\ \simeq\
}X^{\wedge}_{{\tilde{\Omega}}_{Z}^{r}{\tilde{\Sigma}}_{Z}^{r}},\quad\quad(1\leq
r\leq\infty)$

are weak equivalences in retractive pointed spaces over $Z$.

######  Theorem 1.2.

Assume that $Z$ is a 0-connected pointed space. If $F\rightarrow X\rightarrow
Y$ is a fibration sequence in retractive pointed spaces over $Z$ and $X,Y$ are
1-connected (rel. $Z$), then the coaugmentations

$\displaystyle F{\ \simeq\
}F^{\wedge}_{{\tilde{\Omega}}_{Z}^{r}{\tilde{\Sigma}}_{Z}^{r}},\quad\quad(1\leq
r\leq\infty)$

are weak equivalences in retractive pointed spaces over $Z$. More generally,
let

be a homotopy pullback square in retractive pointed spaces over $Z$. If
$A,X,Y$ are 1-connected (rel. $Z$), then the coaugmentations

$\displaystyle F{\ \simeq\
}F^{\wedge}_{{\tilde{\Omega}}_{Z}^{r}{\tilde{\Sigma}}_{Z}^{r}},\quad\quad(1\leq
r\leq\infty)$

are weak equivalences in retractive pointed spaces over $Z$.

In the special case when $Z=*$, Theorem 1.3 is proved in Arone-Kankaanrinta
[1] for $r=\infty$ (using closely related, but different, arguments). We
generalize their result to spaces centered away from $*$ and for $1\leq
r\leq\infty$. Our technical approach is motivated by the work in [28], and the
subsequent development in [9], for operadic algebras in spectra (where the
estimates are different). It’s notable that no use is made of Snaith
splittings—rather (as above) we make extensive use of (the retractive version
of) the homotopical estimates worked out in [4], which are similar in spirit
to the kinds of homotopical estimates appearing in the earlier work of Dundas
[13] and Dundas-Goodwillie-McCarthy [14]; the possibility of giving a proof of
Theorem 1.3 (when $Z=*$) along the lines developed here, was suggested in
[27].

######  Theorem 1.3.

Assume that $Z$ is a 0-connected pointed space. Let $X$ be a retractive
pointed space over $Z$. If $X$ is 0-connected (rel. $Z$), then there are weak
equivalences of the form

$\displaystyle P_{\infty}^{Z}({\mathrm{id}})X{\ \simeq\
}X^{\wedge}_{{\tilde{\Omega}}_{Z}^{r}{\tilde{\Sigma}}_{Z}^{r}},\quad\quad(1\leq
r\leq\infty)$

in retractive pointed spaces over $Z$; here, $P_{n}^{Z}({\mathrm{id}})X$ is
the $n$-excisive approximation to the identity functor ${\mathrm{id}}$ on
retractive pointed spaces over $Z$, evaluated at $X$, and
$P_{\infty}^{Z}({\mathrm{id}})X$ denotes the homotopy limit of the associated
Taylor tower $\\{P_{n}^{Z}({\mathrm{id}})X\\}$ of the identity functor
${\mathrm{id}}$, evaluated at $X$, in Goodwillie’s functor calculus [19].

To keep this paper appropriately concise, we will freely use language from
[4].

##  2\. Proofs of the main results

To get Bousfield-Kan completion into the picture, we work with the kinds of
homotopical resolutions studied in [5]. There are adjunctions of the form

$\displaystyle\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
7.17778pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathsf{S}_{*}^{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
12.69214pt\raise 8.65833pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.13889pt\hbox{$\scriptstyle{{\Sigma}^{r}_{Z}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 31.17778pt\raise
2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
31.17778pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathsf{S}_{*}^{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
12.69215pt\raise-8.65833pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.13889pt\hbox{$\scriptstyle{{\Omega}^{r}_{Z}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 7.1778pt\raise-2.15277pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces\quad\quad\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
7.17778pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathsf{S}_{*}^{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
11.65001pt\raise 8.65833pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.13889pt\hbox{$\scriptstyle{{\Sigma}^{\infty}_{Z}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 31.17778pt\raise
2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
31.17778pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathsf{Sp}^{\mathbb{N}}}({\mathsf{S}_{*}^{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
11.65002pt\raise-8.65833pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.13889pt\hbox{$\scriptstyle{{\Omega}^{\infty}_{Z}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 7.1778pt\raise-2.15277pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces\quad\quad(r\geq
1)$

with left adjoints on top, where ${\Sigma}_{Z}^{r}$ is given by the pointed
spaces action of $S^{r}:=(S^{1})^{\wedge r}\in{\mathsf{S}_{*}}$ on objects in
${\mathsf{S}_{*}^{Z}}$ and ${\mathsf{Sp}^{\mathbb{N}}}({\mathsf{S}_{*}^{Z}})$
denotes Hovey spectra ([23]) on ${\mathsf{S}_{*}^{Z}}$; here,
${\Sigma}_{Z}^{\infty}$ (resp. ${\Omega}_{Z}^{\infty}$) denotes the
stabilization (resp. “0-th object”) functor. Denote by
${\mathrm{id}}\rightarrow\Phi$ and $\Phi\Phi\rightarrow\Phi$ the unit and
multiplication maps of the fibrant replacement monad $\Phi$ on
${\mathsf{S}_{*}^{Z}}$ (see [5, 6.1]) and define
${\tilde{\Omega}}^{r}_{Z}:={\Omega}^{r}_{Z}\Phi$. Similarly, denote by
${\mathrm{id}}\rightarrow F$ and $FF\rightarrow F$ the unit and multiplication
maps of the fibrant replacement monad $F$ on
${\mathsf{Sp}^{\mathbb{N}}}({\mathsf{S}_{*}^{Z}})$ (see [5, 6.1]) and define
${\tilde{\Omega}}_{Z}^{\infty}:={\Omega}^{\infty}_{Z}F$. Since every object in
${\mathsf{S}_{*}^{Z}}$ is cofibrant, ${\Sigma}^{r}$ is already derived and we
define ${\tilde{\Sigma}}^{r}:={\Sigma}^{r}$. If we iterate the comparison map
${\mathrm{id}}\rightarrow{\tilde{\Omega}}^{r}_{Z}{\tilde{\Sigma}}^{r}_{Z}$ it
follows that we can build a cosimplicial resolution of ${\mathrm{id}}$ with
respect to ${\tilde{\Omega}}^{r}_{Z}{\tilde{\Sigma}}^{r}_{Z}$ of the form

