#  A stochastic diffusion process for the Dirichlet distribution
LA-UR-12-26980
Accepted in International Journal of Stochastic Analysis, March 1, 2013

J. Bakosi  J.R. Ristorcelli
<EMAIL_ADDRESS>
Los Alamos National Laboratory  Los Alamos  NM 87545  USA

###### Abstract

The method of potential solutions of Fokker-Planck equations is used to
develop a transport equation for the joint probability of $N$ coupled
stochastic variables with the Dirichlet distribution as its asymptotic
solution. To ensure a bounded sample space, a coupled nonlinear diffusion
process is required: the Wiener-processes in the equivalent system of
stochastic differential equations are multiplicative with coefficients
dependent on all the stochastic variables. Individual samples of a discrete
ensemble, obtained from the stochastic process, satisfy a unit-sum constraint
at all times. The process may be used to represent realizations of a
fluctuating ensemble of $N$ variables subject to a conservation principle.
Similar to the multivariate Wright-Fisher process, whose invariant is also
Dirichlet, the univariate case yields a process whose invariant is the beta
distribution. As a test of the results, Monte-Carlo simulations are used to
evolve numerical ensembles toward the invariant Dirichlet distribution.

###### keywords:

Fokker-Planck equation; Stochastic diffusion; Dirichlet distribution; Monte-
Carlo simulation

##  1 Objective

We develop a Fokker-Planck equation whose statistically stationary solution is
the Dirichlet distribution [1, 2, 3]. The system of stochastic differential
equations (SDE), equivalent to the Fokker-Planck equation, yields a Markov
process that allows a Monte-Carlo method to numerically evolve an ensemble of
fluctuating variables that satisfy a unit-sum requirement. A Monte Carlo
solution is used to verify that the invariant distribution is Dirichlet.

The Dirichlet distribution is a statistical representation of non-negative
variables subject to a unit-sum requirement. The properties of such variables
have been of interest in a variety of fields, including evolutionary theory
[4], Bayesian statistics [5], geology [6, 7], forensics [8], econometrics [9],
turbulent combustion [10], and population biology [11].

##  2 Preview of results

The Dirichlet distribution [1, 2, 3] for a set of scalars
$0\\!\leq\\!Y_{\alpha}$, $\alpha\\!=\\!1,\dots,N-1$,
$\sum_{\alpha=1}^{N-1}Y_{\alpha}\\!\leq\\!1$, is given by

$\mathscr{D}({\mbox{\boldmath$\mathbf{Y}$}},{\mbox{\boldmath$\mathbf{\omega}$}})=\frac{\Gamma\left(\sum_{\alpha=1}^{N}\omega_{\alpha}\right)}{\prod_{\alpha=1}^{N}\Gamma(\omega_{\alpha})}\prod_{\alpha=1}^{N}Y_{\alpha}^{\omega_{\alpha}-1},$
(1)

where $\omega_{\alpha}\\!>\\!0$ are parameters,
$Y_{N}\\!=\\!1-\sum_{\beta=1}^{N-1}Y_{\beta}$, and $\Gamma(\cdot)$ denotes the
gamma function. We derive the stochastic diffusion process, governing the
scalars, $Y_{\alpha}$,

$\mathrm{d}Y_{\alpha}(t)=\frac{b_{\alpha}}{2}\big{[}S_{\alpha}Y_{N}-(1-S_{\alpha})Y_{\alpha}\big{]}\mathrm{d}t+\sqrt{\kappa_{\alpha}Y_{\alpha}Y_{N}}\mathrm{d}W_{\alpha}(t),\qquad\alpha=1,\dots,N-1,$
(2)

where $\mathrm{d}W_{\alpha}(t)$ is an isotropic vector-valued Wiener process
[12], and $b_{\alpha}\\!>\\!0$, $\kappa_{\alpha}\\!>\\!0$, and
$0\\!<\\!S_{\alpha}\\!<\\!1$ are coefficients. We show that the statistically
stationary solution of Eq. (2) is the Dirichlet distribution, Eq. (1),
provided the SDE coefficients satisfy

$\frac{b_{1}}{\kappa_{1}}(1-S_{1})=\dots=\frac{b_{N-1}}{\kappa_{N-1}}(1-S_{N-1}).$
(3)

The restrictions imposed on the SDE coefficients, $b_{\alpha}$,
$\kappa_{\alpha}$, and $S_{\alpha}$, ensure reflection towards the interior of
the sample space, which is a generalized triangle or tetrahedron (more
precisely, a simplex) in $N\\!-\\!1$ dimensions. The restrictions together
with the specification of the $N^{\mathrm{th}}$ scalar as
$Y_{N}\\!=\\!1-\sum_{\beta=1}^{N-1}Y_{\beta}$ ensure

$\sum_{\alpha=1}^{N}Y_{\alpha}=1.$ (4)

Indeed, inspection of Eq. (2) shows that for example when $Y_{1}\\!=\\!0$, the
diffusion is zero and the drift is strictly positive, while if
$Y_{1}\\!=\\!1$, the diffusion is zero ($Y_{N}\\!=\\!0$) and the drift is
strictly negative.

##  3 Development of the diffusion process

The diffusion process (2) is developed by the method of potential solutions.

We start from the Itô diffusion process [12] for the stochastic vector,
$Y_{\alpha}$,

$\mathrm{d}Y_{\alpha}(t)=a_{\alpha}({\mbox{\boldmath$\mathbf{Y}$}})\mathrm{d}t+b_{\alpha\beta}({\mbox{\boldmath$\mathbf{Y}$}})\mathrm{d}W_{\beta}(t),\qquad\alpha,\beta=1,\dots,N-1,$
(5)

with drift, $a_{\alpha}({\mbox{\boldmath$\mathbf{Y}$}})$, diffusion,
$b_{\alpha\beta}({\mbox{\boldmath$\mathbf{Y}$}})$, and the isotropic vector-
valued Wiener process, $\mathrm{d}W_{\beta}(t)$, where summation is implied
for repeated indices. Using standard methods given in [12] the equivalent
Fokker-Planck equation governing the joint probability,
$\mathscr{F}({\mbox{\boldmath$\mathbf{Y}$}},t)$, derived from Eq. (5), is

$\frac{\partial\mathscr{F}}{\partial t}=-\frac{\partial}{\partial
Y_{\alpha}}\big{[}a_{\alpha}({\mbox{\boldmath$\mathbf{Y}$}})\mathscr{F}\big{]}+\frac{1}{2}\frac{\partial^{2}}{\partial
Y_{\alpha}\partial
Y_{\beta}}\big{[}B_{\alpha\beta}({\mbox{\boldmath$\mathbf{Y}$}})\mathscr{F}\big{]},$
(6)

with diffusion $B_{\alpha\beta}\\!=\\!b_{\alpha\gamma}b_{\gamma\beta}$. Since
the drift and diffusion coefficients are time-homogeneous,
$a_{\alpha}({\mbox{\boldmath$\mathbf{Y}$}},t)\\!=\\!a_{\alpha}({\mbox{\boldmath$\mathbf{Y}$}})$
and
$B_{\alpha\beta}({\mbox{\boldmath$\mathbf{Y}$}},t)\\!=\\!B_{\alpha\beta}({\mbox{\boldmath$\mathbf{Y}$}})$,
Eq. (5) is a statistically stationary process and the solution of Eq. (6)
converges to a stationary distribution, [12] Sec. 6.2.2. Our task is to
specify the functional forms of $a_{\alpha}({\mbox{\boldmath$\mathbf{Y}$}})$
and $b_{\alpha\beta}({\mbox{\boldmath$\mathbf{Y}$}})$ so that the stationary
solution of Eq. (6) is $\mathscr{D}({\mbox{\boldmath$\mathbf{Y}$}})$, defined
by Eq. (1).

A potential solution of Eq. (6) exists if

$\frac{\partial\ln\mathscr{F}}{\partial
Y_{\beta}}=B_{\alpha\beta}^{-1}\left(2a_{\alpha}-\frac{\partial
B_{\alpha\gamma}}{\partial
Y_{\gamma}}\right)\equiv-\frac{\partial\phi}{\partial
Y_{\beta}},\qquad\alpha,\beta,\gamma=1,\dots,N-1,$ (7)

is satisfied, [12] Sec. 6.2.2. Since the left hand side of Eq. (7) is a
gradient, the expression on the right must also be a gradient and can
therefore be obtained from a scalar potential denoted by
$\phi({\mbox{\boldmath$\mathbf{Y}$}})$. This puts a constraint on the possible
choices of $a_{\alpha}$ and $B_{\alpha\beta}$ and on the potential, as
$\phi,_{\alpha\beta}=\phi,_{\beta\alpha}$ must also be satisfied. The
potential solution is

$\mathscr{F}({\mbox{\boldmath$\mathbf{Y}$}})=\exp[-\phi({\mbox{\boldmath$\mathbf{Y}$}})].$
(8)

Now functional forms of $a_{\alpha}({\mbox{\boldmath$\mathbf{Y}$}})$ and
$B_{\alpha\beta}({\mbox{\boldmath$\mathbf{Y}$}})$ that satisfy Eq. (7) with
$\mathscr{F}({\mbox{\boldmath$\mathbf{Y}$}})\equiv\mathscr{D}({\mbox{\boldmath$\mathbf{Y}$}})$
are sought. The mathematical constraints on the specification of $a_{\alpha}$
and $B_{\alpha\beta}$ are as follows:

$B_{\alpha\beta}$ must be symmetric positive semi-definite. This is to ensure
that

the square-root of $B_{\alpha\beta}$ (e.g. the Cholesky-decomposition,
$b_{\alpha\beta}$) exists, required by the correspondence of the SDE (5) and
the Fokker-Planck equation (6),

Eq. (5) represents a diffusion, and

$\det(B_{\alpha\beta})\neq 0$, required by the existence of the inverse in Eq.
(7).

For a potential solution to exist Eq. (7) must be satisfied. With
$\mathscr{F}({\mbox{\boldmath$\mathbf{Y}$}})\equiv\mathscr{D}({\mbox{\boldmath$\mathbf{Y}$}})$
Eq. (8) shows that the scalar potential must be

$\phi({\mbox{\boldmath$\mathbf{Y}$}})=-\sum_{\alpha=1}^{N}(\omega_{\alpha}-1)\ln
Y_{\alpha}.$ (9)

It is straightforward to verify that the specifications

$\displaystyle a_{\alpha}({\mbox{\boldmath$\mathbf{Y}$}})$
$\displaystyle=\frac{b_{\alpha}}{2}\big{[}S_{\alpha}Y_{N}-(1-S_{\alpha})Y_{\alpha}\big{]},$
(10) $\displaystyle B_{\alpha\beta}({\mbox{\boldmath$\mathbf{Y}$}})$
$\displaystyle=\left\\{\begin{array}[]{lr}\kappa_{\alpha}Y_{\alpha}Y_{N}&\quad\mathrm{for}\quad\alpha=\beta,\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
0&\quad\mathrm{for}\quad\alpha\neq\beta,\end{array}\right.$ (13)

satisfy the above mathematical constraints, 1. and 2. Here
$b_{\alpha}\\!>\\!0$, $\kappa_{\alpha}\\!>\\!0$, $0\\!<\\!S_{\alpha}\\!<\\!1$,
and $Y_{N}\\!=\\!1-\sum_{\beta=1}^{N-1}Y_{\beta}$. Summation is not implied
for Eqs. (9–13).

