# Equivariant Bordism of 2-Torus Manifolds and Unitary Toric Manifolds

Bo Chen, Zhi Lü, Qiangbo Tan  School of Mathematics and Statistics, Huazhong
University of Science and Technology, Wuhan, China<EMAIL_ADDRESS>School
of Mathematical Science, Fudan University, Shanghai, China<EMAIL_ADDRESS>College of Science, Wuhan University of Science and Technology, Wuhan, China
<EMAIL_ADDRESS>
###### Abstract.

The equivariant bordism classification of manifolds with group actions is an
essential subject in the study of transformation groups. We are interesting in
the action of 2-torus group $\mathbb{Z}_{2}^{n}$ and torus group $T^{n}$, and
study the equivariant bordism of 2-torus manifolds and unitary toric
manifolds. In this paper, we give a new description of the group
$\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n})$ of 2-torus manifolds, and determine the
dimention of $\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n})$ as a
$\mathbb{Z}_{2}$-vector space. With the help of toric topology, Lü and Tan
proved that the bordism groups $\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n})$ are
generated by small covers. We will give a new proof to this result. These
results can be generalized to the equivariant bordism of unitary toric
manifolds, that is, we will give a new description of the group
$\mathcal{Z}_{n}^{U}(T^{n})$ of unitary torus manifolds, and prove that
$\mathcal{Z}_{n}^{U}(T^{n})$ can be generated by quasitoric manifolds with
omniorientations.

###### Key words and phrases:

Equivariant Bordism, 2-Torus Manifold, Unitary Toric Manifold, Small Cover,
Quesitoric Manifold

This work is supported by grants from NSFC11801379

##  1\. Introduction

An $n$-dimensional 2-torus manifold is a smooth closed $n$-dimensional
manifold equipped with an effective smooth $\mathbb{Z}_{2}^{n}$-action, so its
fixed point set is empty or consists of isolated points (see [12, 13]). A
unitary $T^{n}$-manifold is an oriented closed smooth manifold $M^{m}$ with an
effective $T^{n}$-action such that its tangent bundle admits a
$T^{n}$-equivariant stable complex structure, where $T^{n}=(S^{1})^{n}$ is the
compact torus group of dimention $n$. A unitary $T^{n}$-manifold $M^{2n}$ of
dimension 2n with nonempty fixed set is called a unitary torus manifold or
unitary toric manifold (see [15, 10]). All the group actions in this article
are assumed to be effective.

Davis and Januszkiewicz [7] developed the theory of small covers and
quasitoric manifolds, which are topological versions of real toric varieties
and toric varieties, respectively. A small cover is a special case of 2-torus
manifolds. A quasitoric manifold can be equipped with an omniorientation
structure, this omniorientation determines a stable complex structure (see
[2]). So a quasitoric manifold with an omniorientation is a special case of
unitary toric manifolds.

In Section 2, we will focus on the unoriented equivarint bordism of 2-torus
manifolds. In the 1960s Conner and Floyd [4] begun the study of unoriented
equivarint bordism theory of $\mathbb{Z}_{2}^{n}$ manifolds, they introduced a
graded commutative algebra over $\mathbb{Z}_{2}$ with unit,
$\mathcal{Z}_{*}(\mathbb{Z}_{2}^{n})=\bigoplus_{m\geq
0}\mathcal{Z}_{m}(\mathbb{Z}_{2}^{n})$, where the equivariant bordism group
$\mathcal{Z}_{m}(\mathbb{Z}_{2}^{n})$ is consists of
$\mathbb{Z}_{2}^{n}$-equivariant unoriented bordism classes of all
$m$-dimensional 2-torus manifolds with effective action and fixing finite set.
The addition and multiplication on $\mathcal{Z}_{*}(\mathbb{Z}_{2}^{n})$ are
defined to be the disjoint union and the cartesian product with diagonal
$\mathbb{Z}_{2}^{n}$-actions, respectively. For $n\leq 2$, Conner and Floyd
showed that $\mathcal{Z}_{*}(\mathbb{Z}_{2})\cong Z_{2}$ and
$\mathcal{Z}_{*}(\mathbb{Z}_{2}^{2})\cong\mathbb{Z}_{2}[u]$, where $u$ denotes
the class of $\mathbb{R}P^{2}$ with the standard $\mathbb{Z}_{2}^{2}$-action.
The equivariant unoriented bordism group of all $n$-dimensional 2-torus
manifolds is $\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n})$, which was also denoted by
$\mathfrak{M}_{n}$ in [12]. The group $\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n})$
was determined for $n=3$ by Lü [12] and for general $n\geq 3$ by Lü and Tan
[14]. As showed in [14], $\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n})$ is generated by
all small covers of dimention $n$, and each class of
$\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n})$ contains a small cover as its
representative. A computer algorithm was given there to search for the
generators of $\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n})$ and calculate the
dimension $\text{dim}_{\mathbb{Z}_{2}}(\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n}))$.
For $n>4$, the datas are too complex for the algorithm to work out. How to
calculate $\text{dim}_{\mathbb{Z}_{2}}(\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n}))$
remains a problem.

As stated by Davis and Januszkiewicz in [7], there is a universal complex
$X(\mathbb{Z}^{n})$ such that every small cover can be obtained as a pullback
from $X(\mathbb{Z}_{2}^{n})$ via a suitable map. To prove that
$\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n})$ is generated by all small covers, Lü and
Tan [14] defined a boundary operator $d_{*}$ on the dual of the Conner–Floyd
algebra $\mathcal{R}_{*}(\mathbb{Z}_{2}^{n})$, where
$\mathcal{R}_{*}(\mathbb{Z}_{2}^{n})=\bigoplus_{m\geq
0}\mathcal{R}_{m}(\mathbb{Z}_{2}^{n})$ is the graded polynomial algebra over
$\mathbb{Z}_{2}$ generated by all irreducible
$\mathbb{Z}_{2}^{n}$-representations. Then the existence theorem of tom Dieck
is equivalent to the condition $d_{*}f^{*}=0$ ([14] Theorem 7.2). The
differential $d_{*}$ is almost the same as the differential $\partial_{*}$ on
the chain complex $(C_{*}(X(\mathbb{Z}_{2}^{n})),\partial_{*})$ with
coefficient $\mathbb{Z}_{2}$, and $f^{*}$ corresponds to a closed chain, which
represents an homology element in
$H_{n-1}(X(\mathbb{Z}_{2}^{n});\mathbb{Z}_{2})$. With these observations, we
will show the following main theorem for the equivariant bordism of 2-torus
manifolds.

######  Theorem 1.1.

For each $n\geq 1$, the equivariant bordism group
$\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n})$ of 2-torus manifolds is isomorphic to
the homology group $H_{n-1}(X(\mathbb{Z}_{2}^{n});\mathbb{Z}_{2})$ of the
universal $\mathbb{Z}_{2}^{n}$-complex.

$\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n})\cong
H_{n-1}(X(\mathbb{Z}_{2}^{n});\mathbb{Z}_{2})$

It has already been shown in [1] that the homotopy type of
$X(\mathbb{Z}_{2}^{n})$ is the wedge of $A_{n}$ spheres $S^{n-1}$, where
$A_{n}$ is a number related to the dimension $n$. So as a corollary, the
dimention
$\text{dim}_{\mathbb{Z}_{2}}(\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n}))=A_{n}$ can
be given straightforward, this solve the problem left in [14].

######  Corollary 1.2.

The dimention of $\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n})$ as a linear space over
$\mathbb{Z}_{2}$ is

$\dim_{\mathbb{Z}_{2}}(\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n}))=A_{n}=(-1)^{n}+\sum_{i=0}^{n-1}(-1)^{n-1-i}\frac{(2^{n}-2^{i})\cdots(2^{n}-2^{0})}{(i+1)!}$

The homotopy type of $X(\mathbb{Z}_{2}^{n})$ and the isomorphism Theorem 1.1
inspire a new proof to a main theorem in [14].

######  Corollary 1.3 ([14], Theorem 2.5).

The group $\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n})$ is generated by equivariant
unoriented bordism classes of small covers of dimention $n$.

In Section 3, we will deal with the the equivariant bordism of unitary toric
manifolds. There have been many different ways to study the equivariant
unitary bordism (see [3, 5, 8, 11, 16, 17, 9]). However, the explicit ring
structure is still difficult to calculate. Let
$\mathcal{Z}_{*}^{U}(T^{n})=\bigoplus_{m\geq 0}\mathcal{Z}_{m}^{U}(T^{n})$
denote the equivariant unitary bordism ring of unitary $T^{n}$-manifolds with
finite fixed point set, then $\mathcal{Z}_{n}^{U}(T^{n})$ is exactly the
equivariant unitary bordism group of unitary toric manifolds. By analogy with
the 2-torus manifolds, Darby [6] generalized the methods in [14] to the study
of $\mathcal{Z}_{n}^{U}(T^{n})$. To deal with the effect of orientations and
stably complex structures of the unitary manifolds, Darby considered the
exterior algebra $\Lambda_{\mathbb{Z}}^{*}(J_{n}^{\mathbb{C}})$ which is
generated by the exterior product of irreducible $T^{n}$-representations. The
boundary operator $d_{*}$ can be defined on the dual of
$\Lambda_{\mathbb{Z}}^{*}(J_{n}^{\mathbb{C}})$, and a similar existence
theorem ([6] Theorem 8.5) with the condition $d_{*}f^{*}=0$ still holds. But
the unitary case turns out to be more difficult since the existence of an
infinite number of irreducible representations. For $n=1$ and 2, the group
$\mathcal{Z}_{n}^{U}(T^{n})$ was shown to be generated by quasitoric manifolds
with omniorientations, and each class of $\mathcal{Z}_{n}^{U}(T^{n})$ contains
a quasitoric manifold with omniorientation as its representative. But for
$n>2$, it can only be conjectured that each class of
$\mathcal{Z}_{n}^{U}(T^{n})$ contains a quasitoric manifold with
omniorientation as its representative (see [6]).