(3)

for each $1\leq r\leq\infty$; these are the types of homotopical resolutions
studied in [5]; see also [4, 9, 11]. Here, we are only showing the coface
maps. If $X\in{\mathsf{S}_{*}^{Z}}$, the Bousfield-Kan completion of $X$ with
respect to ${\tilde{\Omega}}^{r}_{Z}{\tilde{\Sigma}}^{r}_{Z}$ is the homotopy
limit

(4) $\displaystyle
X^{\wedge}_{{\tilde{\Omega}}^{r}_{Z}{\tilde{\Sigma}}^{r}_{Z}}:=\operatorname*{holim}\nolimits_{\Delta}({\tilde{\Omega}}^{r}_{Z}{\tilde{\Sigma}}^{r}_{Z})^{\bullet+1}(X)$

of the Bousfield-Kan cosimplicial resolution (3) evaluated at $X$. To obtain
the Bousfield-Kan completion tower, we filter $\Delta$ ([4, 5.22]) by its
subcategories $\Delta^{\leq n}\subset\Delta$, $n\geq 0$, and define

$\displaystyle({\tilde{\Omega}}^{r}_{Z}{\tilde{\Sigma}}^{r}_{Z})_{n}:=\operatorname*{holim}\nolimits_{\Delta^{\leq
n}}({\tilde{\Omega}}^{r}_{Z}{\tilde{\Sigma}}^{r}_{Z})^{\bullet+1},\quad\quad
n\geq 0$

to obtain the ${\tilde{\Omega}}^{r}_{Z}{\tilde{\Sigma}}^{r}_{Z}$-completion of
$X$

(5) $\displaystyle
X^{\wedge}_{{\tilde{\Omega}}^{r}_{Z}{\tilde{\Sigma}}^{r}_{Z}}{\ \simeq\
}\operatorname*{holim}\Bigl{(}({\tilde{\Omega}}^{r}_{Z}{\tilde{\Sigma}}^{r}_{Z})_{0}(X)\leftarrow({\tilde{\Omega}}^{r}_{Z}{\tilde{\Sigma}}^{r}_{Z})_{1}(X)\leftarrow({\tilde{\Omega}}^{r}_{Z}{\tilde{\Sigma}}^{r}_{Z})_{2}(X)\leftarrow\cdots\Bigr{)}$

as the homotopy limit of the completion tower, where

$\displaystyle({\tilde{\Omega}}^{r}_{Z}{\tilde{\Sigma}}^{r}_{Z})_{0}(X){\
\simeq\ }({\tilde{\Omega}}^{r}_{Z}{\tilde{\Sigma}}^{r}_{Z})(X)$

For conceptual simplicity and convenience we denote by $*_{Z}:=Z$ the null
object in ${\mathsf{S}_{*}^{Z}}$. It will be useful to denote by
$*^{\prime}_{Z}{\ \simeq\ }*_{Z}$ an appropriately fattened-up version of the
null object $*_{Z}$ in ${\mathsf{S}_{*}^{Z}}$.

######  Proposition 2.1.

Let $k\geq 0$ and $1\leq r\leq\infty$. Let $X$ be a retractive pointed space
over $Z$. If $X$ is $k$-connected (rel. $Z$), then the comparison map
$X\rightarrow{\tilde{\Omega}}_{Z}^{r}{\tilde{\Sigma}}_{Z}^{r}X$ is
$(2k+1)$-connected.

###### Proof.

Consider the case $r=1$. Consider a pushout cofibration 2-cube of the form

in ${\mathsf{S}_{*}^{Z}}$. By assumption we know that the upper and left-hand
1-faces are $(k+1)$-connected. Since the 2-cube is $\infty$-cocartesian, it
follows that the lower and right-hand 1-faces are $(k+1)$-connected. By higher
Blakers-Massey [18, 2.5] for ${\mathsf{S}_{*}}$, we know that the 2-cube is
$l$-cartesian where $l$ is the minimum of

$\displaystyle 1-2+l_{\\{1,2\\}}$ $\displaystyle=-1+\infty$ $\displaystyle
1-2+l_{\\{1\\}}+l_{\\{2\\}}$ $\displaystyle=-1+(k+1)+(k+1)$

Hence $l=2k+1$, the 2-cube is $(2k+1)$-cartesian, and the comparison map
$X\rightarrow{\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z}X$ is $(2k+1)$-connected.
The other cases ($r\geq 2$) follow by repeated application of the $r=1$ case
to
${\tilde{\Sigma}}_{Z}X,{\tilde{\Sigma}}^{2}_{Z}X,{\tilde{\Sigma}}^{3}_{Z}X,\dots$
in the usual way, and finally, for $r=\infty$ by considering the homotopy
colimit of the resulting sequence. ∎

For $k\geq 1$, this pattern persists for the iterative application of
${\mathrm{id}}\rightarrow{\tilde{\Omega}}_{Z}^{r}{\tilde{\Sigma}}_{Z}^{r}$ to
go from 0-cubes to 1-cubes to 2-cubes to 3-cubes, and so forth.