Substituting Eqs. (9–13) into Eq. (7) yields a system with the same functions
on both sides with different coefficients, yielding the correspondence between
the $N$ coefficients of the Dirichlet distribution, Eq. (1), and the Fokker-
Planck equation (6) with Eqs. (10–13) as

$\displaystyle\omega_{\alpha}$
$\displaystyle=\frac{b_{\alpha}}{\kappa_{\alpha}}S_{\alpha},\qquad\alpha=1,\dots,N-1,$
(14) $\displaystyle\omega_{N}$
$\displaystyle=\frac{b_{1}}{\kappa_{1}}(1-S_{1})=\dots=\frac{b_{N-1}}{\kappa_{N-1}}(1-S_{N-1}).$
(15)

For example, for $N\\!=\\!3$ one has
${\mbox{\boldmath$\mathbf{Y}$}}=(Y_{1},Y_{2},Y_{3}=1-Y_{1}-Y_{2})$ and from
Eq. (9) the scalar potential is

$-\phi(Y_{1},Y_{2})=(\omega_{1}-1)\ln Y_{1}+(\omega_{2}-1)\ln
Y_{2}+(\omega_{3}-1)\ln(1-Y_{1}-Y_{2}).$ (16)

Eq. (7) then becomes the system

$\displaystyle\frac{\omega_{1}-1}{Y_{1}}-\frac{\omega_{3}-1}{Y_{3}}$
$\displaystyle=\left(\frac{b_{1}}{\kappa_{1}}S_{1}-1\right)\frac{1}{Y_{1}}-\left[\frac{b_{1}}{\kappa_{1}}(1-S_{1})-1\right]\frac{1}{Y_{3}},$
(17) $\displaystyle\frac{\omega_{2}-1}{Y_{2}}-\frac{\omega_{3}-1}{Y_{3}}$
$\displaystyle=\left(\frac{b_{2}}{\kappa_{2}}S_{2}-1\right)\frac{1}{Y_{2}}-\left[\frac{b_{2}}{\kappa_{2}}(1-S_{2})-1\right]\frac{1}{Y_{3}},$
(18)

which shows that by specifying the parameters, $\omega_{\alpha}$, of the
Dirichlet distribution as

$\displaystyle\omega_{1}$ $\displaystyle=\frac{b_{1}}{\kappa_{1}}S_{1},$ (19)
$\displaystyle\omega_{2}$ $\displaystyle=\frac{b_{2}}{\kappa_{2}}S_{2},$ (20)
$\displaystyle\omega_{3}$
$\displaystyle=\frac{b_{1}}{\kappa_{1}}(1-S_{1})=\frac{b_{2}}{\kappa_{2}}(1-S_{2}),$
(21)

the stationary solution of the Fokker-Planck equation (6) with drift (10) and
diffusion (13) is
$\mathscr{D}({\mbox{\boldmath$\mathbf{Y}$}},{\mbox{\boldmath$\mathbf{\omega}$}})$
for $N\\!=\\!3$. The above development generalizes to $N$ variables, yielding
Eqs. (14–15), and reduces to the beta distribution, a univariate
specialization of $\mathscr{D}$ for $N\\!=\\!2$, where $Y_{1}\\!=\\!Y$ and
$Y_{2}\\!=\\!1-Y$, see [13].

If Eqs. (14–15) hold, the stationary solution of the Fokker-Planck equation
(6) with drift (10) and diffusion (13) is the Dirichlet distribution, Eq. (1).
Note that Eqs. (10–13) are one possible way of specifying a drift and a
diffusion to arrive at a Dirichlet distribution; other functional forms may be
possible. The specifications in Eqs. (10–13) are a generalization of the
results for a univariate diffusion process, discussed in [13, 14], whose
invariant distribution is beta.

The shape of the Dirichlet distribution, Eq. (1), is determined by the $N$
coefficients, $\omega_{\alpha}$. Eqs. (14–15) show that in the stochastic
system, different combinations of $b_{\alpha}$, $S_{\alpha}$, and
$\kappa_{\alpha}$ may yield the same $\omega_{\alpha}$ and that not all of
$b_{\alpha}$, $S_{\alpha}$, and $\kappa_{\alpha}$ may be chosen independently
to keep the invariant Dirichlet.

##  4 Corroborating that the invariant distribution is Dirichlet

For any multivariate Fokker-Planck equation there is an equivalent system of
Itô diffusion processes, such as the pair of Eqs. (5–6) [12]. Therefore, a way
of computing the (discrete) numerical solution of Eq. (6) is to integrate Eq.
(5) in a Monte-Carlo fashion for an ensemble [15]. Using a Monte-Carlo
simulation we show that the statistically stationary solution of the Fokker-
Planck equation (6) with drift and diffusion (10–13) is a Dirichlet
distribution, Eq. (1).

The time-evolution of an ensemble of particles, each with $N=3$ variables
($Y_{1},Y_{2},Y_{3}$), is numerically computed by integrating the system of
equations (5), with drift and diffusion (10–13), for $N=3$ as

$\displaystyle\qquad\mathrm{d}Y^{(i)}_{1}=\frac{b_{1}}{2}\left[S_{1}Y^{(i)}_{3}-(1-S_{1})Y^{(i)}_{1}\right]\mathrm{d}t+\sqrt{\kappa_{1}Y^{(i)}_{1}Y^{(i)}_{3}}\mathrm{d}W^{(i)}_{1}$
(22)
$\displaystyle\qquad\mathrm{d}Y^{(i)}_{2}=\frac{b_{2}}{2}\left[S_{2}Y^{(i)}_{3}-(1-S_{2})Y^{(i)}_{2}\right]\mathrm{d}t+\sqrt{\kappa_{2}Y^{(i)}_{2}Y^{(i)}_{3}}\mathrm{d}W^{(i)}_{2}$
(23) $\displaystyle\enskip\qquad Y^{(i)}_{3}=1-Y^{(i)}_{1}-Y^{(i)}_{2}$ (24)

for each particle $i$. In Eqs. (22–23) $\mathrm{d}W_{1}$ and $\mathrm{d}W_{2}$
are independent Wiener processes, sampled from Gaussian streams of random
numbers with mean ${\langle{\mathrm{d}W_{\alpha}}\rangle}\\!=\\!0$ and
covariance
${\langle{\mathrm{d}W_{\alpha}\mathrm{d}W_{\beta}}\rangle}\\!=\\!\delta_{\alpha\beta}\mathrm{d}t$.
$400,\\!000$ particle-triplets, $(Y_{1},Y_{2},Y_{3})$, are generated with two
different initial distributions, displayed in the upper-left of Figures 1 and
2, a triple-delta and a box, respectively. Each member of both initial
ensembles satisfy $\sum_{\alpha=1}^{3}Y_{\alpha}\\!=\\!1$. Eqs. (22–24) are
advanced in time with the Euler-Maruyama scheme [16] with time step $\Delta
t\\!=\\!0.05$. Table 1 shows the coefficients of the stochastic system
(22–24), the corresponding parameters of the final Dirichlet distribution, and
the first two moments at the initial times for the triple-delta initial
condition case. The final state of the ensembles are determined by the SDE
coefficients, constant for these exercises, also given in Table 1, the same
for both simulations, satisfying Eq. (21).

The time-evolutions of the joint probabilities are extracted from both
calculations and displayed at different times in Figures 1 and 2. At the end
of the simulations two distributions are plotted at the bottom-right of both
figures: the one extracted from the numerical ensemble and the Dirichlet
distribution determined analytically using the SDE coefficients – in excellent
agreement in both figures. The statistically stationary solution of the
developed stochastic system is the Dirichlet distribution.

For a more quantitative evaluation, the time evolution of the first two
moments,
$\mu_{\alpha}\\!=\\!{\langle{Y_{\alpha}}\rangle}\\!=\\!\int_{0}^{1}\int_{0}^{1}Y_{\alpha}\mathscr{F}(Y_{1},Y_{2})\mathrm{d}Y_{1}\mathrm{d}Y_{2}$,
and
${\langle{y_{\alpha}y_{\beta}}\rangle}\\!=\\!{\langle{(Y_{\alpha}\\!-\\!{\langle{Y_{\alpha}}\rangle})(Y_{\beta}\\!-\\!{\langle{Y_{\beta}}\rangle})}\rangle}$,
are also extracted from the numerical simulation with the triple-delta-peak
initial condition as ensemble averages and displayed in Figures 3 and 4. The
figures show that the statistics converge to the precise state given by the
Dirichlet distribution that is prescribed by the SDE coefficients, see Table
1.

The solution approaches a Dirichlet distribution, with non-positive
covariances [2], in the statistically stationary limit, Figure 4(b). Note that
during the evolution of the process, $0\\!<\\!t\\!\lesssim\\!80$, the solution
is not necessarily Dirichlet, but the stochastic variables sum to one at all
times. The point ($Y_{1}$, $Y_{2}$), governed by Eqs. (22–23), can never leave
the $(N\\!-\\!1)$-dimensional (here $N\\!=\\!3$) convex polytope and by
definition $Y_{3}\\!=\\!1-Y_{1}-Y_{2}$. The rate at which the numerical
solution converges to a Dirichlet distribution is determined by the vectors
$b_{\alpha}$ and $\kappa_{\alpha}$.

The above numerical results confirm that starting from arbitrary realizable
ensembles the solution of the stochastic system converges to a Dirichlet
distribution in the statistically stationary state, specified by the SDE
coefficients.

##  5 Relation to other diffusion processes

It is useful to relate the Dirichlet diffusion process, Eq. (2), to other
multivariate stochastic diffusion processes with linear drift and quadratic
diffusion.

A close relative of Eq. (2) is the multivariate Wright-Fisher (WF) process
[11], used extensively in population and genetic biology,

$\mathrm{d}Y_{\alpha}(t)=\frac{1}{2}(\omega_{\alpha}-\omega
Y_{\alpha})\mathrm{d}t+\sum_{\beta=1}^{N-1}\sqrt{Y_{\alpha}(\delta_{\alpha\beta}-Y_{\beta})}\mathrm{d}W_{\alpha\beta}(t),\qquad\alpha=1,\dots,N-1,$
(25)

where $\delta_{\alpha\beta}$ is Kronecker’s delta,
$\omega\\!=\\!\sum_{\beta=1}^{N}\omega_{\beta}$ with $\omega_{\alpha}$ defined
in Eq. (1) and, $Y_{N}\\!=\\!1\\!-\\!\sum_{\beta=1}^{N-1}Y_{\beta}$. Similarly
to Eq. (2), the statistically stationary solution of Eq. (25) is the Dirichlet
distribution [17]. It is straightforward to verify that its drift and
diffusion also satisfy Eq. (7) with $\mathscr{F}\equiv\mathscr{D}$, i.e. WF is
a process whose invariant is Dirichlet and this solution is potential. A
notable difference between Eqs. (2) and (25), other than the coefficients, is
that the diffusion matrix of the Dirichlet diffusion process is diagonal,
while that of the WF process it is full.

Another process similar to Eqs. (2) and (25) is the multivariate Jacobi
process, used in econometrics,

$\mathrm{d}Y_{\alpha}(t)=a(Y_{\alpha}-\pi_{\alpha})\mathrm{d}t+\sqrt{cY_{\alpha}}\mathrm{d}W_{\alpha}(t)-\sum_{\beta=1}^{N-1}Y_{\alpha}\sqrt{cY_{\beta}}\mathrm{d}W_{\beta}(t),\qquad\alpha=1,\dots,N$
(26)

of Gourieroux & Jasiak [9] with $a<0$, $c>0$, $\pi_{\alpha}>0$, and
$\sum_{\beta=1}^{N}\pi_{\beta}=1$.

In the univariate case the Dirichlet, WF, and Jacobi diffusions reduce to

$\mathrm{d}Y(t)=\frac{b}{2}(S-Y)\mathrm{d}t+\sqrt{\kappa
Y(1-Y)}\mathrm{d}W(t),$ (27)

see also [13], whose invariant is the beta distribution, which belongs to the
family of Pearson diffusions, discussed in detail by Forman & Sorensen [14].

##  6 Summary

The method of potential solutions of Fokker-Planck equations has been used to
derive a transport equation for the joint distribution of $N$ fluctuating
variables. The equivalent stochastic process, governing the set of random
variables, $0\\!\leq\\!Y_{\alpha}$, $\alpha\\!=\\!1,\dots,N-1$,
$\sum_{\alpha=1}^{N-1}Y_{\alpha}\\!\leq\\!1$, reads

$\mathrm{d}Y_{\alpha}(t)=\frac{b_{\alpha}}{2}\big{[}S_{\alpha}Y_{N}-(1-S_{\alpha})Y_{\alpha}\big{]}\mathrm{d}t+\sqrt{\kappa_{\alpha}Y_{\alpha}Y_{N}}\mathrm{d}W_{\alpha}(t),\qquad\alpha=1,\dots,N-1,$
(28)

where $Y_{N}\\!=\\!1\\!-\\!\sum_{\beta=1}^{N-1}Y_{\beta}$, and $b_{\alpha}$,
$\kappa_{\alpha}$ and $S_{\alpha}$ are parameters, while
$\mathrm{d}W_{\alpha}(t)$ is an isotropic Wiener process with independent
increments. Restricting the coefficients to $b_{\alpha}\\!>\\!0$,
$\kappa_{\alpha}\\!>\\!0$ and $0\\!<\\!S_{\alpha}\\!<\\!1$, and defining
$Y_{N}$ as above ensure $\sum_{\alpha=1}^{N}Y_{\alpha}\\!=\\!1$ and that
individual realizations of ($Y_{1},Y_{2},\dots,Y_{N})$ are confined to the
($N\\!-\\!1$)-dimensional convex polytope of the sample space. Eq. (28) can
therefore be used to numerically evolve the joint distribution of $N$
fluctuating variables required to satisfy a conservation principle. Eq. (28)
is a coupled system of nonlinear stochastic differential equations whose
statistically stationary solution is the Dirichlet distribution, Eq. (1),
provided the coefficients satisfy

$\frac{b_{1}}{\kappa_{1}}(1-S_{1})=\dots=\frac{b_{N-1}}{\kappa_{N-1}}(1-S_{N-1}).$
(29)

In stochastic modeling, one typically begins with a physical problem, perhaps
discrete, _then_ derives the stochastic differential equations whose solution
yields a distribution. In this paper we reversed the process: we assumed a
desired stationary distribution and derived the stochastic differential
equations that converge to the assumed distribution. A potential solution form
of the Fokker-Planck equation was posited, from which we obtained the
stochastic differential equations for the diffusion process whose
statistically stationary solution is the Dirichlet distribution. We have also
made connections to other stochastic processes, such as the Wright-Fisher
diffusions of population biology and the Jacobi diffusions in econometrics,
whose invariant distributions possess similar properties but whose stochastic
differential equations are different.