There exists a universal complex $X(\mathbb{Z}^{n})$ characterizing the
quasitoric manifolds with omniorientations ([1] Proposition 2.1). As in the
2-torus case, the differential $d_{*}$ on the dual of
$\Lambda_{\mathbb{Z}}^{*}(J_{n}^{\mathbb{C}})$ corresponds to the differential
$\partial_{*}$ on the chain complex $(C_{*}(X(\mathbb{Z}^{n}))$ with
coefficient $\mathbb{Z}$, and $f^{*}$ corresponds to a closed chain, which
represents an homology element in $H_{n-1}(X(\mathbb{Z}^{n});\mathbb{Z})$.
With these observations, we will show the following main theorem for
equivariant bordism of unitary toric manifolds.

######  Theorem 1.4.

For each $n\geq 1$, the equivariant bordism group
$\mathcal{Z}_{2n}^{U}(T^{n})$ of unitary torus manifolds is isomorphic to the
homology group $H_{n-1}(X(\mathbb{Z}^{n});\mathbb{Z})$ of the universal
$\mathbb{Z}^{n}$-complex $X(\mathbb{Z}^{n})$.

$\mathcal{Z}_{2n}^{U}(T^{n})\cong H_{n-1}(X(\mathbb{Z}^{n});\mathbb{Z})$

The homotopy type of $X(\mathbb{Z}^{n})$ was proved to be the wedge of
countably infinite spheres $S^{n-1}$ (see [1]). So for $n\geq 2$, we have
$\mathcal{Z}_{2}^{U}(T^{1})\cong\mathbb{Z}$ and $\mathcal{Z}_{2n}^{U}(T^{n})$
is isomorphic to a countably infinite direct sum of $\mathbb{Z}$ (Corollary
3.8). The homotopy type of $X(\mathbb{Z}^{n})$ and Theorem 1.4 inspire that
$\mathcal{Z}_{2n}^{U}(T^{n})$ is generated by quasitoric manifold with
omniorientations (see Proposition 3.13). The generalized connect sums
introduced by Buchstaber-Ray [2] and Darby [6] suggest that the sum of two
class $[M_{1}]$ and $[M_{2}]$ can still be represented by a quasitoric
manifold with omniorientations, if $[M_{1}]$ and $[M_{2}]$ are represented by
quasitoric manifolds with omniorientations. So we can prove the following
proposition and solve the conjecture that $\mathcal{Z}_{n}^{U}(T^{n})$ is
generated by quasitoric manifolds with omniorientations .

######  Proposition 1.5.

Each class of $\mathcal{Z}_{2n}^{U}(T^{n})$ contains an omnioriented
quasitoric manifold as its representative.

##  2\. Equivariant Bordism of 2-Torus Manifolds

Following [4], let $\mathcal{R}_{m}(\mathbb{Z}_{2}^{n})$ be the linear space
over $\mathbb{Z}_{2}$, generated by the isomorphism classes of all
$m$-dimensional real $\mathbb{Z}_{2}^{n}$-representations. The direct sum of
representations defines a multiplication on
$\mathcal{R}_{*}(\mathbb{Z}_{2}^{n})=\bigoplus_{m\geq
0}\mathcal{R}_{m}(\mathbb{Z}_{2}^{n})$ and makes it into a graded commutative
algebra over $\mathbb{Z}_{2}$, with unit the representation of degree 0.
$\mathcal{R}_{*}(\mathbb{Z}_{2}^{n})$ is called the Conner–Floyd unoriented
$\mathbb{Z}_{2}^{n}$-representation algebra. As pointed out in [12],
$\mathcal{R}_{*}(\mathbb{Z}_{2}^{n})$ is not the Grothendieck ring of
$\mathbb{Z}_{2}^{n}$-representations. Every nontrivial irreducible real
$\mathbb{Z}_{2}^{n}$-representation is 1-dimensional and has the form
$\tau_{\rho}:\mathbb{Z}_{2}^{n}\times\mathbb{R}\to\mathbb{R}$ with
$\tau_{\rho}(g,x)=(-1)^{\rho(g)}$ for
$\rho\in\text{Hom}(\mathbb{Z}_{2}^{n},\mathbb{Z}_{2})$, and $\tau_{\rho}$ is
trivial if $\rho(g)=0$ for all $g\in\mathbb{Z}_{2}^{n}$. Let
$J_{n}^{\mathbb{R}}=\text{Hom}(\mathbb{Z}_{2}^{n},\mathbb{Z}_{2})\setminus\\{0\\}$
and regard it as the set of all nontrivial irreducible real
$\mathbb{Z}_{2}^{n}$-representations, let $\mathbb{Z}_{2}[J_{n}^{\mathbb{R}}]$
denote the free polynomial algebra on $J_{n}^{\mathbb{R}}$ over
$\mathbb{Z}_{2}$, then $\mathcal{R}_{*}(\mathbb{Z}_{2}^{n})$ can be identified
with $\mathbb{Z}_{2}[J_{n}^{\mathbb{R}}]$.

The essential relation between the equivariant unoriented bordism ring
$\mathcal{Z}_{*}(\mathbb{Z}_{2}^{n})=\bigoplus_{m\geq
0}\mathcal{Z}_{m}(\mathbb{Z}_{2}^{n})$ and the Conner–Floyd unoriented
$\mathbb{Z}_{2}^{n}$-representation algebra
$\mathcal{R}_{*}(\mathbb{Z}_{2}^{n})$ is characterized by the Stong [18].

######  Theorem 2.1 (Stong).

The ring homomorphism

$\varphi_{*}:\mathcal{Z}_{*}(\mathbb{Z}_{2}^{n})\to\mathcal{R}_{*}(\mathbb{Z}_{2}^{n})$

defined by $[M]\mapsto\sum_{p\in M^{\mathbb{Z}_{2}^{n}}}[\tau_{p}M]$ is a
monomorphism as algebras over $\mathbb{Z}_{2}$, where $\tau_{p}M$ denotes the
real $\mathbb{Z}_{2}^{n}$-representation on the tangent space at the fixed
point $p\in M^{\mathbb{Z}_{2}^{n}}$.

For each fixed point $p\in M^{\mathbb{Z}_{2}^{n}}$, the tangent
$\mathbb{Z}_{2}^{n}$-representation $\tau_{p}M$ can be decomposed into a
direct sum of 1-dimensional irreducible real representations
$\tau_{p}M=\oplus_{j=1}^{n}\tau_{i,j}$, the set
$\\{[\tau_{p}M]\in\mathcal{R}_{*}(\mathbb{Z}_{2}^{n})|p\in
M^{\mathbb{Z}_{2}^{n}}\\}$ is called the fixed point data of $M$. By Stong’s
theorem, the equivariant bordism class $[M]$ in
$\mathcal{Z}_{m}(\mathbb{Z}_{2}^{n})$ is determined by its fixed point data.
But which polynomials in $\mathcal{R}_{*}(\mathbb{Z}_{2}^{n})$ are polynomial
from fixed point datas? In [8], tom Dieck proved an existence theorem, showed
that the elements in $\mathcal{R}_{*}(\mathbb{Z}_{2}^{n})$ are determined by
their equivariant characteristic numbers. In [14], Lü and Tan gave a simple
proof to this theorem, and formulated it into the following result in terms of
Kosniowski and Stong’s localization formula.

######  Theorem 2.2 (tom Dieck-Kosniowski-Stong).

Let $\\{\tau_{1},\dots,\tau_{l}\\}$ be a collection of $m$-dimensional
faithful $\mathbb{Z}_{2}^{n}$-representations in
$\mathcal{R}_{m}(\mathbb{Z}_{2}^{n})$. Then a necessary and sufficient
condition that $\tau_{1}+\dots+\tau_{l}\in Im(\varphi_{m})$ (or
$\\{\tau_{1},\dots,\tau_{l}\\}$ is the fixed point data of a
$\mathbb{Z}_{2}^{n}$-manifold $M^{m}$) is that for all symmetric polynomial
functions $f(x_{1},\dots,x_{m})$ over $\mathbb{Z}_{2}$,

(2.1)
$\sum_{i=1}^{l}\frac{f(\tau_{i})}{\chi^{\mathbb{Z}_{2}^{n}}(\tau_{i})}\in
H^{*}(B\mathbb{Z}_{2}^{n};\mathbb{Z}_{2})$

where $\chi^{\mathbb{Z}_{2}^{n}}(\tau_{i})$ denotes the equivariant Euler
class of $\tau_{i}$, which is a product of $m$ nonzero elements of
$H^{1}(B\mathbb{Z}_{2}^{n};\mathbb{Z}_{2})$, and $f(\tau_{i})$ means that
variables $x_{1},\dots,x_{m}$ in the function $f(x_{1},\dots,x_{m})$ are
replaced by those $m$ degree-one factors in
$\chi^{\mathbb{Z}_{2}^{n}}(\tau_{i})$.

If ${\tau_{1},\dots,\tau_{l}}$ is the fixed data of a
$\mathbb{Z}_{2}^{n}$-manifold $M^{m}$, then the polynomial 2.1 is exactly an
equivariant Stiefel–Whitney number of $M^{m}$. Although all elements of
$Im(\varphi_{*})$ can be characterized by the above localization formula, the
ring structure of $\mathcal{Z}_{*}(\mathbb{Z}_{2}^{n})$ is still far from
settled. It was showed by Conner and Floyd in [12] that
$\mathcal{Z}_{*}(\mathbb{Z}_{2})\cong\mathbb{Z}_{2}$, and
$\mathcal{Z}_{*}(\mathbb{Z}_{2}^{2})\cong\mathbb{Z}_{2}[u]$, where $u$ denotes
the class of $\mathbb{R}P^{2}$ with the standard
$(\mathbb{Z}_{2})^{2}$-action. But for $n>2$, the structures and generators of
$\mathcal{Z}_{*}(\mathbb{Z}_{2}^{n})$ are still unknown.

When $m=n$, the group $\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n})$ is exactly formed
by the classes of all $n$-dimensional 2-torus manifolds, and the graded ring
is denoted by
$\mathcal{Z}_{*}^{\mathbb{R}}=\bigoplus\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n})$.
The restriction of
$\varphi_{*}|_{\mathcal{Z}_{*}^{\mathbb{R}}}:\mathcal{Z}_{*}^{\mathbb{R}}\to\mathcal{R}_{*}(\mathbb{Z}_{2}^{n})$
is still a monomorphism. Suppose
$[M^{m}]\in\mathcal{Z}_{m}(\mathbb{Z}_{2}^{n})$ and $p\in
M^{\mathbb{Z}_{2}^{n}}$ is a fixed point, then the set
$\\{\tau_{p,1},\dots,\tau_{p,m}\\}$ contains a basis of $J_{n}^{\mathbb{R}}$.
If $m=n$ and $M^{n}$ is a 2-torus manifold, then the monomial
$\tau_{p}=\tau_{p,1}\cdots\tau_{p,n}$ corresponds to a faithful
representation, which means that $\\{\tau_{p,1},\dots,\tau_{p,n}\\}$ is a
basis of $J_{n}^{\mathbb{R}}$.