######  Proposition 2.2.

Let $k\geq 1$ and $1\leq r\leq\infty$. Let $W$ be a finite set and
${\mathcal{X}}$ a $W$-cube in ${\mathsf{S}_{*}^{Z}}$. Let $n=|W|$. If the
$n$-cube ${\mathcal{X}}$ is $(k({\mathrm{id}}+1)+1)$-cartesian in
${\mathsf{S}_{*}^{Z}}$, then so is the $(n+1)$-cube of the form
${\mathcal{X}}\rightarrow{\tilde{\Omega}}_{Z}^{r}{\tilde{\Sigma}}_{Z}^{r}{\mathcal{X}}$.

###### Proof.

These estimates are proved in [4, 1.7, 1.8] for the special case of $Z=*$
using higher Blakers-Massey (and its dual) [18, 2.5, 2.6], together with ideas
closely related to [11, 13, 14]; and similar to the proof of Proposition 2.1,
exactly the same arguments (and estimates) remain true in the more general
context of pointed spaces centered at $Z$; see, for instance, [9] where the
analogous passage to the retractive setting is demonstrated in detail for
operadic algebras in spectra. ∎

###### Proof of Theorem 1.1.

To verify that $X{\ \simeq\
}X^{\wedge}_{{\tilde{\Omega}}^{r}_{Z}{\tilde{\Sigma}}^{r}_{Z}}$, it suffices
to verify that the map of the form
$X\rightarrow({\tilde{\Omega}}^{r}_{Z}{\tilde{\Sigma}}^{r}_{Z})_{n}(X)$ into
the $n$-th stage of the Bousfield-Kan completion tower has connectivity
strictly increasing with $n$. The connectivity of this map is the same as the
cartesian-ness of the coface $(n+1)$-cube ([3, 3.13]) of the coaugmented
Bousfield-Kan cosimplicial resolution which we calculated (Propositions 2.1
and 2.2) to be $(((n+1)+1)+1)=n+3$, which completes the proof. ∎

######  Proposition 2.3.

Let $n\geq 1$. Let $X$ be a retractive pointed space over $Z$. If $X$ is
0-connected (rel. $Z$), then the maps

$\displaystyle({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{k}(X)\xrightarrow{(*)_{n}}P_{n}^{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{k}(X),\quad\quad
k\geq 1$

are $(n+1)$-connected.

###### Proof.

Consider the case of $n=1$. The 1-excisive approximation
$P_{1}^{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{k}(X)$ to the functor
$({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{k}$ on retractive pointed spaces
over $Z$, evaluated at $X$, is the homotopy colimit of

$\displaystyle({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{k}(X)\xrightarrow{(\\#)_{1}}T_{1}^{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{k}(X)\rightarrow
T_{1}^{Z}(T_{1}^{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{k})(X)\rightarrow\cdots$

the indicated sequence ([19, Section 1]); we want to estimate the
connectivities of these maps. It follows, by iteratively applying higher
Blakers-Massey (and its dual) [18, 2.5, 2.6] for ${\mathsf{S}_{*}}$ that the
maps $(\\#)_{1}$ are 2-connected, and the other maps are higher connected. In
more detail: here is the basic idea for the maps $(\\#)_{1}$; estimates for
the other (more highly connected) maps are similar. Consider the
$\infty$-cocartesian 2-cube ${\mathcal{X}}$ of the form

in ${\mathsf{S}_{*}^{Z}}$. Since $X$ is 0-connected (rel. $Z$), we know that
$*_{Z}\rightarrow X$ is 0-connected and hence $X\rightarrow*_{Z}$ is
1-connected ([18, 1.5]); therefore we know that ${\mathcal{X}}$ satisfies: the
1-subcubes are 1-connected and the 2-cube is $\infty$-cocartesian. Then
${\tilde{\Sigma}}_{Z}{\mathcal{X}}$ satisfies: the 1-subcubes are 2-connected
and the 2-subcubes (there is only one) are $\infty$-cocartesian. By higher
Blakers-Massey [18, 2.5] for ${\mathsf{S}_{*}}$, we know the 2-cube is
$k$-cartesian where $k$ is the minimum of

$\displaystyle 1-2+k_{\\{1,2\\}}$ $\displaystyle=-1+\infty$ $\displaystyle
1-2+k_{\\{1\\}}+k_{\\{2\\}}$ $\displaystyle=-1+2+2$

Hence $k=3$, our 2-cube is 3-cartesian, and
${\tilde{\Sigma}}_{Z}{\mathcal{X}}$ satisfies: the 1-subcubes are 2-connected
and the 2-subcubes are 3-cartesian. Then
$({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z}){\mathcal{X}}$ satisfies: the
1-subcubes are 1-connected and the 2-subcubes are 2-cartesian. Hence we have
verifed that the map

$\displaystyle({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})(X)\xrightarrow{(\\#)_{1}}T_{1}^{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})(X)$

is 2-connected. Let’s keep going. By higher dual Blakers-Massey [18, 2.6] for
${\mathsf{S}_{*}}$, we know the 2-cube is $k$-cocartesian where $k$ is the
minimum of

$\displaystyle 2-1+k_{\\{1,2\\}}$ $\displaystyle=1+2$ $\displaystyle
2-1+k_{\\{1\\}}+k_{\\{2\\}}$ $\displaystyle=1+1+1$

Hence $k=3$, our 2-cube is 3-cocartesian, and
$({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z}){\mathcal{X}}$ satisfies: the
1-subcubes are 1-connected and the 2-subcubes are 3-cocartesian. Then
${\tilde{\Sigma}}_{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z}){\mathcal{X}}$
satisfies: the 1-subcubes are 2-connected and the 2-subcubes are
4-cocartesian. By higher Blakers-Massey [18, 2.5] for ${\mathsf{S}_{*}}$, we
know the 2-cube is $k$-cartesian where $k$ is the minimum of