## Acknowledgements

It is a pleasure to acknowledge a series of informative discussions with J.
Waltz. This work was performed under the auspices of the U.S. Department of
Energy under the Advanced Simulation and Computing Program.

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$Y_{1}$$Y_{1}$$t=10$$t=0$$t=20$00.250.50.75100.250.50.75100.250.50.751$Y_{1}$

$Y_{2}$

$Y_{2}$

$Y_{2}$

00.250.50.75100.250.50.75100.250.50.751$Y_{1}$$t=140$00.250.50.751

$Y_{2}$

00.250.50.751$\omega_{1}=5$$\omega_{2}=2$$\omega_{3}=3$$S_{1}=5/8$$S_{2}=2/5$$b_{1}=1/10$$b_{2}=3/2$$\kappa_{1}=1/80$$\kappa_{2}=3/10$

Figure 1: Time evolution of the joint probability, $\mathscr{F}(Y_{1},Y_{2})$,
extracted from the numerical solution of Eqs. (22–24). The initial condition
is a triple-delta distribution, with unequal peaks at the three corners of the
sample space. At the end of the simulation, $t=140$, the solid lines are that
of the distribution extracted from the numerical ensemble, dashed lines are
that of a Dirichlet distribution to which the solution converges in the
statistically stationary state, implied by the constant SDE coefficients,
sampled at the same heights.

$\omega_{1}=5$$\omega_{2}=2$$\omega_{3}=3$$S_{1}=5/8$$S_{2}=2/5$$b_{1}=1/10$$b_{2}=3/2$$\kappa_{1}=1/80$$\kappa_{2}=3/10$$t=0$$t=5$0.250.50.75100.250.50.7510.250.50.75100.250.50.7510.750.25$t=160$0.250.50.751$t=1$0.250.50.75100.510.250.50.7501$Y_{1}$

$Y_{2}$

$Y_{2}$

$Y_{2}$

$Y_{2}$

$Y_{1}$$Y_{1}$$Y_{1}$

Figure 2: Time evolution of the joint probability, $\mathscr{F}(Y_{1},Y_{2})$,
extracted from the numerical solution of Eqs. (22–24). The top-left panel
shows the initial condition: a box with diffused sides. By $t=160$, bottom-
right panel, the distribution converges to the same Dirichlet distribution as
in Figure 1.

02040608010012014000.10.20.30.40.5${\langle{Y_{1}}\rangle}$${\langle{Y_{3}}\rangle}$${\langle{Y_{2}}\rangle}$time

means

Figure 3: Time evolution of the means extracted from the numerically
integrated system of Eqs. (22–24) starting from the triple-delta initial
condition. Dotted-solid lines – numerical solution, dashed lines – statistics
of the Dirichlet distribution determined analytically using the constant
coefficients of the SDE, see Table 1.

-0.12-0.1-0.08-0.06-0.04-0.02000.020.040.060.080.10.120.14020406080100120140020406080100120140$1/44$$21/1100$$4/275$$-3/550$$-1/110$$-3/220$0(a)(b)timetime$\langle{y_{3}^{2}}\rangle$$\langle{y_{1}^{2}}\rangle$$\langle{y_{2}^{2}}\rangle$${\langle{y_{2}y_{3}}\rangle}$${\langle{y_{1}y_{3}}\rangle}$$\langle{y_{1}^{2}}\rangle$$\langle{y_{2}^{2}}\rangle$801001201400.030.020.01-0.01-0.0280100120140${\langle{y_{2}y_{3}}\rangle}$${\langle{y_{1}y_{2}}\rangle}$${\langle{y_{1}y_{3}}\rangle}$$\langle{y_{3}^{2}}\rangle$${\langle{y_{1}y_{2}}\rangle}$

covariances

variances

Figure 4: Time evolution of the second central moments extracted from the numerically integrated system of Eqs. (22–24) starting from the triple-delta initial condition. The legend is the same as in Figure 3. Table 1: Initial and final states of the Monte-Carlo simulation starting from a triple-delta. The coefficients, $b_{1}$, $b_{2}$, $S_{1}$, $S_{2}$, $\kappa_{1}$, $\kappa_{2}$, of the system of SDEs (22–24) determine the distribution to which the system converges. The Dirichlet parameters, implied by the SDE coefficients via Eqs. (19-21), are in brackets. The corresponding statistics are determined by the well-known formulae of Dirichlet distributions [2]. Initial state: triple-delta, see Figure 1 |  SDE coefficients and the statistics of their implied Dirichlet distribution in the stationary state
---|---
|  |  |  |  |  | $b_{1}$ | = | $1/10$ | $b_{2}$ | = | $3/2$ | $(\omega_{1}$ | = | $5)$
|  |  |  |  |  | $S_{1}$ | = | $5/8$ | $S_{2}$ | = | $2/5$ | $(\omega_{2}$ | = | $2)$
|  |  |  |  |  | $\kappa_{1}$ | = | $1/80$ | $\kappa_{2}$ | = | $3/10$ | $\qquad(\omega_{3}$ | = | $3)$
${\langle{Y_{1}}\rangle}_{0}$ | $\approx$ | $0.05$ |  |  |  | ${\langle{Y_{1}}\rangle}_{\mathrm{s}}$ | = | $1/2$ |  |  |  |  |  |
${\langle{Y_{2}}\rangle}_{0}$ | $\approx$ | $0.42$ |  |  |  | ${\langle{Y_{2}}\rangle}_{\mathrm{s}}$ | = | $1/5$ |  |  |  |  |  |
${\langle{Y_{3}}\rangle}_{0}$ | $\approx$ | $0.53$ |  |  |  | ${\langle{Y_{3}}\rangle}_{\mathrm{s}}$ | = | $3/10$ |  |  |  |  |  |
$\langle{y_{1}^{2}}\rangle_{0}$ | $\approx$ | $0.03$ |  |  |  | $\langle{y_{1}^{2}}\rangle_{\mathrm{s}}$ | = | $1/44$ |  |  |  |  |  |
$\langle{y_{2}^{2}}\rangle_{0}$ | $\approx$ | $0.125$ |  |  |  | $\langle{y_{2}^{2}}\rangle_{\mathrm{s}}$ | = | $4/275$ |  |  |  |  |  |
$\langle{y_{3}^{2}}\rangle_{0}$ | $\approx$ | $0.13$ |  |  |  | $\langle{y_{3}^{2}}\rangle_{\mathrm{s}}$ | = | $21/1100$ |  |  |  |  |  |
${\langle{y_{1}y_{2}}\rangle}_{0}$ | $\approx$ | $-0.012$ |  |  |  | ${\langle{y_{1}y_{2}}\rangle}_{\mathrm{s}}$ | = | $-1/110$ |  |  |  |  |  |
${\langle{y_{1}y_{3}}\rangle}_{0}$ | $\approx$ | $-0.017$ |  |  |  | ${\langle{y_{1}y_{3}}\rangle}_{\mathrm{s}}$ | = | $-3/220$ |  |  |  |  |  |
${\langle{y_{2}y_{3}}\rangle}_{0}$ | $\approx$ | $-0.114$ |  |  |  | ${\langle{y_{2}y_{3}}\rangle}_{\mathrm{s}}$ | = | $-3/550$ |  |  |  |  |  |


# Complete pluripolar sets and removable singularities of plurisubharmonic
functions

Xieping Wang  School of Mathematical Sciences, University of Science and
Technology of China, Hefei 230026, Anhui, People’s Republic of China
<EMAIL_ADDRESS>To Li and Rui’an

###### Abstract.

Inspired by Chen-Wu-Wang (Math. Ann. 362: 305–319, 2015), we prove a Hartogs
type extension theorem for plurisubharmonic functions across a compact
complete pluripolar set, which is complementary to a classical theorem of
Shiffman.

###### Key words and phrases:

Removable singularities, pluriharmonic functions, complete pluripolar sets

###### 2020 Mathematics Subject Classification:

32D15, 32D20, 32U30, 32U05

The author was partially supported by NSFC grant 12001513 and NSF of Anhui
Province grant 2008085QA18.

##  1\. Introduction

Let $\Omega$ be a domain in $\mathbb{C}^{n}$, $n\geq 2$, and let $K$ be a
compact subset of $\Omega$ such that $\Omega\\!\setminus\\!K$ is connected.
The famous Hartogs extension theorem states that every holomorphic function on
$\Omega\\!\setminus\\!K$ extends holomorphically to the whole domain $\Omega$.
An analogue for pluriharmonic functions is also valid, as recently discovered
by Chen in [Che17] (see also [Wan22]). When it comes to plurisubharmonic (psh
for short) functions the situation, however, is quite different; for instance,
every bounded domain in $\mathbb{C}^{n}$ with smooth boundary is the domain of
existence of a psh function (see [BT88]). Thus it is meaningful to prove the
following

######  Theorem 1.1.

Let $\Omega$ be a domain in ${\mathbb{C}}^{n}$, $n\geq 2$, and let $K$ be a
compact complete pluripolar subset of $\Omega$. Then every psh function on
$\Omega\\!\setminus\\!K$ admits a unique psh extension to $\Omega$.

This somewhat surprising result is complementary to a classical theorem of
Shiffman [Shi72] that every psh function on a domain in $\mathbb{C}^{n}$
extends plurisubharmonically across a closed set of Hausdorff $(2n-2)$-measure
zero. It is also worth noting that Theorem 1.1 is of global nature, while
Shiffman’s theorem is of local nature.

The special case of Theorem 1.1 where $\Omega$ is the unit polydisc
$\Delta^{n}\subset\mathbb{C}^{n}$ is already contained in the beautiful work
of Chen-Wu-Wang [CWW15], who also dealt with the more general case of $K$
being a closed complete pluripolar subset of $\Delta^{n}$ under certain
reasonable conditions. Chen-Wu-Wang proved their result by using an Ohsawa-
Takegoshi type extension theorem for a single point in bounded complete Kähler
domains, which is also one of the main results in [CWW15] and seems to be
highly nontrivial due to its connection with an open problem posed by Ohsawa
in [Ohs95]. We observe that Theorem 1.1 can be proved by combining this
powerful result with an idea of Shiffman.

One may naturally ask whether Theorem 1.1 remains true when $\mathbb{C}^{n}$
is replaced by a generic Stein manifold of dimension $n\geq 2$. Since it is
not clear to us at this moment whether the Ohsawa-Takegoshi type extension
theorem by Chen-Wu-Wang applies to this more general case, we instead use
other techniques, namely the recently proved Hartogs extension theorem for
pluriharmonic functions in [Wan22] and the Skoda-El Mir extension theorem for
closed positive currents, to prove the following

######  Theorem 1.2.

Let $X$ be a Stein manifold of dimension $n\geq 2$. Suppose $\Omega$ is a
domain in $X$ such that $H^{1}(\Omega,\,\mathcal{O})=0$ and
$H^{2}(\Omega,\,\mathbb{R})=0$, and $K$ is a compact complete pluripolar
subset of $\Omega$. Then every psh function on $\Omega\\!\setminus\\!K$ admits
a unique psh extension to $\Omega$.

######  Remark .

Since $K$ is holomorphically convex in $X$, there always exists a Stein
neighborhood of $K$ contained in a given domain $\Omega\subset X$ so that the
assumption $H^{1}(\Omega,\,\mathcal{O})=0$ is nonessential for the theorem.
Also, it seems that the additional condition $H^{2}(\Omega,\,\mathbb{R})=0$ is
superfluous (and this is the case at least when $X=\mathbb{C}^{n}$, as shown
by Theorem 1.1).

After recalling a fundamental result concerning closed complete pluripolar
sets in Stein manifolds in Section 2, we prove Theorems 1.1 and 1.2 in
Sections 3 and 4, respectively.