######  Definition 2.3.

A homogeneous polynomial $g=\sum_{i}\tau_{i,1}\cdots\tau_{i,m}$ of degree
$m\leq n$ in $\mathbb{Z}_{2}[J_{n}^{\mathbb{R}}]$ is called an essential
$\mathbb{Z}_{2}^{n}$-polynomial if for each $i$, the elements in
$\\{\tau_{i,1},\dots,\tau_{i,m}\\}$ are linearly independent in
$\text{Hom}(\mathbb{Z}_{2}^{n},\mathbb{Z}_{2})$. An essential
$\mathbb{Z}_{2}^{n}$-polynomial of degree $n$ is also called faithful.

For a 2-torus manifold $M$, the image $\varphi_{*}([M])=\sum_{p\in
M^{\mathbb{Z}_{2}^{n}}}[\tau_{p}M]\in\mathbb{Z}_{2}[J_{n}^{\mathbb{R}}]$ is a
faithful polynomial. The space $J_{n}^{\mathbb{R}}$ has a dual space
$J_{n}^{*\mathbb{R}}=\text{Hom}(\mathbb{Z}_{2},\mathbb{Z}_{2}^{n})\setminus\\{0\\}$,
and they are both isomorphic to $\mathbb{Z}_{2}^{n}$, the is defined by the
following composition of homomorphisms:

$<\cdot,\cdot>:J_{n}^{*\mathbb{R}}\times J_{n}^{\mathbb{R}}\to
Hom(\mathbb{Z}_{2},\mathbb{Z}_{2})\cong\mathbb{Z}_{2}$
$<\xi,\rho>=\rho\circ\xi$

Similarly, we can define essential and faithful
$\mathbb{Z}_{2}^{n}$-polynomials in $\mathbb{Z}_{2}[J_{n}^{*\mathbb{R}}]$. If
$g=\sum_{i}\tau_{i,1}\dots\tau_{i,n}$ is a faithful
$\mathbb{Z}_{2}^{n}$-polynomial in $\mathbb{Z}_{2}[J_{n}^{\mathbb{R}}]$, then
each basis $\\{\tau_{i,1},\cdots,\tau_{i,n}\\}$ has a dual basis
$\\{s_{i,1},\cdots,s_{i,n}\\}$, and they determine a unique homogeneous
polynomial $g^{*}=\sum_{i}s_{i,1}\dots s_{i,n}$ in
$\mathbb{Z}_{2}[J_{n}^{*\mathbb{R}}]$, which is called the dual
$\mathbb{Z}_{2}^{n}$-polynomial of $g$. It is obvious that $g$ is faithful in
$\mathbb{Z}_{2}[J_{n}^{\mathbb{R}}]$ if and only if $g^{*}$ is faithful in
$\mathbb{Z}_{2}[J_{n}^{*\mathbb{R}}]$.

In [14], Lü and Tan defined a differential operator $d$ on the graded ring
$\mathbb{Z}_{2}[J_{n}^{*\mathbb{R}}]$ as follows: for each monomial
$\rho_{1}\dots\rho_{k}\in\mathbb{Z}_{2}[J_{n}^{*\mathbb{R}}]$ with $k\geq 1$,
let

(2.2)
$d_{k}(\rho_{1}\dots\rho_{k})=\begin{cases}\sum_{i=1}^{k}\rho_{1}\dots\rho_{i-1}\hat{\rho_{i}}\rho_{i+1}\dots\rho_{k}&\text{if}\
k>1\vspace{2mm}\\\ 1&\text{if}\ k=1\end{cases}$

and $d_{0}(1)=0$, where the symbol $\hat{\rho_{i}}$ means that $\rho_{i}$ is
deleted. It is easy to check that $d^{2}=0$, so it can be regarded as a
boundary operator. For 2-torus manifolds, it was proved in [14] that the
existence Theorem 2.2 of tom Dieck is equivalent to the following.

######  Theorem 2.4 (Lü-Tan).

Let $g=\sum_{i}t_{i,1}\dots t_{i,n}$ be a faithful
$\mathbb{Z}_{2}^{n}$-polynomial in $\mathbb{Z}_{2}[J_{n}^{\mathbb{R}}]$. Then
$g\in Im\varphi_{n}$ if and only if $d_{n}(g^{*})=0$.

Let $0\leq m\leq n-1$, let $\mathcal{E}_{m}(J_{n}^{*\mathbb{R}})$ be the set
of all essential $\mathbb{Z}_{2}^{n}$-polynomials of degree $m+1$ in
$\mathbb{Z}_{2}[J_{n}^{*\mathbb{R}}]$. If we regard an element in
$J_{n}^{*\mathbb{R}}$ as a vertex, and an essential monomial
$\tau_{i,1}\cdots\tau_{i,m}$ as a simplex of dimention $m-1$, then the graded
polynomial ring $\mathcal{E}_{*}(J_{n}^{*\mathbb{R}})=\bigoplus_{0\leq m\leq
n-1}\mathcal{E}_{m}(J_{n}^{*\mathbb{R}})$ can be regarded as a chian complex
with chain map $d_{*}$. This point of view is similar to the definition of
universal $\mathbb{Z}_{2}^{n}$-space $X(\mathbb{Z}_{2}^{n})$, which was
introduced and used to characterized the small covers by Davis and
Januszkiewicz in [7].

######  Definition 2.5.

A subset $\\{v_{1},\dots,v_{m}\\}$ of $\mathbb{Z}_{2}^{n}$ is unimodular if
span$\\{v_{1},\dots,v_{m}\\}$ is a direct summand of $\mathbb{Z}_{2}^{n}$ with
dimension $m$. Note that a subset of a unimodular set is itself unimodular.
The collection of all unimodular subsets of $\mathbb{Z}_{2}^{n}$ forms a
simplicial complex $X(\mathbb{Z}_{2}^{n})$ on unimodular vertices, which is
called the universal $\mathbb{Z}_{2}^{n}$-complex.

As showed in [1], the universal complex $X(\mathbb{Z}_{2}^{n})$ is a pure
simplicial complex and an $(n-1)$-connected Cohen-Macaulay space. It is the
universal simplicial complex classifying small covers of dimention $n$.
Actually, it can be used to classify the equivariant bordism of 2-torus
manifolds as stated in Theorem 1.1 and we now give a proof to that theorem.

###### Proof of Theorem 1.1.

Let $0\leq m\leq n-1$. Let
$r:J_{n}^{*\mathbb{R}}=Hom(\mathbb{Z}_{2},\mathbb{Z}_{2}^{n})\to\mathbb{Z}_{2}^{n}$
be the standard isomorphim. For an essential monomial $s_{i}^{*}=s_{i,1}\cdots
s_{i,m}\in\mathcal{E}_{m}(J_{n}^{*\mathbb{R}})$, the set
$\\{r(s_{i,1}),\dots,r(s_{i,m})\\}$ is unimodular in $\mathbb{Z}_{2}^{n}$, it
spans a simplex of dimention $m-1$ in the universal complex
$X(\mathbb{Z}_{2}^{n})$, denoted by $r(s_{i}^{*})$. Let
$(C_{*}(X(\mathbb{Z}_{2}^{n})),\partial_{*})=\sum_{0\leq m\leq
n-1}(C_{m}(X(\mathbb{Z}_{2}^{n}),\partial_{m})$ be the chain complex of
$X(\mathbb{Z}_{2}^{n})$, then the map $r$ induces a homomorphism

$r_{m}:\mathcal{E}_{m}(J_{n}^{*\mathbb{R}})\to C_{m}(X(\mathbb{Z}_{2}^{n}))$

which maps an essential polynomial $g^{*}=\sum
s_{i}^{*}\in\mathcal{E}_{m}(J_{n}^{*\mathbb{R}})$ to a chain
$r_{m}(g^{*})=\sum r(s_{i}^{*})\in C_{m}(X(\mathbb{Z}_{2}^{n}))$. It is easy
to see that $r$ is an isomorphism of groups and the following is commutative:

$\textstyle{\mathcal{E}_{m}(J_{n}^{*\mathbb{R}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{m}}$$\scriptstyle{r_{m}}$$\textstyle{C_{m}(X(\mathbb{Z}_{2}^{n}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{m}}$$\textstyle{\mathcal{E}_{m-1}(J_{n}^{*\mathbb{R}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{m-1}}$$\textstyle{C_{m-1}(X(\mathbb{Z}_{2}^{n}))}$

where $d_{m}$ is the differential (2.2) defined on
$\mathbb{Z}_{2}[J_{n}^{*\mathbb{R}}]$, and $\partial_{m}$ is the differential
on the chain complex $C_{m}(X(\mathbb{Z}_{2}^{n}))$. Regard
$(\mathcal{E}_{*}(J_{n}^{*\mathbb{R}}),d)$ as a chian complex, then the above
homomorphism $r$ is a chain isomorphism, it induces an isomorphism of homology

$r_{m*}:H_{m}(\mathcal{E}_{*}(J_{n}^{*\mathbb{R}}))\cong
H_{m}(X(\mathbb{Z}_{2}^{n}))$

Let $g$ be a faithful $\mathbb{Z}_{2}^{n}$-polynomial in
$\mathbb{Z}_{2}[J_{n}^{\mathbb{R}}]$, then $g^{*}$ is a faithful
$\mathbb{Z}_{2}^{n}$-polynomial in $\mathbb{Z}_{2}[J_{n}^{*\mathbb{R}}]$. By
Theorem 2.1, the equivariant bordism group
$\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n})$ of 2-torus manifolds is isomorphic to
the image $Im\varphi_{n}$. By Theorem 2.4, $g\in Im\varphi_{n}$ if and only if
$d_{n}(g^{*})=0$, that is, $g^{*}$ is a closed chain in
$\mathcal{E}_{*}(J_{n}^{*\mathbb{R}})$, it represents a homology class
$[g^{*}]\in H_{n-1}(\mathcal{E}_{*}(J_{n}^{*\mathbb{R}}))\cong
H_{n-1}(X(\mathbb{Z}_{2}^{n}))$, so we have

$\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n})\cong Im\varphi_{n}\cong
H_{n-1}(\mathcal{E}_{*}(J_{n}^{*\mathbb{R}});\mathbb{Z}_{2})\cong
H_{n-1}(X(\mathbb{Z}_{2}^{n});\mathbb{Z}_{2})$

∎

The homotopy of the universal complex $X(\mathbb{Z}_{2}^{n})$ was determined
in [1], we recall their result to calculate the dimension of
$\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n})$, the proof is straightforward.