$\displaystyle 1-2+k_{\\{1,2\\}}$ $\displaystyle=-1+4$ $\displaystyle
1-2+k_{\\{1\\}}+k_{\\{2\\}}$ $\displaystyle=-1+2+2$

Hence, $k=3$, our 2-cube is 3-cartesian, and
${\tilde{\Sigma}}_{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z}){\mathcal{X}}$
satisfies: the 1-subcubes are 2-connected and the 2-subcubes are 3-cartesian.
Then $({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{2}{\mathcal{X}}$ satisfies:
the 1-subcubes are 1-connected and the 2-subcubes are 2-cartesian. Hence we
have verifed that the map

$\displaystyle({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{2}(X)\xrightarrow{(\\#)_{1}}T_{1}^{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{2}(X)$

is 2-connected. Let’s keep going. By higher dual Blakers-Massey [18, 2.6] for
${\mathsf{S}_{*}}$, we know the 2-cube is $k$-cocartesian where $k$ is the
minimum of

$\displaystyle 2-1+k_{\\{1,2\\}}$ $\displaystyle=1+2$ $\displaystyle
2-1+k_{\\{1\\}}+k_{\\{2\\}}$ $\displaystyle=1+1+1$

Hence $k=3$, our 2-cube is 3-cocartesian, and
$({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{2}{\mathcal{X}}$ satisfies: the
1-subcubes are 1-connected and the 2-subcubes are 3-cocartesian. Then
${\tilde{\Sigma}}_{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{2}{\mathcal{X}}$
satisfies: the 1-subcubes are 2-connected and the 2-subcubes are
4-cocartesian. By higher Blakers-Massey [18, 2.5] for ${\mathsf{S}_{*}}$, we
know the 2-cube is $k$-cartesian where $k$ is the minimum of

$\displaystyle 1-2+k_{\\{1,2\\}}$ $\displaystyle=-1+4$ $\displaystyle
1-2+k_{\\{1\\}}+k_{\\{2\\}}$ $\displaystyle=-1+2+2$

Hence, $k=3$, our 2-cube is 3-cartesian, and
${\tilde{\Sigma}}_{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{2}{\mathcal{X}}$
satisfies: the 1-subcubes are 2-connected and the 2-subcubes are 3-cartesian.
Then $({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{3}{\mathcal{X}}$ satisfies:
the 1-subcubes are 1-connected and the 2-subcubes are 2-cartesian. Hence we
have verifed that the map

$\displaystyle({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{3}(X)\xrightarrow{(\\#)_{1}}T_{1}^{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{3}(X)$

is 2-connected; notice how the subcube estimates have stabilized at each
respective step. And so forth. Hence it follows that the maps

$\displaystyle({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{k}(X)\xrightarrow{(\\#)_{1}}T_{1}^{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{k}(X),\quad\quad
k\geq 1$

are 2-connected. Consider the case of $n=2$. The 2-excisive approximation
$P_{2}^{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{k}(X)$ to the functor
$({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{k}$ on retractive pointed spaces
over $Z$, evaluated at $X$, is the homotopy colimit of

$\displaystyle({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{k}(X)\xrightarrow{(\\#)_{2}}T_{2}^{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{k}(X)\rightarrow
T_{2}^{Z}(T_{2}^{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{k})(X)\rightarrow\cdots$

the indicated sequence ([19, Section 1]); we want to estimate the
connectivities of these maps. It follows, by iteratively applying higher
Blakers-Massey (and its dual) [18, 2.5, 2.6] for ${\mathsf{S}_{*}}$ that the
maps $(\\#)_{2}$ are 3-connected, and the other maps are higher connected. In
more detail: here is the basic idea for the maps $(\\#)_{2}$; estimates for
the other (more highly connected) maps are similar. Consider a strongly
$\infty$-cocartesian 3-cube ${\mathcal{X}}$ satisfying: the 1-subcubes are
1-connected, the 2-subcubes are $\infty$-cocartesian, and the 3-subcubes
(there is only one) are $\infty$-cocartesian. Then
${\tilde{\Sigma}}_{Z}{\mathcal{X}}$ satisfies: the 1-subcubes are 2-connected,
the 2-subcubes are $\infty$-cocartesian, and the 3-subcubes are
$\infty$-cocartesian. By higher Blakers-Massey [18, 2.5] for
${\mathsf{S}_{*}}$, we know the 3-cube is $k$-cartesian where $k$ is the
minimum of

$\displaystyle 1-3+k_{\\{1,2,3\\}}$ $\displaystyle=-2+\infty$ $\displaystyle
1-3+k_{\\{1,2\\}}+k_{\\{3\\}}$ $\displaystyle=-2+\infty+2$ $\displaystyle
1-3+k_{\\{1\\}}+k_{\\{2\\}}+k_{\\{3\\}}$ $\displaystyle=-2+2+2+2$

Hence $k=4$, our 3-cube is 4-cartesian, and
${\tilde{\Sigma}}_{Z}{\mathcal{X}}$ satisfies: the 1-subcubes are 2-connected,
the 2-subcubes are 3-cartesian, and the 3-subcubes are 4-cartesian. Then
$({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z}){\mathcal{X}}$ satisfies: the
1-subcubes are 1-connected, the 2-subcubes are 2-cartesian, and the 3-subcubes
are 3-cartesian. Hence we have verifed that the map

$\displaystyle({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})(X)\xrightarrow{(\\#)_{2}}T_{2}^{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})(X)$

is 3-connected. Let’s keep going. By higher dual Blakers-Massey [18, 2.6] for
${\mathsf{S}_{*}}$, we know the 3-cube is $k$-cocartesian where $k$ is the
minimum of