Acknowledgements. The author would like to thank Professor Bo-Yong Chen for
kindly explaining his joint work with Wu and Wang [CWW15]. The author would
also like to thank Doctors Yong-Xin Gao and Zhi Li for patiently listening to
his lectures on the basic parts of pluripotential theory and related topics at
the seminar they organized when all three of them were at the Institute of
Mathematics, AMSS, Chinese Academy of Sciences from 2017 to 2019.

##  2\. Complete pluripolar sets and their defining functions

We begin by recalling the notion of complete pluripolarity. Let $X$ be a
complex manifold and $PSH(X)$ denote the set of all psh functions on $X$.

######  Definition .

A subset $E\subset X$ is called complete pluripolar if for every point $z\in
E$ there exists a neighborhood $U$ of $z$ and a function $\varphi\in PSH(U)$
such that

$E\cap U=\varphi^{-1}(-\infty).$

The set of all complex subvarieties of $X$ forms a particularly important
class of (closed) complete pluripolar sets, but complete pluripolar sets are
much more general: for instance, the Cartesian product of finitely many
(possibly different) Cantor type sets in the complex plane of logarithmic
capacity zero is a compact complete pluripolar set in the corresponding
complex Euclidean space (see, e.g., [Ran95]), but far from being complex-
analytic; see also [ElM84] for some other nontrivial examples.

In 1990, Colţoiu proved the following important result concerning the
existence of a global defining function for a closed complete pluripolar set.

######  Theorem 2.1 (see [Col90, Corollary 1]).

Let $X$ be a Stein manifold and $E\subset X$ be a closed complete pluripolar
set. Then there exists a function $\rho\in PSH(X)\cap
C^{\infty}(X\\!\setminus\\!E)$ such that $\rho^{-1}(-\infty)=E$ and
$\sqrt{-1}\partial\bar{\partial}\rho>0$ on $X\\!\setminus\\!E$.

As we shall see later, this result plays a fundamental role in the proofs of
Theorems 1.1 and 1.2.

##  3\. Proof of Theorem 1.1

The idea of the proof is due to Chen-Wu-Wang [CWW15], which in turn was more
or less inspired by the celebrated work of Demailly [Dem92]. The key
ingredient here is an Ohsawa-Takegoshi type extension theorem for a single
point in bounded complete Kähler domains in $\mathbb{C}^{n}$ (see [CWW15,
Theorem 1.3] for details). In order to make use of this theorem, we first need
to prove the following result:

######  Theorem 3.1.

Let $X$ be a Stein manifold and $E\subset X$ be a closed complete pluripolar
set. Then $X\\!\setminus\\!E$ carries a complete Kähler metric.

The important special case of $E\subset X$ being a complex subvariety is well-
known and due to Grauert [Gra56].

###### Proof.

The result is an easy consequence of Theorem 2.1. Let $\varphi$ be a
$C^{\infty}$ strictly psh exhaustion function for $X$ and let
$\psi\\!:X\\!\setminus\\!E\to\mathbb{R}$ be a $C^{\infty}$ function such that

$\psi=-\log(-\rho)\quad\mbox{on}\;\;U\\!\setminus\\!E,$

where $\rho$ is a psh function as in Theorem 2.1 and $U:=\\{\rho<-1\\}$. Then
one can construct a $C^{\infty}$ convex, rapidly increasing function $\chi$ on
$\mathbb{R}$ such that

$\omega:=\sqrt{-1}\partial\bar{\partial}(\chi\circ\varphi)+\sqrt{-1}\partial\bar{\partial}\psi\geq\omega_{0}\quad\mbox{on}\;\;X\\!\setminus\\!E$

for some complete Kähler metric $\omega_{0}$ on $X$.

We claim that $\omega$ is complete on $X\\!\setminus\\!E$. For this, we may
assume without loss of generality that $X$ itself is connected (and so is
$X\\!\setminus\\!E$). Suppose $\\{z_{j}\\}_{j\geq 1}$ is a bounded sequence in
the metric space $(X\\!\setminus\\!E,\,\omega)$. Then there is a sequence of
smooth curves $\\{\gamma_{j}\\}_{j\geq 1}\subset
C^{\infty}([0,1],\,X\\!\setminus\\!E)$ with uniformly bounded lengths with
respect to $\omega$, joining each $z_{j}$ to a (fixed) reference point in
$X\\!\setminus\\!\overline{U}$. Since $\omega\geq\omega_{0}$ on
$X\\!\setminus\\!E$ and $\omega_{0}$ is complete on $X$, we may assume that
the sequence $\\{z_{j}\\}_{j\geq 1}$ itself converges in $X$ by passing to a
subsequence if necessary. What now remains is to show that the limit of
$\\{z_{j}\\}_{j\geq 1}$ lies outside $E$. Suppose the contrary and set

$t_{j}:=\inf\big{\\{}t\in[0,1]\\!:\gamma_{j}([t,1])\subset U\big{\\}},\quad
j\geq 1.$

Clearly $0<t_{j}<1$ and $\gamma_{j}(t_{j})\in\partial U=\\{\rho=-1\\}$ for all
sufficiently large $j$. Observe also that

$\omega\geq\sqrt{-1}\partial\bar{\partial}\big{(}-\log(-\rho)\big{)}\geq\sqrt{-1}\partial\log(-\rho)\wedge\bar{\partial}\log(-\rho)\quad\mbox{on}\;\;U.$

It then follows that

$\begin{split}\sqrt{2}\,{\rm
length}_{\omega}(\gamma_{j})&\geq\int_{t_{j}}^{1}\big{|}\big{(}d\log(-\rho)\big{)}(\gamma^{\prime}(t))\big{|}dt\geq\int_{t_{j}}^{1}\big{(}d\log(-\rho)\big{)}(\gamma^{\prime}(t))dt\\\
&=\log(-\rho(z_{j}))\to\infty\quad{\rm as}\;\;j\to\infty,\end{split}$

contradicting the boundedness of $\\{{\rm
length}_{\omega}(\gamma_{j})\\}_{j\geq 1}$. Therefore the limit of
$\\{z_{j}\\}_{j\geq 1}$ lies outside $E$, and hence $\omega$ is complete on
$X\\!\setminus\\!E$. ∎

We are now ready to prove Theorem 1.1.

###### Proof of Theorem 1.1.

The uniqueness for extension is clear, since two psh functions on $\Omega$
which coincide almost everywhere are actually equal everywhere. So it suffices
to prove the existence part of the theorem.

We first observe that the problem can be reduced to the case when
$\Omega\supset K$ is a bounded pseudoconvex domain. To see this, let $\rho$ be
a psh function on $\mathbb{C}^{n}$, continuous on
$\mathbb{C}^{n}\\!\setminus\\!K$ and satisfying $\rho^{-1}(-\infty)=K$ (see
Theorem 2.1). Choose an open set $U\subset\mathbb{C}^{n}$ such that $K\subset
U\subset\subset\Omega$, and set

$\widetilde{\rho}:=\left\\{\begin{array}[]{lll}\\!\\!\max\big{\\{}\rho,\,\inf\limits_{\partial
U}\rho\big{\\}}\quad\mbox{on}\;\;\mathbb{C}^{n}\\!\setminus\\!\overline{U};\\\
\\!\\!\qquad\quad\rho\quad\quad\quad\;\;\,\mbox{on}\;\;\overline{U}.\end{array}\right.$

Then $\widetilde{\rho}$ is a psh function on $\mathbb{C}^{n}$ with
$\widetilde{\rho}^{-1}(-\infty)=K$. Replacing $\Omega$ by any connected
component of $\\{\widetilde{\rho}<\inf\limits_{\partial U}\rho\\}$, we may
assume in what follows that $\Omega$ is a bounded pseudoconvex domain in
$\mathbb{C}^{n}$.

Let $\varphi\in PSH(\Omega\\!\setminus\\!K)$. To prove the plurisubharmonic
extendibility of $\varphi$ across $K$, it suffices to show that every point of
$K$ admits a small neighborhood on which $\varphi$ is bounded above. Observe
that $\Omega\\!\setminus\\!K$ is a bounded complete Kähler domain, in view of
Theorem 3.1. We can therefore invoke [CWW15, Theorem 1.3] to assign to every
point $z\in\Omega\\!\setminus\\!K$ a holomorphic function $f_{z}$ on
$\Omega\\!\setminus\\!K$ with the property that $f_{z}(z)=e^{\varphi(z)/2}$
and

(3.1) $\int_{\Omega\setminus K}|f_{z}|^{2}e^{-\varphi}\leq{\rm
const}_{n,\,{\rm diam}\,\Omega}.$

By the Hartogs extension theorem, each such $f_{z}$ extends holomorphically to
$\Omega$. With a slight abuse of notation, we denote the extension still by
$f_{z}$.

To proceed the proof, we make use of a result of Shiffman (cf. [Har77, Lemma
2.3] and [Dem12, Chapter 3, Lemma 4.7]). Fix an arbitrary point $z_{0}\in K$
and recall that being a polar subset of $\mathbb{C}^{n}\cong\mathbb{R}^{2n}$,
$K$ has Hausdorff dimension at most $2n-2$ (see, e.g., [AG01, Theorem 5.9.6]).
By suitably selecting affine linear coordinates for $\mathbb{C}^{n}$, we can
find a polydisc
$\Delta^{\prime}\times\Delta^{\prime\prime}\subset\mathbb{C}^{n-1}\times\mathbb{C}$
centered at $z_{0}=:(z^{\prime}_{0},z^{\prime\prime}_{0})$ such that

$(\Delta^{\prime}\times\partial\Delta^{\prime\prime})\cap K=\emptyset.$

By shrinking $\Delta^{\prime}$ if necessary, we further arrive at

$\big{(}\Delta^{\prime}\times(\Delta^{\prime\prime}\\!\setminus\\!(1-\varepsilon)\overline{\Delta^{\prime\prime}})\big{)}\cap
K=\emptyset$

for some sufficiently small $\varepsilon>0$. Now choose $R,\,r>0$ such that
$1-\varepsilon<r<R<1$ and ball $B\subset\subset\Delta^{\prime}$ centered at
$z^{\prime}_{0}$. Then the Cauchy estimate and inequality (3.1) imply

$\begin{split}e^{\varphi(z)}=|f_{z}(z)|^{2}&\leq{\rm
const}_{B,\,R,\,r,\,\varepsilon}\int_{B\times(R\Delta^{\prime\prime}\setminus
r\Delta^{\prime\prime})}|f_{z}|^{2}\\\ &\leq{\rm
const}_{B,\,R,\,r,\,\varepsilon}\sup_{B\times(R\Delta^{\prime\prime}\setminus
r\Delta^{\prime\prime})}e^{\varphi}\int_{B\times(R\Delta^{\prime\prime}\setminus
r\Delta^{\prime\prime})}|f_{z}|^{2}e^{-\varphi}\\\ &\leq
C\sup_{B\times(R\Delta^{\prime\prime}\setminus
r\Delta^{\prime\prime})}e^{\varphi}\end{split}$

for all
$z\in\big{(}(1-\varepsilon)(B\times\Delta^{\prime\prime})\big{)}\setminus K$,
where $C>0$ is a constant independent of $z$. Consequently, $\varphi$ is
bounded above on $(1-\varepsilon)(B\times\Delta^{\prime\prime})\ni z_{0}$
outside $K$. This completes the proof. ∎

##  4\. Proof of Theorem 1.2

We start with the following result, which is essentially due to Sibony
[Sib85].

######  Proposition 4.1.

Let $X$ be a Stein manifold of dimension $n\geq 2$ and $K\subset X$ be a
compact complete pluripolar set. Then every closed positive $(p,p)$-current on
$X\\!\setminus\\!K$ with $p\leq n-1$ has finite mass near $K$.

###### Proof.

As in the proof of Theorem 1.1, we can construct a strongly pseudoconvex
neighborhood $\Omega\subset\subset X$ of $K$ with $C^{\infty}$-boundary. Let
$u_{K}$ denote the relative extremal function of $K$ in $\Omega$, that is

$u_{K}=\sup\Big{\\{}u\in PSH(\Omega)\cap C(\Omega)\\!:u<1\;{\rm
on}\;\Omega,\,u\leq 0\;{\rm on}\;K\Big{\\}}.$

Clearly $u_{K}=0$ on $K$. Moreover according to [Sib85, Proposition 1.4], the
product $u_{K}T$ of $u_{K}$ and every closed positive $(p,p)$-current $T$ on
$\Omega\\!\setminus\\!K$ with $p\leq n-1$ has finite mass near $K$. (This is
true for all compact sets $K\subset\Omega$, regardless of the complete
pluripolarity of $K$.)