######  Theorem 2.6.

[1] The simplicial complex $X(\mathbb{Z}_{2}^{n})$ is homotopy equivalent to
the wedge of $A_{n}$ spheres $S^{n-1}$, where

$A_{n}=(-1)^{n}+\sum_{i=0}^{n-1}(-1)^{n-1-i}\frac{(2^{n-1}-2^{i})\cdots(2^{n}-2^{0})}{(i+1)!}$

###### Proof of Corollary 1.2.

The proof is straightforward by Theorem 1.1 and Theorem 2.6. ∎

It was proved in [14] that the generators of
$\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n})$ can be chosen to be small covers over
products of simplices. Small covers are important models for
$\mathbb{Z}_{2}^{n}$-manifolds, which are defined and studied by Davis and
Januszkiewicz in [7], they are topological versions of real toric varieties.
We recall their definitions and basic properties.

######  Definition 2.7.

An $n$-dimensional small cover $\pi:M^{n}\to P^{n}$ is a smooth closed
$n$-dimensional manifold $M^{n}$ with a locally standard
$\mathbb{Z}_{2}^{n}$-action such that its orbit space is homeomorphic to a
$n$-dimensional simple convex polytope $P^{n}$, here locally standard means
that the action is locally isomorphic to a faithful representation of
$\mathbb{Z}_{2}^{n}$ on $\mathbb{R}^{n}$.

Each small cover $\pi:M^{n}\to P^{n}$ determines a characteristic function
$\lambda:\mathcal{F}(P^{n})\to J_{n}^{*\mathbb{R}}$, where
$\mathcal{F}(P^{n})$ consists of all facets (i.e., $(n-1)$-dimensional faces)
of $P^{n}$. Given a vertex $v$ of $P^{n}$, since $P$ is simple, there are $n$
facets $F_{1},\dots,F_{n}$ in $\mathcal{F}(P^{n})$ such that
$v=F_{1}\cap\dots\cap F_{n}$, characteristic function has the property that
the elements $\\{\lambda(F_{1}),\dots,\lambda(F_{n})\\}$ are linearly
independent, so their product $\lambda_{v}=\lambda(F_{1})\cdots\lambda(F_{n})$
is essential, and the polynomial $f^{*}_{\lambda}(P)=\sum_{v\in
V(P)}\lambda_{v}$ is a faithful polynomial of degree $n$ in
$\mathbb{Z}_{2}[J_{n}^{*\mathbb{R}}]$. We call the map $\lambda$ a
$J_{n}^{*\mathbb{R}}$-coloring of $P^{n}$ here, and the $f^{*}_{\lambda}(P)$
is called the $J_{n}^{*\mathbb{R}}$-coloring polynomial  of $(P^{n},\lambda)$.

For a small cover $\pi:M^{n}\to P^{n}$, the image
$f(P^{n},\lambda)=\varphi_{n}([M])$ under the Stong homomorphism is a faithful
polynomial in $\mathbb{Z}_{2}[J_{n}^{\mathbb{R}}]$, and its dual is just the
$J_{n}^{*\mathbb{R}}$-coloring polynomial $f^{*}_{\lambda}(P)$. We recall the
result about the generators of $\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n})$, and give
a new proof to it.

###### Proof of Corollary 1.3.

By Theorem 2.6, the universal complex $X(\mathbb{Z}_{2}^{n})$ is homotopy to
the wedge of $A_{n}$ spheres $S^{n-1}$, we call such spheres basic spheres of
$X(\mathbb{Z}_{2}^{n})$. By Theorem 1.1, each equivariant bordism class of
$\mathcal{Z}_{n}(\mathbb{Z}_{2}^{n})$ is corresponding to an element in
$H_{n-1}(X(\mathbb{Z}_{2}^{n});\mathbb{Z}_{2})$, which are generalized by the
homology class of basic spheres. Since $X(\mathbb{Z}_{2}^{n})$ is pure, the
dual of a basic sphere $S^{n-1}$ is the boundary $\partial{P^{n}}$ of a simple
polytope $P^{n}$, and the unimodular subsets of $S^{n-1}$ induce a
$J_{n}^{*\mathbb{R}}$-coloring $\lambda$ on $P^{n}$. By Davis–Januszkiewicz’s
theory of small covers, the pair $(P^{n},\lambda)$ determines a unique small
cover $\pi:M^{n}\to P^{n}$ such that $\lambda$ is the characteristic function.
So the equivariant bordism class corresponding to the baisc sphere $S^{n-1}$
is represented by a small cover $M$. ∎

##  3\. Equivariant Bordism of Unitary Torus Manifolds

A unitary manifold $M$ is a smooth manifold endowed with a complex structure
on the stable tangent bundle of $M$. If the complex structure is given on the
tangent bundle $TM$ of $M$, M is called an almost complex manifold. Let $G$ be
a compact Lie group, if the action of $G$ preserves the tangential stably
complex structure of unitary manifold $M$, then $M$ is called a unitary
$G$-manifold.

Let $\Omega^{U,G}$ denote the equivariant unitary bordism ring of unitary
$G$-manifolds. There have been many different ways to study the equivariant
unitary bordism, however, the explicit ring structure of $\Omega^{U,G}$ is
still difficult to calculate, only partial results are known. In [8], tom
Dieck gave a commutative diagram

${\Omega_{*}^{U,G}}$${MU_{*}[e_{V},e_{V}^{-1},Y_{V,d}]}$${MU_{*}^{G}}$${MU_{*}[e_{V},e_{V}^{-1},Y_{V,d}]}$$\scriptstyle{\varphi_{\Omega}}$

where $MU_{*}^{T^{n}}$ is the homotopy theoretic equivariant unitary bordism
ring, and $MU_{*}$ is the ordinary homotopy theoretic unitary bordism ring,
$e_{V}$ are the Euler classes of nontrivial irreducible complex
$T^{n}$-representations V, $Y_{V,d}$ are the classes of degree $2d$ $(2\leq
d\leq\infty)$ represented by the $T^{n}$-bundle $E\otimes
V\to\mathbb{C}P^{d-1}$ with $E\to\mathbb{C}P^{d-1}$ being the hyperplane line
bundle. When $G=T^{n}$ is the compact torus of rank $n$, Hanke [9] showed that
the image $\text{Im}(\varphi_{\Omega})\subset MU_{*}[e_{V}^{-1},Y_{V,d}]$, and
proved that the following

${\Omega_{*}^{U,T}}$${MU_{*}[e_{V}^{-1},Y_{V,d}]}$${MU_{*}^{T}}$${MU_{*}[e_{V},e_{V}^{-1},Y_{V,d}]}$$\scriptstyle{\varphi_{\Omega}}$

is a pullback square and has all maps injective.

For a unitary $T^{n}$-manifold $M$, each component of the fixed point set
$M^{T}$ is again a unitary $T$-manifold. If $M$ has an isolated fixed point,
then $\text{dim}M$ is even and the tangent space $T_{p}M$ is a complex
$T^{n}$-representation. Let $\mathcal{Z}_{*}^{U}(T^{n})=\bigoplus_{m\geq
0}\mathcal{Z}_{m}^{U}(T^{n})$ denote the subring of $\Omega_{*}^{U,T}$
consists of all classes that can be represented by unitary $T^{n}$-manifolds
with finite fixed point set. In his Ph.D’s thesis [6], Darby refined Hanke’s
result to the following pull back square with all maps injective.

${\mathcal{Z}_{*}^{U}(T^{n})}$${MU_{*}[e_{V}^{-1}]}$${MU_{*}^{T}}$${MU_{*}[e_{V},e_{V}^{-1},Y_{V,d}]}$$\scriptstyle{\varphi_{\Omega}}$

Let $M^{2m}$ be a unitary $T^{n}$-manifold with finite fixed point set, for
each isolated fixed point $p\in M^{T}$, there are two orientations on the
tangent space $T_{p}M$: the one induced from the orientation of $M$ and the
other induced from the complex structure on $T_{p}M$. Define the sign
$\varepsilon(p)$ of $p$

(3.1) $\varepsilon(p)=\begin{cases}+1&\text{if these two orientations
coincide}\vspace{2mm}\\\ -1&\text{otherwise}\end{cases}$

The tangent space $T_{p}M$ is a complex $T^{n}$-representation and can be
decomposed into a direct sum of irreducible representations
$\tau_{p}M=\tau_{p,1}\oplus\dots\oplus\tau_{p,m}$. Let
$J_{n}^{\mathbb{C}}=\text{Hom}(T^{n},S^{1})\setminus\\{0\\}$ and regard it as
the set of all irreducible complex $T^{n}$-representations. Let
$\mathbb{Z}[J_{n}^{\mathbb{C}}]$ be the polynomial algebras on
$J_{n}^{\mathbb{C}}$ over $\mathbb{Z}$, each element in $J_{n}^{\mathbb{C}}$
is of degree 2. The set $\\{[\tau_{p}M]\in\mathbb{Z}[J_{n}^{\mathbb{C}}]|p\in
M^{T}\\}$ is called the fixed point data. The homomorphism $\varphi_{\Omega}$
restrict to $\mathcal{Z}_{*}^{U}(T^{n})$ can be written as

$\varphi_{\Omega}:\mathcal{Z}_{*}^{U}(T^{n})\to\mathbb{Z}[J_{n}^{\mathbb{C}}]$
$[M]\mapsto\sum_{p\in M^{T^{n}}}\varepsilon(p)\prod_{i=1}^{m}\tau_{p,i}$

By Darby’s pullback square above, the $\varphi_{\Omega}$ is injective. To
compare with Stong’s Theorem 2.1 in the 2-torus case, we restate the Darby’s
result in the following theorem.