$\displaystyle 3-1+k_{\\{1,2,3\\}}$ $\displaystyle=2+3$ $\displaystyle
3-1+k_{\\{1,2\\}}+k_{\\{3\\}}$ $\displaystyle=2+2+1$ $\displaystyle
3-1+k_{\\{1\\}}+k_{\\{2\\}}+k_{\\{3\\}}$ $\displaystyle=2+1+1+1$

Hence $k=5$, our 3-cube is 5-cocartesian, and
$({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z}){\mathcal{X}}$ satisfies: the
1-subcubes are 1-connected, the 2-subcubes are 3-cocartesian, and the
3-subcubes are 5-cocartesian. Then
${\tilde{\Sigma}}_{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z}){\mathcal{X}}$
satisfies: the 1-subcubes are 2-connected, the 2-subcubes are 4-cocartesian,
and the 3-subcubes are 6-cocartesian. By higher Blakers-Massey [18, 2.5] for
${\mathsf{S}_{*}}$, we know the 3-cube is $k$-cartesian where $k$ is the
minimum of

$\displaystyle 1-3+k_{\\{1,2,3\\}}$ $\displaystyle=-2+6$ $\displaystyle
1-3+k_{\\{1,2\\}}+k_{\\{3\\}}$ $\displaystyle=-2+4+2$ $\displaystyle
1-3+k_{\\{1\\}}+k_{\\{2\\}}+k_{\\{3\\}}$ $\displaystyle=-2+2+2+2$

Hence, $k=4$, our 3-cube is 4-cartesian, and
${\tilde{\Sigma}}_{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z}){\mathcal{X}}$
satisfies: the 1-subcubes are 2-connected, the 2-subcubes are 3-cartesian, and
the 3-subcubes are 4-cartesian. Then
$({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{2}{\mathcal{X}}$ satisfies: the
1-subcubes are 1-connected, the 2-subcubes are 2-cartesian, and the 3-subcubes
are 3-cartesian. Hence we have verifed that the map

$\displaystyle({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{2}(X)\xrightarrow{(\\#)_{2}}T_{2}^{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{2}(X)$

is 3-connected. Let’s keep going. By higher dual Blakers-Massey [18, 2.6] for
${\mathsf{S}_{*}}$, we know the 3-cube is $k$-cocartesian where $k$ is the
minimum of

$\displaystyle 3-1+k_{\\{1,2,3\\}}$ $\displaystyle=2+3$ $\displaystyle
3-1+k_{\\{1,2\\}}+k_{\\{3\\}}$ $\displaystyle=2+2+1$ $\displaystyle
3-1+k_{\\{1\\}}+k_{\\{2\\}}+k_{\\{3\\}}$ $\displaystyle=2+1+1+1$

Hence $k=5$, our 3-cube is 5-cocartesian, and
$({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{2}{\mathcal{X}}$ satisfies: the
1-subcubes are 1-connected, the 2-subcubes are 3-cocartesian, and the
3-subcubes are 5-cocartesian. Then
${\tilde{\Sigma}}_{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{2}{\mathcal{X}}$
satisfies: the 1-subcubes are 2-connected, the 2-subcubes are 4-cocartesian,
and the 3-subcubes are 6-cocartesian. By higher Blakers-Massey [18, 2.5] for
${\mathsf{S}_{*}}$, we know the 3-cube is $k$-cartesian where $k$ is the
minimum of

$\displaystyle 1-3+k_{\\{1,2,3\\}}$ $\displaystyle=-2+6$ $\displaystyle
1-3+k_{\\{1,2\\}}+k_{\\{3\\}}$ $\displaystyle=-2+4+2$ $\displaystyle
1-3+k_{\\{1\\}}+k_{\\{2\\}}+k_{\\{3\\}}$ $\displaystyle=-2+2+2+2$

Hence, $k=4$, our 3-cube is 4-cartesian, and
${\tilde{\Sigma}}_{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{2}{\mathcal{X}}$
satisfies: the 1-subcubes are 2-connected, the 2-subcubes are 3-cartesian, and
the 3-subcubes are 4-cartesian. Then
$({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{3}{\mathcal{X}}$ satisfies: the
1-subcubes are 1-connected, the 2-subcubes are 2-cartesian, and the 3-subcubes
are 3-cartesian. Hence we have verifed that the map

$\displaystyle({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{3}(X)\xrightarrow{(\\#)_{2}}T_{2}^{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{3}(X)$

is 3-connected; notice how the subcube estimates have stabilized at each
respective step. And so forth. Hence it follows that the maps

$\displaystyle({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{k}(X)\xrightarrow{(\\#)_{2}}T_{2}^{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{k}(X),\quad\quad
k\geq 1$

are 3-connected. And so forth. ∎

######  Proposition 2.4.

Let $n\geq 1$ and $1\leq r\leq\infty$. Let $X$ be a retractive pointed space
over $Z$. If $X$ is 0-connected (rel. $Z$), then the maps

$\displaystyle({\tilde{\Omega}}^{r}_{Z}{\tilde{\Sigma}}^{r}_{Z})^{k}(X)\xrightarrow{(*)_{n}}P_{n}^{Z}({\tilde{\Omega}}^{r}_{Z}{\tilde{\Sigma}}^{r}_{Z})^{k}(X),\quad\quad
k\geq 1$

are $(n+1)$-connected.