It remains to show that $u_{K}$ is no other than the characteristic function
$\chi_{\Omega\setminus K}$ of $\Omega\\!\setminus\\!K$, provided
$K\subset\Omega$ is further assumed to be complete pluripolar. For this let
$\rho$ be a negative psh function on $\Omega$, continuous on
$\Omega\\!\setminus\\!K$ and satisfying $\rho^{-1}(-\infty)=K$, and set

$\rho_{t}:=\max\big{\\{}\rho/t+1,\,0\big{\\}},\quad t>0.$

Then $\\{\rho_{t}\\}_{t>0}$ forms a family of candidates for the supremum
defining $u_{K}$, hence $\rho_{t}\leq u_{K}$ for all $t>0$. Letting
$t\to\infty$ yields $u_{K}=\chi_{\Omega\setminus K}$, as desired. ∎

Now we are in a position to prove Theorem 1.2.

###### Proof of Theorem 1.2.

As pointed out in the proof of Theorem 1.1, it suffices to prove the existence
part of the theorem.

Given a function $\varphi\in PSH(\Omega\\!\setminus\\!K)$, we consider the
associated closed positive $(1,1)$-current
$T:=\sqrt{-1}\partial\bar{\partial}\varphi$ on $\Omega\\!\setminus\\!K$. By
virtue of Proposition 4.1 and the Skoda-El Mir extension theorem (see [ElM84,
Théorème II.1] or [Sib85, Dem12]), $T$ extends to a closed positive
$(1,1)$-current on $\Omega$, which we denote by $\widetilde{T}$. Since
$H^{1}(\Omega,\,\mathcal{O})=0$ and $H^{2}(\Omega,\,\mathbb{R})=0$, a standard
argument shows that $\widetilde{T}$ admits a global potential
$\widetilde{\varphi}\in PSH(\Omega)$, i.e.,
$\sqrt{-1}\partial\bar{\partial}\widetilde{\varphi}=\widetilde{T}$. One can
then write

$\widetilde{\varphi}=\varphi+h\quad{\rm on}\ \,\Omega\\!\setminus\\!K$

with $h$ being a pluriharmonic function on $\Omega\\!\setminus\\!K$, in view
of Weyl’s lemma. On the other hand, the Hartogs extension theorem for
pluriharmonic functions (see [Wan22, Theorem 1.1]) implies that $h$ admits a
pluriharmonic extension $\widetilde{h}$ to $\Omega$. It now follows that
$\widetilde{\varphi}-\widetilde{h}$ is a psh function on $\Omega$ that extends
$\varphi$. ∎

## References

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#  The cube axiom and resolutions in homotopy theory

Manfred Stelzer

###### Abstract

We show that a version of the cube axiom holds in cosimplicial unstable
coalgebras and cosimplicial spaces equipped with a resolution model structure.
As an application, classical theorems in unstable homotopy theory are extended
to this context.

##  1 Introduction

The cube axiom holds in a model category $\mathcal{M}$ if in every commutative
cube the top square is a homotopy pushout if the bottom square is a homotopy
pushout and the vertical faces are homotopy pullbacks. It was proved by Mather
that the cube axiom is valid in topological spaces with the Strøm structure
[M]. In many other model categories it is known to hold. Examples include
every stable model category, every $\infty$-topos and the model category of
motivic spaces [Ho, Proposition 3.15].

In the model categorical axiomatic approach to Lusternik-Schnirelmann
category, due to Doerane, a variant of the cube axiom is to be assumed
[Do],[Do2]. If it is satisfied many of the classical topological results on
Lusternik-Schnirelmann category extend.
To know that the cube axiom holds brings some more benefits. There is a good
theory of fiberwise localization [C.S]. Moreover, in a recent paper of
Devalpurka and Haine it is shown that a form of the James splitting and a weak
form of the Hilton-Milnor theorem hold under some assumptions in an
$\infty$-category [D.H]. The main assumption made is the cube axiom.

In this paper, we investigate in which form the cube axiom holds in two
resolution model categories of cosimplicial objects. The first is the one of
cosimplicial spaces $c\mathcal{S}^{\mathcal{G}}$ and the second the one of
cosimplicial unstable coalgebras $c\mathcal{CA}^{\mathcal{E}}$ for a prime
field $\mathbb{F}$. The classes of group objects $\mathcal{G}$ and
$\mathcal{E}$ which define the resolutions are the Eilenberg-MacLane spaces of
type $\mathbb{F}$ and their homology respectively [Bou]. These model
categories give home to the constructions of unstable Adams spectral sequences
of Bousfield and Kan [Bou2],[BK]. For this reason, they also play a central
role in the obstruction theory for realizations of unstable coalgebras and the
construction of moduli spaces for such realizations [Bl], [BRS]. Since
$c\mathcal{CA}^{\mathcal{E}}$ and $c\mathcal{S}^{\mathcal{G}}$ are not known
to be cofibrantly generated, the general results of [Re] and [R.S.] do not
apply. In fact, it is the class of fibrations which is defined by cocell
attachments.
As an application we see that versions of classical theorems of James, Hilton-
Milnor and Ganea hold in $c\mathcal{CA}^{\mathcal{E}}$ and
$c\mathcal{S}^{\mathcal{G}}$.
For the rest of this paper $\mathcal{M}$ stands for $\mathcal{CA}$ or
$\mathcal{S}$ and $\mathcal{F}$ for $\mathcal{E}$ or $\mathcal{G}$.
We show the following:

######  Theorem 1.1.

The cube axiom holds in $c\mathcal{M}^{\mathcal{F}}$ for cubes

---
$\textstyle{E_{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

in which the map $B\to A$ is $0$-connected.

######  Remark 1.2.

That one has to make assumptions on the maps in the cube is not uncommon.
Several examples of categories in which such a restriction is necessary for
the cube axiom or its dual to hold can be found in [Do, Appendix]. A
particular important example is the category of commutative differential
graded algebras over the rationals. This serves as one model for rational
homotopy theory.

Let $Z$ be a fibrant and cofibrant object in $c\mathcal{CA}^{\mathcal{E}}$ or
$c\mathcal{S}^{\mathcal{G}}$ and write $c\mathcal{CA}^{\mathcal{E}}_{Z}$ and
$c\mathcal{S}^{\mathcal{G}}_{Z}$ for the model categories of objects over $Z$
with coaugmentation under $Z$. These are pointed model categories with
$0$-object $Z$.

All suspensions, loop spaces, products, smash and wedge products in
$c\mathcal{M}_{Z}^{\mathcal{F}}$ are taken in the derived sense and over and
under the $0$-object $Z$.
Combining 1.1 with some of the main theorems in [D.H] we obtain the following
application.

######  Theorem 1.3.

Let $X$, $Y$ be objects in $c\mathcal{M}_{Z}^{\mathcal{F}}$. Assume that the
structure maps $X\to Z$ and $Y\to Z$ are $0$-connected. Then there are
equivalences in $c\mathcal{M}^{\mathcal{F}}$

  * (1)

$\Sigma(X\times Y)\simeq(\Sigma X\wedge Y)\vee(\Sigma X)\vee(\Sigma Y)$

  * (2)

(James splitting theorem) $\Sigma\Omega\Sigma X\simeq\bigvee_{i\geq 1}\Sigma
X^{\wedge i}$

  * (3)

(Hilton-Milnor splitting )
$\Omega\Sigma(X\vee Y)\simeq\Omega\Sigma X\times\Omega\Sigma
Y\times\Omega(\bigvee_{i,j\geq 1}\Sigma X^{\wedge i}\wedge Y^{\wedge j})$

and there are homotopy fiber sequences

  * (4)

$\Sigma\Omega X\to X\vee X\to X$

  * (5)

$\Sigma(\Omega X\wedge\Omega Y)\to X\vee Y\to X\times Y$

  * (6)

$(\Omega Y\times X)/\Omega Y\to X\vee Y\to Y$

Let $F\to E\to X$ be a fibration sequence and write $E\cup CF$ for the
homotopy cofiber of the inclusion of the homotopy fiber $F$ in $E$. Assume
that $F\to Z$ or $F\to E$ is $0$-connected. Then there is a homotopy fiber
sequence

  * (7)

(Ganea’s theorem ) $\Sigma F\wedge\Omega X\to E\cup CF\to X.$

In the usual form of the Hilton-Milnor theorem the right hand side of (3) in
1.3 is further decomposed into an infinite product. We prove such a version in
1.4 below. This needs a convergence property which is implied by some
connectivity estimates which are derived from the dual Blakers-Massey theorem
in [BRS].

######  Theorem 1.4.

Let $X$, $Y$ be as in 1.3. Then there is an equivalence in
$c\mathcal{M}^{\mathcal{F}}$

$\Omega\Sigma(X\vee Y)\simeq\prod_{w}\Omega\Sigma w(X,Y)$

where $w$ runs through all basic words in $(x,y)$ and $w(X,Y)$ is the iterated
smash power of $X,Y$ defined by $w$.

######  Remark 1.5.

Under the additional assumptions that the homotopy groups
$\pi_{*}(X),\pi_{*}(Y)$ are free $\pi_{0}(Z)$-modules and that $\pi_{*}(Z)$ is
concentrated in degree $0$ a (dual) Hilton-Milnor theorem was established by
Goerss in the category of simplicial unstable algebras [G]. He noted that
possibly this restriction may be omitted by restructuring the proof along the
lines of Milnor’s original argument [Mi], [Wh]. That is what we did.
Computational applications of his theorem where given in [G2] and [G3]. Many
of them generalize due to 1.4. In particular, 1.4 helps in the computation of
André-Quillen cohomology of coabelian objects in $c\mathcal{CA}^{\mathcal{E}}$
which shows up in the $E_{2}$-term of the unstable Adams spectral sequence.

######  Remark 1.6.

Many of our arguments extend to resolution model categories defined by the
spaces in the $\Omega$ spectrum of Morava K-theory. The unstable homology
operations are also known due to the determination of the Hopf ring of Morava
K-theory by Wilson [W]. One ingredient which is missing so far is the homotopy
excision theorem.

The paper is organized as follows. In section 2 we give background information
on the model categories $c\mathcal{CA}^{\mathcal{E}}$ and
$c\mathcal{S}^{\mathcal{G}}$. The proof of 1.1 is given in section 3. Section
4 is devoted to the proof of 1.3. Connectivity estimates for relative infinite
products, wedge and smash products are derived in section 5. These are needed
in the proof of 1.4 which is the content of section 6.

###  1.1 Conventions

Throughout the paper we fix a prime field $\mathbb{F}$. We will use the
following notation.

  * •

$\mathcal{S}$ = the category of simplicial sets;

  * •

$Vec$ = the category of non-negatively graded $\mathbb{F}$-vector spaces

  * •

$\mathcal{CA}$ = the category of unstable coalgebras over the Steenrod algebra
$\mathcal{A}$ over $\mathbb{F}$

For any category $\mathcal{C}$ we let

  * •

$c\mathcal{C}$ = the category of cosimplicial objects over $\mathcal{C}$;

We use freely notions and standard facts about model categories. Besides the
founding [Q] some widely used references are [Hi],[H] and [GJ]. In section 5
we use the language of $\infty$-categories. The needed definitions and results
are to be found in [L].

###  1.2 Acknowledgments

I would like to thank Hadrian Heine and Markus Spitzweck for patiently
answering my questions on $\infty$-categorical issues. Finally, I would like
to thank the Deutsche Forschungsgemeinschaft for support through the
Schwerpunktprogramm 1786 “Homotopy theory and algebraic geometry”.

##  2 Cosimplicial spaces and unstable coalgebras

###  2.1 Resolution model categories

The categories $\mathcal{S}$ and $\mathcal{CA}$ are equipped with the standard
Quillen and the discrete model structure respectively. Let $K(\mathbb{F},m)$
denote the Eilenberg-MacLane space of type $\mathbb{F}$. A product of those
spaces will be called an $\mathbb{F}$-gem. Let
$\mathcal{G}=\\{K(\mathbb{F},m)|m\geq 0\\}$ and
$\mathcal{E}=\\{H_{*}(K(\mathbb{F},m))|m\geq 0\\}.$ The categories
$c\mathcal{S}$ of cosimplicial spaces and cosimplicial unstable coalgebras
$c\mathcal{CA}$ carry simplicial resolution model category structures relative
to $\mathcal{G}$ and $\mathcal{E}$ respectively [D.K.S.], [Bou]. We explain
the basic facts about the model categories $c\mathcal{S}^{\mathcal{G}}$ and
$c\mathcal{CA}^{\mathcal{E}}$. As any resolution model category they carry an
external simplicial structure.
The weak equivalences are the maps of cosimplicial objects

$X\to Y$ (2.1.1)

which induce an isomorphism on $\pi_{*}[-,F]$ for every $F\in\mathcal{F}$.
This can be equivalently described as maps which induce an isomorphism on
$\pi^{*}H_{*}(-)$ and $\pi^{*}(-)$ respectively. For $\mathcal{E}$ this
equivalence is due to the fact that objects in $\mathcal{E}$ are cofree
unstable coalgebras.