######  Theorem 3.1 ([6]).

The ring homomorphism
$\varphi_{\Omega}:\mathcal{Z}_{*}^{U}(T^{n})\to\mathbb{Z}[J_{n}^{\mathbb{C}}]$
is a monomorphism as algebras over $\mathbb{Z}$.

Unitary toric manifolds was introduced by Masuda in [15], which is a special
case of unitary $T^{n}$-manifolds. The equivariant bordism group
$\mathcal{Z}_{n}^{U}(T^{n})$ is formed by the classes of all $2n$-dimensional
unitary toric manifolds, and the graded ring is
$\mathcal{Z}_{*}^{\mathbb{C}}=\bigoplus_{n\geq 0}\mathcal{Z}_{n}(T^{n})$. To
eliminate the sign $\varepsilon(p)$ of a fixed point $p$ and absorb the effect
of orientations in the image $\varphi_{\Omega}([M])$, consider the exterior
algebra $\Lambda_{\mathbb{Z}}^{*}(J_{n}^{\mathbb{C}})$ as in Darby [6].

######  Definition 3.2.

The exterior algebra $\Lambda_{\mathbb{Z}}^{*}(J_{n}^{\mathbb{C}})$ is the
quotient algebra
$\Lambda_{\mathbb{Z}}^{*}(J_{n}^{\mathbb{C}}):=\mathbb{Z}[J_{n}^{\mathbb{C}}]/I$,
where $I$ is the ideal generated by all elements of the form $\tau^{2}$ and
expressions of the form $\tau_{1}\tau_{2}+\tau_{2}\tau_{1}$. The exterior
product is $\tau_{1}\wedge\tau_{2}=\tau_{1}\tau_{2}\text{ mod }I$, and
$\Lambda_{\mathbb{Z}}^{*}(J_{n}^{\mathbb{C}})=\bigoplus_{k\geq
0}\Lambda_{\mathbb{Z}}^{k}(J_{n}^{\mathbb{C}})$ is a graded algebra such that
$\Lambda_{\mathbb{Z}}^{k}[J_{n}^{\mathbb{C}}]\wedge\Lambda_{\mathbb{Z}}^{l}[J_{n}^{\mathbb{C}}]\subset\Lambda_{\mathbb{Z}}^{k+l}[J_{n}^{\mathbb{C}}]$.

Suppose $M^{2n}$ is a unitary torus manifold and $p\in M^{T}$ is a fixed
point, the tangent $T^{n}$-representaion at $p$ can be decomposed as a product
of irreducible $T^{n}$-representations $\tau_{p}=\tau_{p,1}\dots\tau_{p,n}$.
By [15] Lemma 1.3 (1), the set $\\{\tau_{p,1},\dots,\tau_{p,n}\\}$ is a basis
of $J_{n}^{\mathbb{C}}$. For each fixed poit $p$, rearrange the basis
$\\{\tau_{p,1},\dots,\tau_{p,n}\\}$ if necessary such that

$\det[\tau_{p,1}\dots\tau_{p,n}]=\varepsilon(p)$

where the $n\times n$ integral matrix $[\tau_{p,1}\dots\tau_{p,n}]$ is formed
by taking the ith column to be the vector $\tau_{p,n}\in
J_{n}^{\mathbb{C}}\cong\mathbb{Z}^{n}$. Then $\varphi_{\Omega}$ induces a
homomorphism
$\varphi_{*}^{U}:\mathcal{Z}_{*}^{\mathbb{C}}\to\Lambda_{\mathbb{Z}}^{*}(J_{n}^{\mathbb{C}})$
such that

$\varphi_{*}^{U}([M])=\sum_{p\in
M^{T^{n}}}\tau_{p,1}\wedge\dots\wedge\tau_{p,n}$

By Theorem 3.1, the following result is straightforward.

######  Corollary 3.3 ([6]).

The ring homomorphism
$\varphi_{*}^{U}:\mathcal{Z}_{*}^{\mathbb{C}}\to\Lambda_{\mathbb{Z}}^{*}(J_{n}^{\mathbb{C}})$
is a monomorphism as algebras over $\mathbb{Z}$.

To describe the image of $\varphi_{*}^{U}$, we introduce the definitions of
essential and faithful polynomials as in the 2-torus case.

######  Definition 3.4.

An exterior polynomial $g=\sum_{i}\tau_{i,1}\wedge\cdots\wedge\tau_{i,m}$ of
degree $2m\leq 2n$ in $\mathbb{Z}_{2}[J_{n}^{\mathbb{C}}]$ is called an
essential exterior $T^{n}$-polynomial if for each $i$, the elements in
$\\{\tau_{i,1},\dots,\tau_{i,m}\\}$ are linearly independent in
$\text{Hom}(T^{n},S^{1})$. An essential exterior $T^{n}$-polynomial of degree
$2n$ is also called faithful.

For a unitary toric manifold $M^{2n}$, the image $\varphi_{*}^{U}([M])$ is a
faithful extorior polynomial. Let
$J_{n}^{*\mathbb{C}}=\text{Hom}(S^{1},T^{n})$ be the dual space of
$J_{n}^{\mathbb{C}}$, both $J_{n}^{\mathbb{C}}$ and $J_{n}^{*\mathbb{C}}$ are
isomorphic to $\mathbb{Z}^{n}$, where the dual homomorphism is defined by a
composition as in the 2-torus case. Similarly, we can define essential and
faithful $T^{n}$-polynomials in $\mathbb{Z}_{2}[J_{n}^{*\mathbb{C}}]$. Each
faithful polynomial $g$ in $\Lambda_{\mathbb{Z}}^{*}(J_{n}^{\mathbb{C}})$ has
a dual $g^{*}$ which is faithful in
$\Lambda_{\mathbb{Z}}^{*}(J_{n}^{*\mathbb{C}})$. A differential operator
$d_{*}$ on $\Lambda_{\mathbb{Z}}^{*}(J_{n}^{*\mathbb{C}})$ was defined in [6]
as follows: for each monomial
$\rho_{1}\wedge\dots\wedge\rho_{k}\in\Lambda[J_{n}^{*\mathbb{C}}]$ with $k\geq
1$, define

(3.2)
$d_{k}(\rho_{1}\dots\rho_{k})=\begin{cases}\sum_{i=1}^{k}(-1)^{i+1}\rho_{1}\dots\rho_{i-1}\hat{\rho_{i}}\rho_{i+1}\dots\rho_{k}&\text{if}\
k>1\vspace{2mm}\\\ 1&\text{if}\ k=1\end{cases}$

and $d_{0}(1)=0$, where the symbol $\hat{\rho_{i}}$ means that $\rho_{i}$ is
deleted. It is easy to check that $d^{2}=0$. For unitary torus manifolds, the
existence theorem similar to Theorem 2.4 can be stated as follows.

######  Theorem 3.5 ([6] Theorem 8.5).

Let $g=\sum_{i}t_{i,1}\wedge\dots\wedge t_{i,n}$ be a faithful extorior
$T^{n}$-polynomial in $\Lambda_{\mathbb{Z}}^{*}(J_{n}^{\mathbb{C}})$. Then
$g\in Im\varphi_{n}^{U}$ if and only if $d(g^{*})=0$.

Let $0\leq m\leq n-1$, let $\mathcal{E}_{m}(J_{n}^{*\mathbb{R}})$ be the set
of all essential extorior polynomials of degree $2(m+1)$ in
$\mathbb{Z}[J_{n}^{*\mathbb{C}}]$. As in the 2-torus case, the graded
polynomial ring $\mathcal{E}_{*}(J_{n}^{*\mathbb{C}})=\bigoplus_{0\leq m\leq
n-1}\mathcal{E}_{m}(J_{n}^{*\mathbb{C}})$ can be regarded as a chian complex
with chain map $d_{*}$. This point of view is similar to the definition of
universal $\mathbb{Z}^{n}$-space $X(\mathbb{Z}^{n})$ ([1]).

######  Definition 3.6.

A subset $\\{v_{1},\dots,v_{m}\\}$ of $\mathbb{Z}^{n}$ is unimodular if
span$\\{v_{1},\dots,v_{m}\\}$ is a direct summand of $\mathbb{Z}^{n}$ with
rank $m$. Note that a subset of a unimodular set is itself unimodular. The
collection of all unimodular subsets of $\mathbb{Z}^{n}$ forms a simplicial
complex $X(\mathbb{Z}^{n})$ on unimodular vertices, which is called the
universal $\mathbb{Z}^{n}$-complex.

Note that this universal $\mathbb{Z}^{n}$-complex $X(\mathbb{Z}^{n})$ is
different from the universal complex characterizing the quasitoric manifolds
by Davis and Januszkiewicz in [7]. It is clear that the simplicial complexes
$X(\mathbb{Z}^{n})$ is $(n-1)$-dimensional and pure. As showed in [1], the
universal complex $X(\mathbb{Z}_{2}^{n})$ is $(n-1)$-connected Cohen-Macaulay.
As in the 2-torus case, the universal simplicial complex $X(\mathbb{Z}^{n})$
can be used to classify the equivariant bordism of unitary torus manifolds.