###### Proof.

A detailed proof of the $r=1$ case is given above (Proposition 2.3), and the
other cases are similar. In the case of $r=\infty$, several of the steps are
easier since ${\tilde{\Sigma}}_{Z}^{\infty}$ preserves cocartesian-ness,
${\tilde{\Omega}}_{Z}^{\infty}$ preserves cartesian-ness, and the stable
estimates in [10, 3.10] are available for each estimate step following the
application of ${\tilde{\Sigma}}_{Z}^{\infty}$. ∎

######  Proposition 2.5.

Let $n\geq 1$. Let $X$ be a retractive pointed space over $Z$. If $X$ is
0-connected (rel. $Z$), then the maps

$\displaystyle({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})_{k}(X)\xrightarrow{(**)_{n}}P_{n+k}^{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})_{k}(X),\quad\quad
k\geq 0$

are $(n+1)$-connected.

###### Proof.

Consider the case of $k=0$. Then the map $(**)_{n}$ is $(n+1)$-connected by
Proposition 2.4. Consider the case of $k=1$. By definition,
$({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})_{1}X$ fits into an
$\infty$-cartesian 2-cube of the form ([4, 5.26])

and therefore the map $(**)_{n}$ fits into a 3-cube of the form

(14)

Several applications of [18, 1.6] show that the map $(**)_{n}$ is
$(n+1)$-connected. In more detail: the back 2-face is $\infty$-cartesian,
hence the front 2-face is $\infty$-cartesian ([19, 1.7]). Therefore, the
3-cube is $\infty$-cartesian. By Proposition 2.4, the maps $(*)_{n+1}$ are
$(n+2)$-connected, hence the right-hand 2-face is $(n+1)$-cartesian. Since the
3-cube is $\infty$-cartesian, we therefore know the left-hand 2-face is
$(n+1)$-cartesian; and hence, since the map $(*)_{n+1}$ is $(n+2)$-connected,
therefore we know the map $(**)_{n}$ is $(n+1)$-connected. Consider the case
of $k=2$. By definition, $({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})_{2}X$
fits into an $\infty$-cartesian 3-cube ${\mathcal{X}}$ of the form

and therefore the map $(**)_{n}$ fits into a 4-cube of the form
${\mathcal{X}}\rightarrow P_{n+2}^{Z}{\mathcal{X}}$. Several applications of
[18, 1.6] show that the map $(**)_{n}$ is $(n+1)$-connected. In more detail:
${\mathcal{X}}$ is $\infty$-cartesian, hence $P_{n+2}^{Z}{\mathcal{X}}$ is
$\infty$-cartesian ([19, 1.7]). Therefore, the 4-cube is $\infty$-cartesian.
Consider the 3-face of the form

By Proposition 2.4, the maps $(*)_{n+2}$ are $(n+3)$-connected, hence the top
and bottom 2-faces are $(n+2)$-cartesian, and therefore the 3-face is
$(n+1)$-cartesian. Since the 4-cube is $\infty$-cartesian, we therefore know
the opposite 3-face of the form

is $(n+1)$-cartesian; and hence, since the maps $(*)_{n+2}$ are
$(n+3)$-connected, we know the bottom 2-face is $(n+2)$-cartesian, it follows
that the top face is $(n+1)$-cartesian, and since the map $(*)_{n+2}$ is
$(n+3)$-connected, therefore we know the map $(**)_{n}$ is $(n+1)$-connected.
The other cases similarly follow by repeated applications of [18, 1.6]. ∎

######  Proposition 2.6.

Let $n\geq 1$ and $1\leq r\leq\infty$. Let $X$ be a retractive pointed space
over $Z$. If $X$ is 0-connected (rel. $Z$), then the maps

$\displaystyle({\tilde{\Omega}}_{Z}^{r}{\tilde{\Sigma}}_{Z}^{r})_{k}(X)\xrightarrow{(**)_{n}}P_{n+k}^{Z}({\tilde{\Omega}}_{Z}^{r}{\tilde{\Sigma}}_{Z}^{r})_{k}(X),\quad\quad
k\geq 0$

are $(n+1)$-connected.

###### Proof.

A detailed proof of the $r=1$ case is given above (Proposition 2.5), and the
other cases are similar; the estimates are identical (Proposition 2.4). ∎

###### Proof of Theorem 1.3.

We follow the basic proof strategy in [28], and the subsequent development in
[9], for operadic algebras in spectra (where the estimates are different).
Here is the basic idea. Consider the case of $r=1$. We start with the
Bousfield-Kan completion tower of the form

and resolve each term by its Taylor tower to produce the tower of towers
diagram

(21)

By our uniformity estimates (Propositions 2.1 and 2.2), it follows immediately
that ${\mathrm{id}}\rightarrow({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})_{n}$
satisfies $O_{n+1}$ ([19, 1.2]) for each $n\geq 0$; in other words, via this
map the functors ${\mathrm{id}}$ and
$({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})_{n}$ agree to order $n+1$ and
hence the maps

$\displaystyle P_{m}^{Z}({\mathrm{id}})(X)$ $\displaystyle\xrightarrow{{\
\simeq\
}}P_{m}^{Z}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})_{n}(X),\quad\quad 1\leq
m\leq n+1$

are weak equivalences for every $n\geq 0$. It follows that
$\operatorname*{holim}$ applied horizontally produces the Taylor tower of the
identity functor $\\{P_{n}^{Z}({\mathrm{id}})\\}$ and hence applying
$\operatorname*{holim}$ first horizontally and then vertically produces

$\displaystyle\operatorname*{holim}_{\mathsf{vert}}\operatorname*{holim}_{\mathsf{horiz}}\eqref{eq:tower_of_towers_exotic}{\
\simeq\ }P_{\infty}^{Z}({\mathrm{id}})(X)$

What about the other way? By our estimates (Proposition 2.6), it follows
immediately that $\operatorname*{holim}$ applied vertically produces the
Bousfield-Kan completion tower

and hence applying $\operatorname*{holim}$ first vertically and then
horizontally produces

$\displaystyle\operatorname*{holim}_{\mathsf{horiz}}\operatorname*{holim}_{\mathsf{vert}}\eqref{eq:tower_of_towers_exotic}{\
\simeq\ }X^{\wedge}_{{\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z}}$

Hence we have verified that

$\displaystyle P_{\infty}^{Z}({\mathrm{id}})(X){\ \simeq\
}X^{\wedge}_{{\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z}}$

The other cases are similar; the estimates are identical (Proposition 2.6). ∎

######  Proposition 2.7.

If $n\geq-1$, then $({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})$ preserves
$({\mathrm{id}}+1)$-cartesian $(n+1)$-cubes in ${\mathsf{S}_{*}^{Z}}$.