The cofibrations are the Reedy cofibrations such that the induced homomorphism

$[Y,F]\to[X,F]$ (2.1.2)

is a fibration of simplicial groups. A concrete description of the fibrations
can be found and will be of importance later on.

######  Definition 2.1.

A map $i:A\to B$ in the homotopy category $Ho(\mathcal{M})$ is called
$\mathcal{F}$-monic if

$i^{*}:[B,F]\to[A,F]$

is surjective for al $F\in\mathcal{F}$.
An object $Y\in Ho(\mathcal{M})$ is called $\mathcal{F}$-injective if

$i^{*}:[B,Y]\to[A,Y]$

is surjective for all $\mathcal{F}$-monic $i:A\to B$.

For $D\in c\mathcal{M}$ and $K\in\mathcal{S}$ we write $D\otimes K$ and
$D^{K}$ for the tensor and cotensor. Now we can describe the fibrations in
$c\mathcal{M}^{\mathcal{F}}$. Recall from [BRS, 2.3.].

######  Definition 2.2.

A map $f:X\to Y$ in $c\mathcal{M}$ is quasi-$\mathcal{F}$-cofree if there is a
sequence of fibrant $\mathcal{F}$-injective objects $(F_{s})_{s\geq 0}$ such
that for all $s\geq 0$, there are homotopy pullback diagrams in the Reedy
model structure of $c\mathcal{M}$

$\textstyle{cosk_{s}(f)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{s-1}(f)}$$\textstyle{(cF_{s})^{\Delta^{s}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{s}^{*}}$$\textstyle{cosk_{s-1}(f)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(cF_{s})^{\partial\Delta^{s}}}$

where $i_{s}:\partial\Delta^{s}\to\Delta^{s}$ is the canonical inclusion.

######  Definition 2.3.

Let $F$ be a fibrant $\mathcal{F}$-injective. A a Reedy homotopy pullback
diagram in $c\mathcal{M}$

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{(cF)^{\Delta^{s}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{s}^{*}}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{(cF)^{\partial\Delta^{s}}}$

is called a quasi-cocell attachment of dimension $s$. The map $f$ is called
the coattaching map.

A map in $c\mathcal{M}^{\mathcal{F}}$ is a fibration if and only if it is a
Reedy fibration and a retract of a quasi-cofree map [BRS, corollary 2.3.12].
We recall the definition of the external loop functor.

######  Definition 2.4.

The collapsed boundary gives the $n$-sphere
$S^{n}:=\Delta^{n}/\partial\Delta^{n}$ a canonical basepoint. If $X$ is
pointed in $c\mathcal{M}$ we define the $s$-th external loop object by

$\Omega^{n}_{ext}(X)=\overline{hom}(\Delta^{n}/\partial\Delta^{n},X)$

where

$\overline{hom}(\Delta^{n}/\partial\Delta^{n},X)=fib[hom(\Delta^{n}/\partial\Delta^{n},X)\to\hom(\Delta^{0},X)\cong
Y]\in c\mathcal{M}_{*}$

as the fiber taken at the basepoint of $X$ of the map induced by the basepoint
of $\Delta^{n}/\partial\Delta^{n}$.

We will often omit the subscript ext when the reference to the external
structure is clear from the context.

The following observation from [BRS] will be useful to us later on. Let $E$ be
as in 2.3. Then each cosimplicial degree $t$ there is an isomorphism

$E^{t}\cong D^{t}\times\Omega^{s}(cF)^{t}.$ (2.1.3)

As a consequence of homotopy excision the model category
$c\mathcal{M}^{\mathcal{F}}$ is proper [BRS, 4.4.2 6.2.5].
We recall the notion of cosimplicial connectivity.

######  Definition 2.5.

Let $n\geq 0$.

  * (a)

A map $f:X\to Y$ in $c\mathcal{M}^{\mathcal{F}}$ is called $n$-connected if
the induced map

$\pi_{s}[Y,F]_{\mathcal{M}}\to\pi_{s}[X,F]_{\mathcal{M}}$

is an isomorphism for $0<s<n$ and an epimorphism for $s=n$ for all
$F\in\mathcal{F}$

  * (b)

A pointed object $(C,\ast)$ of $c\mathcal{M}^{\mathcal{F}}$ is cosimplicially
$n$-connected if the map $\ast\to C$ is cosimplicially $n$-connected.

  * (c)

An arbitrary object $X$ of $c\mathcal{M}^{\mathcal{F}}$ is called
$n$-connected if the canonical map $X\to\ast$ is cosimplicially $n$-connected.

A pointed object $(C,\ast)$ of $c\mathcal{M}^{\mathcal{F}}$ is $n$-connected
if and only if $C$ is cosimplicially $(n-1)$-connected. Every pointed object
is cosimplicially $0$-connected.

The next result from [BRS] is used to reduce questions in
$c\mathcal{S}^{\mathcal{G}}$ to questions in the simpler category
$c\mathcal{CA}^{\mathcal{E}}$.

######  Lemma 2.6.

The homology functor $H_{*}:c\mathcal{S}^{\mathcal{G}}\to
c\mathcal{CA}^{\mathcal{E}}$ sends homotopy pullbacks into homotopy pullbacks
and homotopy puhouts along maps one of which is $0$-connected to homotopy
pushouts [BRS, Proposition 6.17.,Corollary 6.2.3.].

Objects in $\mathcal{M}$ satisfy the following $\mathcal{F}$\- flatness
property with respect to the product.

######  Lemma 2.7.

Let $X\in\mathcal{M}$ and $i:A\to B$ an $\mathcal{F}$-injective map in
$Ho(\mathcal{M})$. Then $i\times 1_{X}:A\times X\to B\times X$ is
$\mathcal{F}$-injective.

Proof:  Suppose first that $\mathcal{M}=\mathcal{CA}$. In this case a map is
$\mathcal{E}$-injective if and only if it is injective. This follows from the
fact that the homology of an Eilenberg-MacLane space of type $\mathbb{F}$ is
cofree. The product in $\mathcal{CA}$ is the tensor product which is an exact
functor of the underlying graded vector spaces.
In case $\mathcal{M}=\mathcal{S}$ note that a map $i:A\to B$ in
$Ho(\mathcal{S})$ is $\mathcal{G}$-injective if and only if $H_{*}(i)$ is
$\mathcal{E}$-injective in $Ho(\mathcal{CA})$ [BRS, p.68].$\square$

##  3 The cube theorem in $c\mathcal{CA}^{\mathcal{E}}$ and
$c\mathcal{S}^{\mathcal{G}}$

######  Definition 3.1.

Let $\mathcal{N}$ be a model category. We say that the cube axiom holds in
$\mathcal{N}$ if in any commutative cube in $\mathcal{N}$

---
$\textstyle{E_{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

whose bottom square is a homotopy pushout and whose vertical squares are
homotopy pull backs the top square is a homotopy push out.

######  Proposition 3.2.

Let $C$ be a cube as in 3.1 where $p:E_{D}\to D$ be a quasi-cocell attachment
in $c\mathcal{M}^{\mathcal{F}}$ with coattaching map $f$ and cocell
$(cF)^{\Delta^{s}}$. Assume that $B\to A$ is $0$-connected.

Then $E_{D}$ is the homotopy pushout of $E_{A}\to E_{B}\leftarrow E_{C}$ in
$c\mathcal{M}^{\mathcal{F}}$. Moreover, $j:E_{B}\to E_{A}$ is $0$-connected.

Proof:  We may assume that all the vertical squares are pullbacks along
fibrations and that $D\cong A\sqcup_{B}C$ is a pushout along two cofibrations
in $c\mathcal{M}^{\mathcal{F}}$ [Do, Theorem A.1]. Let us start with the case
$c\mathcal{M}^{\mathcal{F}}=c\mathcal{CA}^{\mathcal{E}}$. Recall that
pullbacks in $\mathcal{CA}$ are given by the cotensor product [BRS, remark
p.46] and that colimits are computed in the underlying vector spaces. There is
a natural map

$q:E_{A}\oplus_{E_{B}}E_{C}\to E_{D}.$

In each cosimplcial degree $n$ there are isomorphisms

$((cE)^{\Delta^{s}})^{n}\cong\otimes_{\sigma\in\Delta^{s}_{n}}E\cong(\otimes_{\sigma\in\partial\Delta^{s}_{n}}E)\otimes\Omega^{s}(E))$

In each cosimplicial degree $n$ the map $q^{n}$ is the chain of isomorphism

$\scriptstyle(E_{A})^{n}\oplus_{(E_{B})^{n}}(E_{C})^{n}\cong$
$\scriptstyle((A^{n})\square_{((cF)^{\partial\Delta^{s}})^{n}}((cF)^{\Delta^{s}})^{n})\oplus_{((B^{n})\square_{((cF)^{\partial\Delta^{s}})^{n}}((cF)^{\Delta^{s}})^{n})}((C^{n})\square_{((cF)^{\partial\Delta^{s}})^{n}}((cF)^{\Delta^{s}})^{n})\cong$
$\scriptstyle((A^{n})\otimes(\Omega^{s}(F))^{n})\oplus_{((B^{n})\otimes(\Omega^{s}(F))^{n})}((C^{n})\otimes(\Omega^{s}(F))^{n})\cong$
$\scriptstyle((A^{n})\oplus_{(B^{n})}(C^{n}))\otimes(\Omega^{s}(F))^{n})\cong
E_{D}^{n}$

We claim that $j:E_{B}\to E_{A}$ is a $0$-connected cofibration in
$c\mathcal{CA}^{\mathcal{E}}$.

We have to show that $j^{n}$ is $\mathcal{E}$\- injective that is to say
injective for all $n$. But $j^{n}$ is isomorphic to the map

$A^{n}\otimes\Omega^{s}(E)^{n}\to B^{n}\otimes\Omega^{s}(E)^{n}$

which is injective since $A\to B$ is. So $E_{D}\cong E_{A}\oplus_{E_{B}}E_{C}$
is a homotopy pushout. This completes the proof for
$c\mathcal{CA}^{\mathcal{E}}$.

We turn to $c\mathcal{S}^{\mathcal{G}}$. Factor $j$ as
$E_{B}\stackrel{{\scriptstyle k}}{{\to}}\bar{E}_{A}\stackrel{{\scriptstyle
w}}{{\to}}E_{B}$ in $c\mathcal{S}^{\mathcal{G}}$ in a cofibration $k$ and a
trivial fibration $w$. We have to see that the induced map from the
(homotopy)pushout $P=\bar{E_{A}}\sqcup_{E_{B}}E_{C}\to E_{D}$ is a weak
equivalence.

Apply the homology functor to the given cube $\mathcal{C}$. The cube
$H_{*}\mathcal{C}$ so obtained in $c\mathcal{CA}^{\mathcal{E}}$ has bottom
square a homotopy pushout with $0$-connected $H_{*}(B)\to H_{*}(A)$ and all
vertical squares homotopy pullbacks by 2.6 . The top square is a homotopy
pushout, as we just have seen above, and the map $H_{*}(E_{B})\to
H_{*}(E_{A})$ is $0$-connected. Hence the maps $E_{B}\to E_{A}$ and $k$ are
$0$-connected.
By 2.6 again $H_{*}(P)$ is a homotopy pushout. It follows that the map
$H_{*}(P)\to H_{*}(E_{D})$ is a weak equivalence which is what we want.
$\square$

######  Definition 3.3.

We say that the restricted cube axiom holds in if the assertion in 3.1 holds
for all cubes in which the map $B\to A$ is $0$-connected.

######  Theorem 3.4.

The restricted cube axiom holds in $c\mathcal{M}^{\mathcal{F}}$.

Proof:  Using the fact that fibrations are retracts of quasi-cofree maps the
assertion follows from the coskeletal tower by means of 3.2. $\square$

We record some invariance properties of $0$-connected maps.

######  Lemma 3.5.

Let

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B}$

be a homotopy pullback diagram in $c\mathcal{M}^{\mathcal{F}}$. If p is
$0$-connected so is $q$.

Proof:  The assertion for $c\mathcal{CA}^{\mathcal{E}}$ is in [BRS, Lemma
4.6.3.]. For the proof in $c\mathcal{S}^{\mathcal{G}}$, note that a map $f$ in
$c\mathcal{S}^{\mathcal{G}}$ is $0$-connected if and only if $H_{*}(f)$ is
$0$-connected in $c\mathcal{CA}^{\mathcal{E}}$ and that homotopy pullbacks are
preserved under $H_{*}$.$\square$.