###### Proof of Theorem 1.4.

The proof here is just the unitary version of the proof of Theorem 1.1. Let
$0\leq m\leq n-1$. Let
$r:J_{n}^{*\mathbb{C}}=\text{Hom}(S^{1},T^{n})\to\mathbb{Z}^{n}$ be the
standard isomorphim. For an essential exterior monomial
$s_{i}^{*}=s_{i,1}\cdots s_{i,m}\in\mathcal{E}_{m}(J_{n}^{*\mathbb{C}})$, the
set $\\{r(s_{i,1}),\dots,r(s_{i,m})\\}$ is unimodular in $\mathbb{Z}^{n}$, it
spans a simplex of dimention $m-1$ in the universal complex
$X(\mathbb{Z}^{n})$, denoted by $r(s_{i}^{*})$. Let
$(C_{*}(X(\mathbb{Z}^{n})),\partial_{*})=\sum_{0\leq m\leq
n-1}(C_{m}(X(\mathbb{Z}^{n})),\partial_{m})$ be the chain complex of
$X(\mathbb{Z}^{n})$, then the map $r$ induces a homomorphism

$r_{m}:\mathcal{E}_{m}(J_{n}^{*\mathbb{C}})\to C_{m}(X(\mathbb{Z}^{n}))$

which maps an essential polynomial $g^{*}=\sum
s_{i}^{*}\in\mathcal{E}_{m}(J_{n}^{*\mathbb{C}})$ to a chain
$r_{m}(g^{*})=\sum r(s_{i}^{*})\in C_{m}(X(\mathbb{Z}^{n}))$. It is easy to
see that $r$ is an isomorphism of groups and the following is commutative:

$\textstyle{\mathcal{E}_{m}(J_{n}^{*\mathbb{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{m}}$$\scriptstyle{r_{m}}$$\textstyle{C_{m}(X(\mathbb{Z}^{n}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{m}}$$\textstyle{\mathcal{E}_{m-1}(J_{n}^{*\mathbb{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{m-1}}$$\textstyle{C_{m-1}(X(\mathbb{Z}^{n}))}$

where $d_{m}$ is the differential (3.2) defined on
$\Lambda_{\mathbb{Z}}^{m}[J_{n}^{*\mathbb{R}}]$, and $\partial_{m}$ is the
differential on the chain complex $C_{m}(X(\mathbb{Z}^{n}))$. Regard
$(\mathcal{E}_{*}(J_{n}^{*\mathbb{R}}),d)$ as a chian complex, then the above
homomorphism $r$ is a chain isomorphism, it induces an isomorphism of homology

$r_{m*}:H_{m}(\mathcal{E}_{*}(J_{n}^{*\mathbb{R}}))\cong
H_{m}(X(\mathbb{Z}_{2}^{n}))$

Let $g$ be a faithful extorior polynomial in $\mathbb{Z}[J_{n}^{\mathbb{C}}]$,
then $g^{*}$ is a faithful extorior polynomial in
$\mathbb{Z}_{2}[J_{n}^{*\mathbb{R}}]$. By Corollary3.3, the equivariant
bordism group $\mathcal{Z}_{n}^{U}(T^{n})$ of unitary torus manifolds is
isomorphic to the image $Im\varphi_{n}^{U}$. By Theorem 3.5, $g\in
Im\varphi_{n}^{U}$ if and only if $d(g^{*})=0$, that is, $g^{*}$ is a closed
chain in $\mathcal{E}_{*}(J_{n}^{*\mathbb{C}})$, it represents a homology
class $[g^{*}]\in H_{n-1}(\mathcal{E}_{*}(J_{n}^{*\mathbb{C}}))\cong
H_{n-1}(X(\mathbb{Z}^{n}))$, so we have

$\mathcal{Z}_{n}^{U}(T^{n})\cong Im\varphi_{n}^{U}\cong
H_{n-1}(\mathcal{E}_{*}(J_{n}^{*\mathbb{C}});\mathbb{Z})\cong
H_{n-1}(X(\mathbb{Z}^{n});\mathbb{Z})$

∎

The homotopy of the universal complex $X(\mathbb{Z}^{n})$ was determined in
[1], we recall their result to calculate the equivariant bordism groups
$\mathcal{Z}_{n}^{U}(\mathbb{Z}^{n})$, the proof is straightforward.

######  Theorem 3.7 ([1] Theorem 1.3).

For $n\geq 2$, the simplicial complex $X(\mathbb{Z}^{n})$ is homotopy
equivalent to a countably infinite wedge of $(n-1)$-spheres.

######  Corollary 3.8.

The group $\mathcal{Z}_{2}^{U}(T^{1})\cong\mathbb{Z}$ and
$\mathcal{Z}_{2n}^{U}(T^{n})$ is isomorphic to a countably infinite direct sum
of $\mathbb{Z}$ for $n\geq 2$.

Quasitoric manifolds are important models for $T^{n}$-manifolds, which are
defined and studied by Davis and Januszkiewicz in [7], they are topological
versions of toric varieties. We recall their definitions and basic properties.

######  Definition 3.9.

An $2n$-dimensional quasitoric manifold $\pi:M^{2n}\to P^{n}$ is a smooth
closed $2n$-dimensional manifold $M^{2n}$ with a locally standard
$T^{n}$-action such that its orbit space is homeomorphic to a $n$-dimensional
simple convex polytope $P^{n}$, here locally standard means that the action is
locally isomorphic to a faithful representation of $T^{n}$ on
$\mathbb{C}^{n}$.

A quasitoric manifold $\pi:M^{2n}\to P^{n}$ determines a characteristic
function $\lambda:\mathcal{F}(P^{n})\to J_{n}^{*\mathbb{C}}$, where
$\mathcal{F}(P^{n})$ is the set of all facets of $P^{n}$. For each facet
$F\in\mathcal{F}(P^{n})$, the preimage $\pi^{-1}(F)$ is submanifold of
dimention $2(n-1)$, and itself is also a quasitoric manifold. Given a vertex
$v=F_{1}\cap\dots\cap F_{n}$ of $P^{n}$, where
$F_{1},\dots,F_{n}\in\mathcal{F}(P^{n})$, then the elements
$\lambda(F_{1}),\dots,\lambda(F_{n})$ form a basis of $J_{n}^{*\mathbb{C}}$.
Quasitoric manifolds can be equipped with an additional structure called
omniorientation, providing a combinatorial description for unitary structures.

######  Definition 3.10.

An omniorientation of a quasitoric manifold $M^{2n}$ consists of a choice of
an orientation for $M^{2n}$ and for every facial submanifold
$M^{2(n-1)}=\pi^{-1}(F_{i}),i=1,...,m$.

Note that a choice of orientation for $P^{n}$ is equivalent to a choice of
orientation for $M^{2n}$, and an omniorientation on $M^{2n}$ is equivalent to
a choice of orientations for $P^{n}$ and all its facets. As proved by
Buchstaber and Ray in [2] Proposition 4.5, every omniorientation of a
quasitoric manifold $M^{2n}$ determines a stably complex structure on it,
which is compatible with the action of the torus. So a quasitoric manifold
$M^{2n}$ with an omniorientation is a special case of unitary toric manifolds.
If we forget the group action, then the bordism class $[M]$ is an element of
the ordinary unitary bordism group $\Omega_{2n}^{U}$. For $n>1$, every complex
cobordism class in $\Omega_{2n}^{U}$ contains a quasitoric manifold with an
omniorientation ([2] Theorem 5.9).

######  Definition 3.11.

A quasitoric pair $(P^{n},\Lambda)$ consists of a combinatorial oriented
simple $n$-polytope $P^{n}$ and an integral $(n\times m)$-matrix $\Lambda$
whose columns $\lambda_{i}$ satisfy

$\det(\lambda_{i_{1}}\cdots\lambda_{i_{n}})=\pm 1\text{, whenever
}v=F_{i_{1}}\cap\dots\cap F_{i_{n}}\text{is a vertex of }P^{n}$

where $F_{i_{j}}\in\mathcal{F}(P^{n}),j=1,\dots,n,$ are some facets of
$P^{n}$.

Two quasitoric pairs $(P_{1},\Lambda_{1})$ and $(P_{2},\Lambda_{2})$ are
equivalent if and only if $P_{1}=P_{2}$ and there exists an $(m\times
m)$-permutation matrix $\Sigma$ such that $\Lambda_{1}=\Lambda_{2}\Sigma$, the
matrix $\Sigma$ may be thought as a reordering of the facets of $P^{n}$. The
following was showed by Buchstaber, Panov and Ray in [2], see also [6]
Corollary 5.28.

######  Lemma 3.12 ([2][6]).

There is a 1-1 correspondence between equivalence classes of omnioriented
quasitoric manifolds and combinatorial quasitoric pairs.

######  Proposition 3.13.

The group $\mathcal{Z}_{2n}^{U}(T^{n})$ is generated by equivariant unitary
bordism classes of $2n$-dimensional omnioriented quasitoric manifolds.

###### Proof.

By Theorem 3.7, the universal complex $X(\mathbb{Z}^{n})$ is homotopy to the
wedge of countably infinite spheres $S^{n-1}$, we call such spheres basic
spheres of $X(\mathbb{Z}^{n})$. By Theorem 1.4, each equivariant bordism class
of $\mathcal{Z}_{n}(T^{n})$ corresponds to an element in
$H_{n-1}(X(\mathbb{Z}^{n});\mathbb{Z})$, which is generalized by the homology
class of basic spheres. Since $X(\mathbb{Z}^{n})$ is pure, the dual of a basic
sphere $S^{n-1}$ is the boundary $\partial{P^{n}}$ of a simple polytope
$P^{n}$, and the unimodular subsets of $S^{n-1}$ induce a
$J_{n}^{*\mathbb{R}}$-coloring $\lambda$ of $P^{n}$. By Lemma 3.12, the
quasitoric pair $(P^{n},\lambda)$ determines a unique omnioriented quasitoric
manifold $\pi:M^{2n}\to P^{n}$ such that $\lambda$ is the characteristic
function. So the equivariant bordism class corresponding to the baisc sphere
$S^{n-1}$ is represented by an omnioriented quasitoric manifold. ∎

###### Proof of Proposition 1.5.

The case for $n=1$ is clear and was shown by [6] Corollary 8.8. For $n>1$, we
consider equivarint cobordism classes $[M_{1}]$ and $[M_{2}]$ in
$\mathcal{Z}_{2n}^{U}(T^{n})$, represented by omnioriented quasitoric
manifolds over quotient polytopes $P_{1}$ and $P_{2}$, respectively. It then
suffices to construct a third such manifold $M$ such that
$[M]=[M_{1}]+[M_{2}]$. The connect sum in the proof of [2] Theorem 5.9 can be
used to define a omnioriented quasitoric manifold $M=M_{1}\\#M_{2}$ over the
connect sum $P^{n}=P_{1}\\#P_{2}$. Or we can just use [6] Proposition 8.9 to
get a quasitoric pair $(P,\lambda)$ and the corresponding omnioriented
quasitoric manifold $M$ such that
$\varphi_{n}^{U}(M)=\varphi_{n}^{U}(M_{1})+\varphi_{n}^{U}(M_{2})$. Since
$\varphi_{n}^{U}$ is monomorphic by Corollary 3.3, the equivariant bordism
class $[M]=[M_{1}]+[M_{2}]\in\mathcal{Z}_{2n}^{U}(T^{n})$, so the quasitoric
manifold $M$ is what we need. ∎

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††footnotetext:  File: mod20231221.tex, printed: 2024-8-27, 23.45

#  Conformally invariant metrics and lack of Hölder continuity

Rahim Kargar∗ R. Kargar: Department of Mathematics and Statistics, University
of Turku, Turku, Finland<EMAIL_ADDRESS>
ORCID ID: http://orcid.org/0000-0003-1029-5386 and  Oona Rainio  O. Rainio:
Department of Mathematics and Statistics, University of Turku, Turku, Finland
<EMAIL_ADDRESS>
ORCID ID: http://orcid.org/0000-0002-7775-7656

###### Abstract.

The modulus metric between two points in a subdomain of $\mathbb{R}^{n},n\geq
2,$ is defined in terms of moduli of curve families joining the boundary of
the domain with a continuum connecting the two points. This metric is one of
the conformally invariant hyperbolic type metrics that have become a standard
tool in geometric function theory. We prove that the modulus metric is not
Hölder continuous with respect to the hyperbolic metric.