###### Proof.

The cases for $n=-1,0$ are trivial. Now that we know the desired behavior is
satisfied on 0-subcubes, we will not continue to indicate their estimates
below when verifying the $({\mathrm{id}}+1)$-cartesian property. Consider the
case of $n=1$. Assume that ${\mathcal{X}}$ is an $({\mathrm{id}}+1)$-cartesian
2-cube in ${\mathsf{S}_{*}^{Z}}$. Then ${\mathcal{X}}$ satisfies: the
1-subcubes are 2-connected, and the 2-subcubes (there is only one) are
3-cartesian. By higher dual Blakers-Massey [18, 2.6] for ${\mathsf{S}_{*}}$,
we know the 2-cube is $k$-cocartesian where $k$ is the minimum of

$\displaystyle 2-1+k_{\\{1,2\\}}$ $\displaystyle=1+3$ $\displaystyle
2-1+k_{\\{1\\}}+k_{\\{2\\}}$ $\displaystyle=1+2+2$

Hence $k=4$, our 3-cube is 4-cocartesian, and ${\mathcal{X}}$ satisfies: the
1-subcubes are 2-connected, and the 2-subcubes are 4-cocartesian. Then
${\tilde{\Sigma}}_{Z}{\mathcal{X}}$ satisfies: the 1-subcubes are 3-connected,
and the 2-subcubes are 5-cocartesian. By higher Blakers-Massey [18, 2.5] for
${\mathsf{S}_{*}}$, we know the 2-cube is $k$-cartesian where $k$ is the
minimum of

$\displaystyle 1-2+k_{\\{1,2\\}}$ $\displaystyle=-1+5$ $\displaystyle
1-2+k_{\\{1\\}}+k_{\\{2\\}}$ $\displaystyle=-1+3+3$

Hence, $k=4$, our 2-cube is 4-cartesian, and
${\tilde{\Sigma}}_{Z}{\mathcal{X}}$ satisfies: the 1-subcubes are 3-connected,
and the 2-subcubes are 4-cartesian. Then
${\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z}{\mathcal{X}}$ satisfies: the
1-subcubes are 2-connected, and the 2-subcubes are 3-cartesian. Hence we have
verified that $({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z}){\mathcal{X}}$ is
$({\mathrm{id}}+1)$-cartesian in ${\mathsf{S}_{*}^{Z}}$. Consider the case of
$n=2$. Assume that ${\mathcal{X}}$ is an $({\mathrm{id}}+1)$-cartesian 3-cube
in ${\mathsf{S}_{*}^{Z}}$. Then ${\mathcal{X}}$ satisfies: the 1-subcubes are
2-connected, the 2-subcubes are 3-cartesian, and the 3-subcubes (there is only
one) are 4-cartesian. By higher dual Blakers-Massey [18, 2.5] for
${\mathsf{S}_{*}}$, we know the 3-cube is $k$-cocartesian where $k$ is the
minimum of

$\displaystyle 3-1+k_{\\{1,2,3\\}}$ $\displaystyle=2+4$ $\displaystyle
3-1+k_{\\{1,2\\}}+k_{\\{3\\}}$ $\displaystyle=2+3+2$ $\displaystyle
3-1+k_{\\{1\\}}+k_{\\{2\\}}+k_{\\{3\\}}$ $\displaystyle=2+2+2+2$

Hence $k=6$, our 3-cube is 6-cocartesian, and ${\mathcal{X}}$ satisfies: the
1-subcubes are 2-connected, the 2-subcubes are 4-cocartesian, and the
3-subcubes are 6-cocartesian. Then ${\tilde{\Sigma}}_{Z}{\mathcal{X}}$
satisfies: the 1-subcubes are 3-connected, the 2-subcubes are 5-cocartesian,
and the 3-subcubes are 7-cocartesian. By higher Blakers-Massey [18, 2.5] for
${\mathsf{S}_{*}}$, we know the 3-cube is $k$-cartesian where $k$ is the
minimum of

$\displaystyle 1-3+k_{\\{1,2,3\\}}$ $\displaystyle=-2+7$ $\displaystyle
1-3+k_{\\{1,2\\}}+k_{\\{3\\}}$ $\displaystyle=-2+5+3$ $\displaystyle
1-3+k_{\\{1\\}}+k_{\\{2\\}}+k_{\\{3\\}}$ $\displaystyle=-2+3+3+3$

Hence, $k=5$, our 3-cube is 5-cartesian, and
${\tilde{\Sigma}}_{Z}{\mathcal{X}}$ satisfies: the 1-subcubes are 3-connected,
the 2-subcubes are 4-cartesian, and the 3-subcubes are 5-cartesian. Then
$({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z}){\mathcal{X}}$ satisfies: the
1-subcubes are 2-connected, the 2-subcubes are 3-cartesian, and the 3-subcubes
are 4-cartesian. Hence we have verified that
$({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z}){\mathcal{X}}$ is
$({\mathrm{id}}+1)$-cartesian in ${\mathsf{S}_{*}^{Z}}$. And so forth. ∎

######  Proposition 2.8.

Let $1\leq r\leq\infty$. If $n\geq-1$, then
$({\tilde{\Omega}}^{r}_{Z}{\tilde{\Sigma}}^{r}_{Z})$ preserves
$({\mathrm{id}}+1)$-cartesian $(n+1)$-cubes in ${\mathsf{S}_{*}^{Z}}$.