######  Lemma 3.6.

Let

$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{j}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{j}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{i}}$$\textstyle{P}$

be a homotopy pushout diagram in $c\mathcal{M}^{\mathcal{F}}$ If $i$ is
$0$-connected so is $\bar{i}$.

Proof:  We start with the case $c\mathcal{CA}^{\mathcal{E}}$ and may assume
that $i$ is a cofibration. So the map

$i^{*}:[C,H_{*}(K(\mathbb{F},m))]\to[B,H_{*}(K(\mathbb{F},m))]$

is a fibration of simplicial groups for each $m$. By the cofreenes of
$H_{*}(K(\mathbb{F},m))$, this sequence can be identified with the induced map
on the dual in degree $m$

$i^{*}:(C^{m})^{*}\to(B^{m})^{*}.$

Because $i^{*}$ is $0$-connected it is a $\pi_{0}$-surjective Kan fibration
and hence, surjective. It follows that $i$ is injective. Then clearly
$\bar{i}$ is injective as well since colimits are formed in the underlying
cosimplicial vector spaces or equivalently in the underlying cochain complex.
But then the restriction to the cocycles

$\bar{i}:Z^{0}(C)=H^{0}(C)\to Z^{0}(P)=H^{0}(P)$

is injective.
Suppose that the cube is in $c\mathcal{S}^{\mathcal{G}}$. Apply $H_{*}$ to get
a cube in $c\mathcal{CA}^{\mathcal{E}}$ which is a homotopy pushout by 2.6.
Since $H_{*}(i)$ is $0$-connected $H_{*}(\bar{i})$ is $0$-connected and hence
so is $\bar{i}$.$\square$

##  4 Applications of the cube theorem

Let $\mathcal{N}$ be a model category and $Z$ an object of $\mathcal{N}$.
Denote the categories of objects over and under $Z$ in $\mathcal{N}$ by
$(\mathcal{N}\downarrow Z)$ and $(\mathcal{N}\uparrow Z)$ respectively. If the
category $\mathcal{M}$ has an initial objects $\emptyset$ then $\emptyset\to
Z$ is initial in $(\mathcal{N}\downarrow X)$ and $1_{Z}$ is terminal. If
$\mathcal{N}$ has a terminal object $1$ then $Z\to 1$ is terminal in
$(\mathcal{N}\uparrow Z)$ and $1_{Z}$ is initial.
There are model structures on these categories in which a morphism is defined
to be a weak equivalence, cofibration or fibration if its image under the
forget functor to $\mathcal{N}$ is in the corresponding class. Moreover, the
property of being left proper, right proper or proper is inherited from the
model category $\mathcal{N}$ [Hi2, Theorem 1.7.(3), Theorem 2.8.(3)]. Pushouts
and pullbacks in $(\mathcal{N}\downarrow Z)$ and $(\mathcal{N}\uparrow Z)$ are
formed in $\mathcal{M}$ [Hi2, Lemma1.3., Lemma 2.3.].

######  Definition 4.1.

Let $\mathcal{N}$ be a category and $X\in\mathcal{N}$. The category objects
coaugmented under $Z$ is defined as

$\mathcal{N}_{Z}:=((\mathcal{N}\downarrow X)\uparrow 1_{Z})$

Objects of $\mathcal{N}_{Z}$ are pairs of maps $Z\to Y\to Z$ which compose to
$1_{Z}$. In case $\mathcal{M}$ is a (proper) model category so is
$\mathcal{M}_{Z}$. The object $(1_{Z},1_{Z})$ is initial and terminal in
$\mathcal{N}_{Z}$. For short we denote it by $X$.

######  Lemma 4.2.

The restricted cube axiom holds in $(c\mathcal{M}^{\mathcal{F}}\downarrow Z)$,
$(c\mathcal{M}^{\mathcal{G}}\uparrow Z)$ and in
$c\mathcal{M}_{Z}^{\mathcal{F}}$ for each $Z\in c\mathcal{M}^{\mathcal{F}}$.

Proof:  This follows from 3.4 and the fact all the relevant structure is
defined by the forget functor to $c\mathcal{M}^{\mathcal{F}}$.$\square$ Recall
that one says that homotopy pushouts are universal in an $\infty$-category if
homotopy pushouts are stable under homotopy pullbacks.

######  Lemma 4.3.

Homotopy pushouts along pairs of maps one of which is $0$-connected are
universal in $c\mathcal{M}^{\mathcal{F}}$.

Proof: The proof of [D.H, Lemma 2.5.] applies word for word.$\square$

The following result which is key for us can be found except for the items (4)
and (6) in [D.H].

######  Theorem 4.4.

Let $\mathcal{M}$ be an $\infty$-category with finite limits, pushouts in
which the cube axiom holds.
Let $X$ and $Y$ be pointed objects in $\mathcal{M}_{*}$ and $F\to E\to X$ a
fibration sequence. Then all the assertions made in 1.3 hold true in
$\mathcal{M}$ where for (2) and (3) we assume in addition that countable
coproducts exist in $\mathcal{M}$.$\square$

Proof:  The assertions in (1), (2), (3), (5) and (7) are proved in 2.21.2,
2.10., 3.2., 2.24 and 3.5. of [D.H] respectively. For (4) note that by [D.H,
Lemma 2.5.] pushouts are universal in $\mathcal{M}$. Point (4) follows from an
application of this universality to the diagram

$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\vee
X}$

and the two maps $X\vee X\stackrel{{\scriptstyle(1,1)}}{{\to}}X$ and $\ast\to
X$. For (6) apply universality of pushouts to

$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\vee
X}$

and the two maps $Y\vee X\stackrel{{\scriptstyle(1,\ast)}}{{\to}}Y$ and
$\ast\to X$. $\square$

######  Remark 4.5.

In the context of model categories, (3) and (5) were proved by Doerane in
[Do2] as we learned from [R.S.] where a version of (4) is given.

Now we apply 4.4 to the model category $c\mathcal{M}_{Z}^{\mathcal{F}}$. Since
it is model category enriched over pointed simplicial sets $\mathcal{S}_{*}$
the categories of cofibrant and fibrant objects define pointed
$\infty$-categories in the model of simplicial categories. We denote the
underlying pointed $\infty$-categories by $cM_{Z}^{\mathcal{F}}$. They have
all small limits and colimits. The restricted cube axiom holds in
$(cM^{\mathcal{F}})_{Z}$ by 3.4 and 4.2.

Proof:  of (1.3).
One simply checks that under the connectivity assumptions made the proofs in
[D.H] go through in the $\infty$-category $cM_{Z}^{\mathcal{F}}$ since the
restricted cube axiom is sufficient in all applications of the cube axiom
made. For (4) and (6) one uses 4.3.$\square$

##  5 Connectivity estimates

In this section we provide the results on cosimplicial connectivity needed in
the proof of the Hilton-Milnor theorem. The proofs flow from three sources.
First the fact that colimits in $\mathcal{CA}_{Z}^{E}$ are formed in the
underlying cosimplicial vector spaces. Second 2.6 which enables us to reduce
the proofs in $\mathcal{S}_{Z}^{G}$ to the ones in $\mathcal{CA}_{Z}^{E}$.
Third the dual Blakers-Massey theorem which replaces the generalized Blakers-
Massey theorem of [A.B.F.J.] valid in $\infty$-topoi.

Recall that finite products in $\mathcal{CA}_{Z}^{E}$ are defined by the
cotensor product over $Z$. Infinte products are described as the filtered
limit of all finite subproducts. Now filtered limits are given as the colimit
of all unstable subcoalgebras of finite dimension in the limit of the
underlying vector spaces. So all in all

$\square_{I}C_{i}=colim_{D_{a}}lim^{Vec}_{J}(\square)_{J}C_{j}$

where $J$ runs through the finite subsets of $I$ and the colimit over the
finite dimensional subcoalgebras $D_{a}$ in the vector space limit. This may
be seen as in the absolute case to be found for example in [G.4, 1.1.b].
Another way to the construction of infinite products is via the cofree functor
right adjoint to the forget functor $I$:

$I:\mathcal{CA}_{C}\rightleftarrows(Vec\uparrow J(C)):G_{C}$

as in [A].

######  Proposition 5.1.

Let $I$ be a discrete category and $D_{i}\in c\mathcal{M}_{Z}^{\mathcal{F}}$
fibrant and $D_{i}\to Z$ s-connected for all i. Then $\square_{I}D_{i}\to Z$
is s-connected.

Proof:  First note that finite products $c\mathcal{CA}_{C}$ are defined by the
degreewise cotensor product over $C$ and this functor has the asserted
property by the spectral sequence [BRS, Theorem 4.5.1 (b)]. Since $D_{i}$ is
fibrant there is an isomorphism

$D_{i}^{n}\cong Z^{n}\otimes E_{i}^{n}$

with $E_{i}\in\mathcal{E}$ and consequently an isomorphism

$\square_{Z^{n}}(D^{n}_{i},D^{n}_{j})\cong E^{n}_{i}\otimes E_{j}^{n}\otimes
Z^{n}.$

So the limit under consideration is just an ordinary infinite tensor product.
The structure maps of the system in degree $n$ are given by the projections

$E^{n}_{i_{1}}\otimes\ldots\otimes E_{i_{k}}^{n}\otimes Z^{n}\to
E^{n}_{i_{1}}\otimes\ldots\otimes\hat{E^{n}_{i_{s}}}\otimes\ldots\otimes
E_{i_{k}}^{n}\otimes Z^{n}$

which are clearly surjective. By [BRS, Proposition 3.4.3] we may assume that
$E_{i}^{n}=\mathbb{F}$ for $n\leq s$. Consequently $D_{i}^{n}=Z^{n}$ for
$n\leq s$. By [We, p.84] there is a short exact sequence for each r

$Lim^{Vec,1}\pi^{r-1}(\otimes_{J}E_{j}^{*}\otimes
Z^{*})\to\pi^{r}(Lim^{Vec}(\otimes_{J}E_{j}^{*}\otimes Z^{*})\to
Lim^{Vec}\pi^{r}(\otimes_{J}E_{j}^{*}\otimes Z^{*}).$

For $r-1\leq s$ the derived functor is trivial since in this case the system
is constant. So there are isomorphisms for $r\leq s$

$\pi^{r}(Lim^{cVec}(\otimes_{J}E_{j}\otimes Z))\cong
Lim^{Vec}\pi^{r}(\otimes_{J}E_{j}\otimes Z)\cong\pi^{r}(Z)$

The actual limits in coalgebras are then defined by the colimits over the
filtered system of finite dimensional subcoalgebras. Colimits are computed in
the underlying cosimplicial vector spaces or equivalently cochain complexes.
Hence, this colimit commutes with $\pi^{*}(-)$. This implies the assertion in
$c\mathcal{CA}^{\mathcal{E}_{C}}$ because every unstable coalgebra is the
colimit of its finite subcoalgebras .
The claim in $c\mathcal{S}^{\mathcal{G}}$ may be reduced to the case just
proved as follows. By the same argument as above the products which show up
are the ordinary products of the form $(\times_{i\in I}G_{i}^{n})\times Z^{n}$
with $G_{i}\in\mathcal{G}$. We have to estimate the connectivity of the
functor $\pi^{*}H_{\ast}$ applied to these products. Now recall the fact that
homology commutes with infinite products of Eilenberg-MacLane spaces of type
$\mathbb{F}$ and combine this with the assertion in
$c\mathcal{CA}^{\mathcal{E}}$ just seen. For finite characteristic this fact
is a consequence of [Bou3, 4.4.] and in characteristic zero it may be deduced
from the Milnor-Moore theorem as in the proof of [BRS, Theorem 4.2.2.]. This
completes the proof.$\square$

######  Lemma 5.2.

Let $X_{n}\in c\mathcal{M}_{Z}^{\mathcal{F}}$ with

$q_{n}:X_{n}\to Z$

$k_{n}$-connected such that $k_{n}\geq 0$ for all $n\in\mathbb{N}$. Then $\vee
X_{n}\to Z$ is at least $m$-connected with $m=min(k_{n}|n\in\mathbb{N})$.

Proof:  We may assume that the maps $i_{n}:Z\to X_{n}$ are cofibrations.
Consider the case of $c\mathcal{CA}^{\mathcal{E}}_{Z}$ first. The maps
$i_{n}:Z\to X_{n}$ are injective as they are $0$-connected cofibrations.
Consider the exact sequence of cochain complexes for $X_{1},X_{2}$.