###### Key words and phrases:

Conformal modulus; Hyperbolic metric; Modulus metric; Hölder continuity.

###### 2020 Mathematics Subject Classification:

51M09; 30F45

##  1\. Introduction

In the geometric function theory, one of the important subjects of study is
the modulus of a curve family in ${\mathbb{R}}^{2}$ introduced in 1950 by L.
Ahlfors and A. Beurling. Its definition was extended to the Euclidean space
${\mathbb{R}}^{n}$ with dimension $n\geq 2$ by B. Fuglede, after which it was
quickly adopted as a standard tool to study mappings during the early 1960s by
F.W. Gehring [5] and J. Väisälä [17]. This conformal invariant has numerous
applications in the current research, see [3, 6, 8, 16].

Due to its invariance properties, the conformal modulus of a curve family is
often used to study the distortion of distances between points under
quasiconformal mappings. At times, it is enough to use crude estimates for
simple curve families combined with majorization or minorization. However, the
use of crude estimates has two drawbacks: Firstly, it requires experience with
the moduli of curve families and, what is more unfortunate, crude estimates
lead to loss of information.

Instead of using crude estimates, it is often useful to reduce the estimation
problem to classical extremal problems and to use these systematically. We now
discuss two classical extremal problems which have been applied in this way.
Namely, these two problems have already been studied by H. Grötzsch and O.
Teichmüller, who both were pioneers of conformal invariant from 1920s and
1930s on.

Let $G$ be a domain in the extended real space
$\overline{{\mathbb{R}}}^{n}=\mathbb{{\mathbb{R}}}^{n}\cup\\{\infty\\}$ such
that ${\operatorname{card}\,}(\overline{{\mathbb{R}}}^{n}\backslash G)\geq 2$.
Our first extremal problem is [6, (10.2), p. 174]

(1.1)
$\displaystyle\mu_{G}(x,y)=\inf_{C_{xy}}\mathsf{M}(\Delta(C_{xy},\partial
G;G)),$

where the infimum is taken over all continua $C_{xy}$ joining the points $x$
and $y$ in $G$, $\Delta(E,F;G)$ is the family of all such curves in $G$ that
join $E$ and $F$, and $\mathsf{M}(\Gamma)$ denotes the conformal modulus of a
curve family $\Gamma$ in $G$. The second extremal problem is

(1.2)
$\displaystyle\lambda_{G}(x,y)=\inf_{C_{x},C_{y}}\mathsf{M}(\Delta(C_{x},C_{y};G)),$

where infimum is taken over all pairs of continua $C_{x}$ and $C_{y}$ in
$\overline{G}$ such that $x\in C_{x}$, $y\in C_{y}$,
$\overline{C_{x}}\cap\partial G\neq\varnothing$ and
$\overline{C_{y}}\cap\partial G\neq\varnothing$.

By Ahlfors [1, p. 72], the extremal problem (1.1) was studied with $n=2$ and
$G=\mathbb{B}^{2}$ by H. Grötzsch while the problem (1.2) was considered with
$n=2$ and $G={\mathbb{R}}^{2}\backslash\\{0\\}$ by O. Teichmüller. For further
notes on the literature, see [6, 10.30, p. 186], where the contributions of
I.S. Gál, T. Kuusalo and J. Lelong-Ferrand are cited. As stated in the
following two theorems [6, 19, 20], the functions $\mu_{G}$ and $\lambda_{G}$
of the extremal problems (1.1) and (1.2) can also be used to define metrics,
out of which the first one is called the _modulus metric_.

######  Theorem 1.3.

(1) If $\mathrm{cap}\,(\partial G)>0$, then $\mu_{G}$ is a metric.
(2) $\lambda_{G}^{p}$ is a metric if and only if $p\in[-1/(n-1),0)$.

The proof of Theorem 1.3(1) is straightforward, whereas part (2) has an
interesting history. It was proved with $p=-1/n$ by Ferrand [10] and in the
special case $G=\mathbb{B}^{2}$ by G.D. Anderson, M.K. Vamanamurthy and M.
Vuorinen [2]. As an open problem, part (2) was formulated in [18, p. 193] and
J. Ferrand, J.A. Jenkins, and A.Y. Solynin found solutions independently; see
[6, p. 453].

Theorem 1.3 has numerous applications, based on the next two theorems.

######  Theorem 1.4.

For $D\in\\{{\mathbb{H}}^{n},\mathbb{B}^{n}\\}$ and all distinct points
$x,y\in D$,

  1. (1)

$\mu_{D}(x,y)=\gamma_{n}(1/{\rm th}(\rho_{D}(x,y)/2))$;

  2. (2)

$\lambda_{D}(x,y)=2^{-n}\gamma_{n}({\rm ch}(\rho_{D}(x,y)/2)),$

where $\rho$ stands for the hyperbolic metric and $\gamma_{n}$ is a special
function called the Grötzsch capacity.

######  Theorem 1.5.

Let $f:G\rightarrow G^{\prime}$ be a $K$-quasiconformal homeomorphism. Then
for all distinct $x,y\in G$,

$\mu_{G}(x,y)/K\leq\mu_{G^{\prime}}(f(x),f(y))\leq K\mu_{G}(x,y)$

and

$\lambda_{G}(x,y)/K\leq\lambda_{G^{\prime}}(f(x),f(y))\leq K\lambda_{G}(x,y).$

In other words, written as mappings between metric spaces,

$\displaystyle
f:(G,\mu_{G})\to(G^{\prime},\mu_{G^{\prime}}),\quad\text{or}\quad
f:(G,\lambda_{G}^{1/(1-n)})\to(G^{\prime},\lambda_{G^{\prime}}^{1/(1-n)})$

the mapping $f$ is bi-Lipschitz. For the case of $\mu_{G}$, we assume that
$\mathrm{cap}\,(\partial G)>0$ and, for the case of $\lambda_{G}$, let
${\operatorname{card}\,}(\overline{{\mathbb{R}}}^{n}\backslash G)\geq 2$.

The metrics $\rho_{D}$, $\mu_{D}$ and $\lambda_{D}^{1/(1-n)}$ in
$D\in\\{{\mathbb{H}}^{n},\mathbb{B}^{n}\\}$ are all conformally invariant, so
it is natural to ask whether they are comparable in some other fashion.
Theorem 1.5 implies that quasiconformal mappings satisfy a version of the
Schwarz lemma, according to which mappings are Hölder continuous with respect
to the hyperbolic and the Euclidean metrics. As well known [6, Thm 16.3, Thm
16.21] for dimensions $n\geq 3$, both the variants of the Theorem 1.5 yield
different Hölder exponents, while Theorem 1.5 itself speaks for bi-Lipschitz
continuity in the respective two metric spaces.

The above two theorems show that quasiconformal mappings are bi-Lipschitz in
the respective metric spaces. However, because the local structure of these
spaces depends on the special function $\gamma_{n},$ it is desirable to
compare metrics to Euclidean and hyperbolic metrics and this is what we will
do here. Surprisingly, it turns out that the $\mu_{\mathbb{B}^{2}}$ metric is
not Hölder equivalent to the hyperbolic metric $\rho_{\mathbb{B}^{2}},$ but
nevertheless we can conclude sharp modulus of continuity estimates from the
above theorems. We also prove various other results for these metrics.

######  Theorem 1.6.

The $\mu_{D}$ metric, $D\in\\{{\mathbb{H}}^{2},\mathbb{B}^{2}\\},$ is not
Hölder continuous with respect to $\rho_{D}.$

Here, it should be noted that the same result holds for
$\lambda^{-1}_{D}(x,y)$ metric as proven in Corollary 3.14. For the Hölder
non-equivalence of $\mu_{D}$ and, $\lambda_{D}^{1/(1-n)}$ see [2, 16.6].

Very recently, $\mu_{G}$ isometries were thoroughly studied by D. Betsakos- S.
Pouliasis [3], X. Zhang [21], and S. Pouliasis- A. Solynin [11]. In this case,
it turns out that $\mu_{G}$ isometries are, in fact, conformal mappings. One
might expect that a similar result also holds for the $\lambda_{G}^{1/(1-n)}$
metric, but this seems to be an open problem. Isometries of some other metrics
were studied by P. Hästö, Z. Ibragimov, D. Minda, S. Ponnusamy, and S. Sahoo
[7].

The structure of this article is as follows. In Section 2 we give some
necessary definitions and notations. In Section 3 we prove that the modulus
metric is not Hölder continuous with respect to the hyperbolic metric.