###### Proof.

A detailed proof of the $r=1$ case is given above (Proposition 2.7), and the
other cases are similar. In the case of $r=\infty$, several of the steps are
easier since ${\tilde{\Sigma}}_{Z}^{\infty}$ preserves cocartesian-ness,
${\tilde{\Omega}}_{Z}^{\infty}$ preserves cartesian-ness, and the stable
estimates in [10, 3.10] are available for each estimate step following the
application of ${\tilde{\Sigma}}_{Z}^{\infty}$. ∎

###### Proof of Theorem 1.2.

We follow the basic proof strategy in [26], and the subsequent development in
[9], for operadic algebras in spectra (where the estimates are different).
Here is the basic idea. Consider the case $r=1$. Start with the Bousfield-Kan
cosimplicial resolution

(24)

of the identity functor and consider the fibration sequence $F\rightarrow
E\rightarrow B$ in ${\mathsf{S}_{*}^{Z}}$. Since we know that $E,B$ are
1-connected (rel. $Z$) by assumption, this means that we have the homotopical
estimates in Propositions 2.1 and 2.2 available. With this in mind, let’s
resolve $E,B$ with respect to the Bousfield-Kan cosimplicial resolution

and take homotopy fibers vertically to define the coaugmented cosimplicial
diagram of the form $F\rightarrow\tilde{F}$. By construction the columns are
homotopy fiber sequences in ${\mathsf{S}_{*}^{Z}}$, and since $E,B$ are
1-connected (rel $Z$), we know from Theorem 1.1 that

$\displaystyle E$ $\displaystyle\xrightarrow{{\ \simeq\
}}\operatorname*{holim}\nolimits_{\Delta}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{\bullet+1}(E)$
$\displaystyle B$ $\displaystyle\xrightarrow{{\ \simeq\
}}\operatorname*{holim}\nolimits_{\Delta}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{\bullet+1}(B)$

Since homotopy limits commute with homotopy fibers, it follows that

$\displaystyle F{\ \simeq\ }\operatorname*{holim}\nolimits_{\Delta}\tilde{F}$

We want to show that $F{\ \simeq\
}F^{\wedge}_{{\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z}}$. To get Bousfield-Kan
completion into the picture, let’s resolve each term in
$F\rightarrow\tilde{F}$ with respect to (24) to obtain the cosimplicial
resolution of coaugmented cosimplicial diagrams of the form

(33)

We know by our homotopical estimates (Propositions 2.1 and 2.2) that the
coface $(n+1)$-cubes ([3, 3.13]) associated to

$\displaystyle E$
$\displaystyle\rightarrow({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{\bullet+1}(E)$
$\displaystyle B$
$\displaystyle\rightarrow({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{\bullet+1}(B)$

are $(({\mathrm{id}}+1)+1)$-cartesian in ${\mathsf{S}_{*}^{Z}}$, and hence it
follows by several applications of [18, 1.6, 1.18] that the coface
$(n+1)$-cubes associated to $F\rightarrow\tilde{F}$ are
$({\mathrm{id}}+1)$-cartesian in ${\mathsf{S}_{*}^{Z}}$. We know, by
Proposition 2.8, that $({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})$ preserves
$({\mathrm{id}}+1)$-cartesian $(n+1)$-cubes in ${\mathsf{S}_{*}^{Z}}$ for each
$n\geq-1$. Therefore, the coface $(n+1)$-cubes ([3, 3.13]) associated to

$\displaystyle({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{k}F\rightarrow({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{k}\tilde{F},\quad\quad
k\geq 0$

are $({\mathrm{id}}+1)$-cartesian in ${\mathsf{S}_{*}^{Z}}$ for each
$n\geq-1$, and hence each of the maps

$\displaystyle({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{k}F\rightarrow\operatorname*{holim}\nolimits_{\Delta^{\leq
n}}({\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z})^{k}\tilde{F},\quad\quad k\geq 0$

is $(n+2)$-connected. Therefore applying $\operatorname*{holim}_{\Delta}$
horizontally to the maps $(\\#)$ induces a weak equivalence, and hence
applying $\operatorname*{holim}_{\Delta}$ first horizontally and then
vertically produces

$\displaystyle\operatorname*{holim}_{\mathsf{vert}}\operatorname*{holim}_{\mathsf{horiz}}\eqref{eq:nice_array_of_stuff}{\
\simeq\ }F^{\wedge}_{{\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z}}$

What about the other way? Since the $(**)$ columns have extra codegeneracy
maps $s^{-1}$ [15, 6.2] (by formal reasons: ${\tilde{\Omega}}_{Z}$ commutes
with homotopy fibers), applying $\operatorname*{holim}_{\Delta}$ vertically
produces [12, 3.16] the coaugmented cosimplicial diagram

and hence applying $\operatorname*{holim}_{\Delta}$ first vertically and then
horizontally produces

$\displaystyle\operatorname*{holim}_{\mathsf{horiz}}\operatorname*{holim}_{\mathsf{vert}}\eqref{eq:nice_array_of_stuff}{\
\simeq\ }F$

Hence we have verified that the coaugmentation

$\displaystyle F{\ \simeq\
}F^{\wedge}_{{\tilde{\Omega}}_{Z}{\tilde{\Sigma}}_{Z}}$

is a weak equivalence. The other cases are similar (the estimates are
identical). Consider the case of the homotopy pullback square; then
$F\rightarrow\tilde{F}$ is constructed by taking homotopy pullbacks instead of
homotopy fibers and the above arguments complete the proof. ∎

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