$0\to Z\stackrel{{\scriptstyle(i_{1},-i_{2})}}{{\longrightarrow}}X_{1}\oplus
X_{2}\to X_{1}\oplus_{Z}X_{2}\to 0$ (5.0.1)

Because

$H^{*}(i_{j}):H^{*}(Z)\to H^{*}(X_{j})$

is injective for $j\leq 2$ the long exact cohomology sequence of 5.0.1 of
decomposes in short exact sequences. Since $H^{k_{j}\leq}(i_{j})$ is an
isomorphism it follows that there is an isomorphism

$(i_{1}\oplus i_{2})^{*}:H^{\leq m}(Z)\cong H^{\leq m}(X_{1}\oplus_{Z}X_{2}).$

By an easy induction the assertion holds for any finite subset of
$\mathbb{N}$.
Now there is an isomorphism

$\oplus X_{n}\cong colim_{|J|<\infty}(\oplus)_{J}{X_{j}}$

and since filtered colimits are exact the assertion holds in
$c\mathcal{CA}^{\mathcal{E}}_{Z}$.
We turn to the case $c\mathcal{S}^{\mathcal{G}}_{Z}$. By [BRS, Proposition
6.1.7] the statement for finite subsets of the natural numbers hols there as
well. The infinite coproduct $c\mathcal{S}^{\mathcal{G}}_{Z}$ is again the
colimit of its finite subcoproducts and since singular homology commutes with
filtered colimits of cofibrations the assertion holds in
$c\mathcal{S}^{\mathcal{G}}_{Z}$.$\square$

We turn to smash powers now. The next result is a consequence of the homotopy
excision theorem in $c\mathcal{M}^{\mathcal{F}}$[BRS, Theorems 4.6.5, 6.2.6.].

######  Lemma 5.3.

Let $X,Y\in c\mathcal{M}_{Z}^{\mathcal{F}}$ with $X\to Z$ and $Y\to Z$ $k$ and
$l$ connected respectively. Then $X\wedge Y$ is $k+l+1$-connected as a pointed
object.

Proof:  We consider the case $c\mathcal{CA}_{Z}^{\mathcal{E}}$. Note that in
the abelian category of coaugmented cosimplicial $Z$-comodules $cComod(Z)_{Z}$
coproducts and products coincide. Consider

$X\oplus Y\stackrel{{\scriptstyle i}}{{\to}}X\square Y\stackrel{{\scriptstyle
q}}{{\to}}X\oplus Y$

where the map on the left is the canonical inclusion and the map on the right
is the canonical map of $Z$-comodules. The composition is the identity and $q$
is $k+l+1$-connected by [BRS, 4.6.5. b)]. It follows that $i$ is
$k+l+1$-connected as well. By the homotopy pushout square

$\textstyle{X\oplus
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\square
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\wedge
Y}$

$X\wedge_{Z}$ is $k+l+1$-connected as a pointed object which was to be seen.
We consider the case $c\mathcal{S}^{\mathcal{G}}$. Since $H_{*}$ commutes with
homotopy pullbacks [BRS, 6.2.3.] there are isomorphisms

$H_{*}(X\times Y)\cong H_{*}(X)\square H_{*}(Y).$

Since $H_{*}$ commutes with homotopy pushouts along pairs of maps one of which
is $0$-connected 2.6, there are isomorphisms

$H_{*}(X\vee Y)\cong H_{*}(X)\oplus H_{*}(Y).$

Apply the same fact again to the homotopy pushout square

$\textstyle{X\vee
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\times
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\wedge
X}$

and find an isomorphism

$H_{*}(Y\wedge X)\cong H_{*}(X)\wedge H_{*}(Y).$

Now the assertion follows from the proved statement in
$c\mathcal{CA}^{\mathcal{E}}_{Z}$.$\square$

######  Lemma 5.4.

Let $X\in c\mathcal{M}^{\mathcal{F}}_{Z}$ and assume that $X\to Z$ is
$k$-connected. Then $\Omega\Sigma(X)\to Z$ is also $k$-connected.

Proof:  We start as usual by considering $c\mathcal{CA}^{\mathcal{E}}_{Z}$.
The assertion is true in the abelian category $cComod(Z)_{Z}$ since the loop
and suspension functors $\Omega^{Comod}$ and $\Sigma^{Comod}=\Sigma$ are
inverse to each other. By [BRS, Theorem 4.6.5. b)] the natural map

$\Omega\Sigma(X)\to\Omega^{Comod}\Sigma(X)$

is $(2k+1)$-connected. This proves the assertion in
$c\mathcal{CA}^{\mathcal{E}}_{Z}$.
It holds in $c\mathcal{S}^{\mathcal{G}}_{Z}$ by an application of
2.6.$\square$

##  6  The Hilton-Milnor theorem in $c\mathcal{M}^{\mathcal{F}}_{Z}$

We start with a preparatory lemma established by Gray in the category of
topological spaces [Gr].

######  Lemma 6.1.

Let $X,Y\in c\mathcal{M}^{\mathcal{F}}_{Z}$ with $X\to Z$ and $Y\to Z$
$0$-connected. Then there is an equivalence

$Y\times\Sigma X/Y\simeq\Sigma X\vee\Sigma(X\wedge Y)$

Proof:  The following diagram

$\textstyle{X\times
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma
X\times Y}$

is a homotopy pushout by 4.3 as is its composition with

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma
X\times
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma
X\vee\Sigma(X\wedge Y)}$

by [D.H, Corollary 2.21.1]. The pasting law for homotopy pushouts implies that
the second diagram is a homotopy pushout as well.$\square$

######  Corollary 6.2.

Let $\Sigma X,Y\in c\mathcal{M}_{Z}^{\mathcal{F}}$ be $1$-connected as pointed
objects. Then there is a homotopy fiber sequence

$\Sigma X\vee\Sigma(X\wedge\Omega Y)\to\Sigma X\vee Y\to Y$

Proof:  This follows from 6.1 together with (6) in 1.3 .$\square$

Using this we arrive at.

######  Lemma 6.3.

Let $\Sigma X,Y\in c\mathcal{M}_{Z}^{\mathcal{F}}$ be $1$-connected as pointed
objects. Then there is an equivalence

$\Omega(\Sigma X\vee Y)\simeq\Omega(Y)\times\Omega(\Sigma
X\vee\Sigma(X\wedge\Omega Y))$

Proof:  The equivalence is induced by the inclusion $Y\to\Sigma X\vee Y$ and
the map $\Sigma X\vee\Sigma(X\wedge\Omega Y)\to\Sigma X\vee Y$. The loops of
these maps can be multiplied in $\Omega(\Sigma X\vee Y)$ which is a group
object in the homotopy category of $c\mathcal{M}_{Z}^{\mathcal{F}}$ and this
map is the searched for equivalence.
$\square$

As a consequence of the James splitting in 1.3 we get:

######  Lemma 6.4.

Let $\Sigma X,\Sigma Y\in c\mathcal{M}_{Z}^{\mathcal{F}}$ be $1$-connected as
pointed objects. Then there is an equivalence

$\Omega(\Sigma X\vee\Sigma
Y)\simeq\Omega\Sigma(X)\times\Omega\Sigma(\bigvee_{0\leq i<\infty}Y\wedge
X^{\wedge i})$

The proof of the Hilton-Milnor theorem below proceeds from here on essentially
as in the classical sources [Mi], [Wh] but amplified by the connectivity
estimates proved in section 5. These are needed in order to control the
convergence of a series of maps which define the asserted equivalence.

We recall the classical notion of basic products on $k$ generators.

######  Definition 6.5.

Let $T$ be the free non-associative algebra on $k$ generators
$x_{1},\ldots,x_{k}$. Basic products of weight $w$ and rank $r$ are given
inductively. We define $x_{1},\ldots,x_{k}$ to be the basic products of weight
$1$ and rank $0$. Suppose the basic products of weight $n$ have been found as
$x_{1},\ldots,x_{m}$ together with a rank function $r$ which satisfies
$r(x_{i})<i$. The basic products of weight $n+1$ are formed as all
$x_{i}x_{j}$ with $w(x_{i})+w(x_{j})=n+1$ and $r(x_{i})\leq j<i$. Order them
in any way $x_{m+1},\ldots,x_{l}$ and extend the function $r$ as $r(x_{h})=j$
for $x_{h}=x_{i}x_{j}$.

For more details we refer to [Wh]. By a theorem of Hall [Ha] the basic
products form a basis for the free Lie ring on generators
$x_{1},\ldots,x_{k}$.

######  Definition 6.6.

Let $w=x_{m}$ be the m-th basic product. Define

$w(X,Y)=X_{m}=X^{a_{x}}\wedge Y^{a_{y}}$

where $a_{x}$ is the number of occurrences of $x$ in the word $x_{m}$ and
$a_{y}$ the number of occurrences of $y$.

Proof: of the Hilton-Milnor theorem 1.4.
We define

$R_{m}=\bigvee_{\stackrel{{\scriptstyle i\geq m}}{{r(x_{i})<m}}}X_{i},\quad
R^{{}^{\prime}}_{m}=\bigvee_{\stackrel{{\scriptstyle
i>m}}{{r(x_{i})<m}}}X_{i}$

, so that $R_{m}=X_{m}\vee R^{{}^{\prime}}_{m}$. We have equivalences

$\Omega\Sigma(X_{m}\vee
R^{{}^{\prime}}_{m})\simeq\Omega\Sigma(X_{m})\times\Omega\Sigma(\bigvee_{{i\geq
0}}(R^{{}^{\prime}}_{m}\wedge X_{m}^{\wedge i})\simeq$
$\Omega\Sigma(X_{m})\times\Omega\Sigma(\bigvee_{i\geq
0}\bigvee_{\stackrel{{\scriptstyle j>m}}{{r(w_{j})<m}}}(X_{j}\wedge
X_{m}^{\wedge i})\simeq\Omega\Sigma(X_{m})\times\Omega\Sigma(R_{m+1})$

by 6.4 and definition of $R^{{}^{\prime}}_{m}$ where the last equivalence
holds by a combinatorical identity for basic products to be found in [Wh, 6.1.
p.512].
For $x_{m}$ a basic monomial of weight $w$ the connectivity of $X_{m}$ is
bigger or equal to $w-1$ by 5.3. As there are only finitely many basic
monomials of a given weight, the connectivity of $X_{m}$ tends to infinity
with $m$.
Due to 5.2 the connectivity of $R_{m}$ tends with $m$ to infinity as well.
Let

$f_{m}:\Omega\Sigma(X_{m})\times\Omega\Sigma(R_{m+1})\to\Omega\Sigma(R_{m})$

be a weak equivalence. Let $g_{m}=1\times f_{m}$ then

$g_{m}:(\Omega\Sigma(X_{1})\times\ldots\times\Omega\Sigma(X_{m-1}))\times\Omega\Sigma(X_{m})\times\Omega\Sigma(R_{m+1})\to(\Omega\Sigma(X_{1})\times\ldots\times\Omega\Sigma(X_{m-1}))\times\Omega\Sigma(R_{m})$

is a weak equivalence as follows from the spectral sequence of [BRS, Theorem
4.5.1 (b)].
Let $h_{m}=g_{1}\circ g_{2}\circ\ldots\circ
g_{m}|:\Omega\Sigma(X_{1})\times\ldots\times\Omega\Sigma(X_{m})$.
Because the pointed object $P=\prod_{\stackrel{{\scriptstyle
m=1}}{{}}}^{\infty}\Omega\Sigma(X_{m})$ is filtered by the objects
$P_{n}=\prod_{\stackrel{{\scriptstyle i=1}}{{}}}^{n}\Omega\Sigma(X_{i})$ and
$h_{n}|P_{n-1}=h_{n-1}$ the collection of $h_{m}$ define a map

$h:P\to\Omega\Sigma(R_{1})=\Omega\Sigma(X\vee Y).$

We show that $h$ induces an isomorphism on $\pi^{r}()$ for each $r$. Choose
$N$ such that $X_{m}$ and $R_{m+1}$ are $r$-connected for all $m\geq N$.
Consider the obvious maps

$i_{N}:P_{N}\to P$

and

$h_{N}:P_{N}\to P_{N}\times\Omega\Sigma(R_{N+1}).$

As a consequence of 5.1, 5.4 and an application of the Künneth spectral
sequence , they induce isomorphisms on $\pi^{r}(-)$. Inspection of the
commutative diagram

$\textstyle{\pi^{r}(P_{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{r}_{N}}$$\scriptstyle{i^{r}_{N}}$$\textstyle{\pi^{r}(P_{N}\times\Omega\Sigma(R_{n+1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{r}_{N}}$$\textstyle{\pi^{r}(P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{r}}$$\textstyle{\pi^{r}(\Omega\Sigma(X\vee
Y))}$

gives the assertion.$\square$

######  Remark 6.7.

Porter has extended the classical Hilton-Milnor theorem by decomposing the
loop space of the fat wedge of suspension spaces [P]. Such an extension also
holds in $c\mathcal{M}_{Z}^{\mathcal{F}}$.

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Manfred Stelzer, Universität Osnabrück.