##  2\. Preliminaries

Let us first introduce our notations. For all points $x\in{\mathbb{R}}^{n}$
and any radius $r>0$, we can define an open Euclidean ball
$B^{n}(x,r)=\\{y\in{\mathbb{R}}^{n}\text{ }:\text{ }|x-y|<r\\}$ and its
boundary sphere $S^{n-1}(x,r)=\\{y\in{\mathbb{R}}^{n}\text{ }:\text{
}|x-y|=r\\}$. For the unit ball and sphere, we use the simplified notations
$\mathbb{B}^{n}=B^{n}(0,1)$ and $S^{n-1}=S^{n-1}(0,1)$. Denote also the
$(n-1)$-dimensional surface area of $S^{n-1}$ by $\omega_{n-1}$ and define the
following constant $c_{n}$ (see Ref. [2, p. 41]):

$\displaystyle c_{2}=\frac{2}{\pi},\quad
c_{n}=2^{1-n}\omega_{n-2}\left(\int^{\pi/2}_{0}\sin^{\frac{2-n}{n-1}}t\,dt\right)^{1-n}\text{
for }n\geq 2.$

The _hyperbolic metric_ $\rho$ can be defined in the unit ball with the
formula [6, (4.16), p. 55]

$\displaystyle{\rm
sh}^{2}\left(\frac{\rho_{\mathbb{B}^{n}}(x,y)}{2}\right)=\frac{|x-y|^{2}}{(1-|x|^{2})(1-|y|^{2})},\quad
x,y\in\mathbb{B}^{n}$

and, by the conformal invariance of this metric, we can compute its value in
any such domain that can be mapped conformally onto the unit ball. A
hyperbolic ball $B_{\rho}(x,M)$ defined in the unit ball $\mathbb{B}^{n}$ is
equal to the Euclidean ball $B^{n}(y,r)$, where [6, (4.20), p. 56]

$\displaystyle y=\frac{x(1-t^{2})}{1-|x|^{2}t^{2}},\quad
r=\frac{(1-|x|^{2})t}{1-|x|^{2}t^{2}},\quad t={\rm th}(M/2).$

The _modulus of a curve family_ $\Gamma$ in ${\mathbb{R}}^{n}$ is [6, (7.1),
p. 104]

$\displaystyle\mathsf{M}(\Gamma)=\inf_{\rho\in\mathcal{F}(\Gamma)}\int_{{\mathbb{R}}^{n}}\rho^{n}dm,$

where $\mathcal{F}(\Gamma)$ consists of all non-negative Borel-measurable
functions $\rho:{\mathbb{R}}^{n}\to\overline{{\mathbb{R}}}^{n}$ such that
$\int_{\gamma}\rho ds\geq 1$ for each locally rectifiable curve
$\gamma\in\Gamma$ and $m$ stands for the $n$-dimensional Lebesgue measure.
Denote the family of all curves joining two non-empty sets $E$ and $F$ in a
domain $G$ by $\Delta(E,F;G)$. Now, for the annular ring
$D=\overline{B}^{n}(0,b)\backslash B^{n}(0,a)$ with $0<a<b$, it holds that [6,
(7.4), p. 107]

$\displaystyle\mathsf{M}(\Delta(S^{n-1}(0,a),S^{n-1}(0,b);D))=\omega_{n-1}\left(\log\frac{b}{a}\right)^{1-n}.$

Any domain $G$ and its compact subset $F\subset G$ form a _condenser_ $(G,F)$
and the _capacity_ of this condenser is [6, Thm 9.6, p. 152]

$\displaystyle\mathrm{cap}\,(G,F)=\mathsf{M}(\Delta(F,\partial G;G)).$

A compact set $E$ in $\mathbb{R}^{n}$ is said to be of capacity zero, denoted
${\rm cap}(E)=0$, if there exists a bounded open set $A$ containing $E$ with
the capacity of the pair $(A,E)$ is equal to zero. A compact set
$E\subset\overline{\mathbb{R}}^{n}$, $E\neq\overline{\mathbb{R}}^{n}$ is said
to be of capacity zero if $E$ can be mapped by a Möbius transformation onto a
bounded set of capacity zero. If ${\rm cap}(E)=0$ does not hold, we express it
as ${\rm cap}(E)>0$.

Define next two decreasing homeomorphisms called the Grötzsch capacity
$\gamma_{n}:(1,\infty)\to(0,\infty)$ and the Teichmüller capacity
$\tau_{n}:(0,\infty)\to(0,\infty)$ with the following formula [6, (7.17), p.
121]:

$\gamma_{n}(s)=\mathsf{M}(\Delta(\overline{\mathbb{B}}^{n},[se_{1},\infty];{\mathbb{R}}^{n})),\quad
s>1$

and

$\tau_{n}(s)=\mathsf{M}(\Delta([-e_{1},0],[se_{1},\infty];{\mathbb{R}}^{n})),\quad
s>0,$

where $e_{1}=(1,0,\ldots,0)$ is a unit vector in ${\mathbb{R}}^{n}$. For
$s>1$, $\gamma_{n}(s)=2^{n-1}\tau_{n}(s^{2}-1)$. If $n=2$, the capacities can
be computed with the formula [6, (7.18), p. 122],

(2.1) $\gamma_{2}(1/r)=\frac{2\pi}{\mu(r)},\quad 0<r<1,$

where

$\mu(r)=\frac{\pi}{2}\frac{\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt
K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp
K}}(\sqrt{1-r^{2}})}{\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt
K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp
K}}(r)},\quad{\text{and}}\quad\mathchoice{\hbox{\,\fFt K}}{\hbox{\,\fFt
K}}{\hbox{\,\fFa K}}{\hbox{\,\fFp
K}}(r)=\int^{1}_{0}\frac{dx}{\sqrt{(1-x^{2})(1-r^{2}x^{2})}}.$

Moreover, for $s>1$ [6, (7.20)]

(2.2) $\mu(1/s)\mu\left(\frac{s-1}{s+1}\right)=\frac{\pi^{2}}{2},$

and for $0<r<1$ (see, [2, 5.30])

(2.3) ${\rm arth}\,\sqrt[4]{r^{\prime}}<\mu(r)<\frac{\pi^{2}}{4\,{\rm
arth}\sqrt[4]{r}},\quad r^{\prime}=\sqrt{1-r^{2}}.$

For approximation of $\mu(r)$, see [9]. The Grötzsch capacity has the
following well-known estimates [6, Thm 9.17(2), p. 160]

(2.4) $\displaystyle
2^{n-1}c_{n}\log\left(\frac{s+1}{s-1}\right)\leq\gamma_{n}(s)\leq
2^{n-1}c_{n}\mu\left(\frac{s-1}{s+1}\right)<2^{n-1}c_{n}\log\left(4\frac{s+1}{s-1}\right).$

where the second inequality holds as an identity for $n=2$ by (2.2).

A homeomorphism $f:G\to G^{\prime}$ between two domains
$G,G^{\prime}\subset{\mathbb{R}}^{n}$, $n\geq 2$, is called _$K$
-quasiconformal_ with some constant $K\geq 1$, if the two-sided inequality

$\displaystyle\mathsf{M}(\Gamma)/K\leq\mathsf{M}(f(\Gamma))\leq
K\mathsf{M}(\Gamma)$

holds for every curve family $\Gamma$ in $G$.

J. Ferrand [10] posed the question whether $\mu_{G}$ bi-Lipschitz
homeomorphisms are quasiconformal. This question was studied in [4] where it
was proved that these mappings are locally Hölder continuous, but they need
not be quasiconformal and thus Ferrand’s question was solved in the negative
in [4]. Furthermore, the radial mapping $g:\mathbb{B}^{n}\to\mathbb{B}^{n}$
defined as

$g(x)=\left\\{\begin{array}[]{ll}|x|^{\alpha-1}x,&\hbox{$x\in\mathbb{B}^{n}\backslash\\{0\\}$;}\\\
\\\ 0,&\hbox{$x=0$,}\end{array}\right.$

is quasiconformal, as noted in [17, 16.2], and Hölder continuous but not
Lipschitz with respect to the Euclidean metric. Thus, the bi-Lipschitz
condition of the modulus metric under $K$-quasiconformal mappings does not
imply the same property for the Euclidean metric.

##  3\. Upper and lower bounds for modulus metric

Recently, the modulus metric $\mu_{G}$ has been studied in [16] where a
characterization of its completeness is given. Also in [16], a new lower bound
for $\mu_{G}$ was found in terms of the Möbius invariant metric $\delta_{G}$
(also, $\delta_{G}$ is called the Seittenranta metric [6, p. 75 & p. 199]): If
the boundary $\partial G$ is uniformly perfect, then

(3.1) $\displaystyle\mu_{G}(x,y)\geq c\delta_{G}(x,y),$

where the constant $c$ only depends on the dimension $n$ and the parameters of
the uniform perfectness. The estimates in (2.4) also yield

(3.2) $\displaystyle 2^{n-1}c_{n}\rho_{D}(x,y)\leq\mu_{D}(x,y)\leq
2^{n-1}c_{n}\mu(1/e^{\rho_{D}(x,y)})<2^{n-1}c_{n}(\rho_{D}(x,y)+\log 4),$

where $D\in\\{{\mathbb{H}}^{n},\mathbb{B}^{n}\\}$. A similar inequality can be
also written for $\lambda_{D}$:

$\displaystyle
c_{n}\log(t)\leq\lambda_{D}(x,y)\leq\frac{c_{n}}{2}\mu\left(t^{-2}\right)<c_{n}\log(2t),\quad\text{with}\quad
t=\frac{e^{\rho_{D}(x,y)/2}+1}{e^{\rho_{D}(x,y)/2}-1}.$

Because $\rho_{\mathbb{B}^{n}}=\delta_{\mathbb{B}^{n}}$, the lower bound in
(3.2) is compatible with (3.1) up to a constant factor, but it can be still
improved for small values of the hyperbolic distance
$\rho_{\mathbb{B}^{2}}(x,y)$ in the two-dimensional case. Note that, for
$n=2$, it follows from the inequality (3.2) that

(3.3)
$\displaystyle\frac{4}{\pi}\rho_{D}(x,y)\leq\mu_{D}(x,y)\leq\frac{4}{\pi}(\rho_{D}(x,y)+\log
4),$

and consider the following preliminary result:

######  Lemma 3.4.

For all $t>0$ and $p>0$, the expression $[{\rm arth}\left(({\rm
th}\,t)^{1/p}\right)]^{p}$ is strictly increasing with respect to $p$.

###### Proof.

Since we are interested in the expression above with respect to $p$ only, we
can substitute $u={\rm th}\,t$ and study $[{\rm arth}\,(u^{1/p})]^{p}$ for
$0<u<1$ and $p>0$ instead. By differentiation,

(3.5) $\displaystyle\frac{\partial}{\partial p}[{\rm arth}\,(u^{1/p})]^{p}$
$\displaystyle=[{\rm arth}\,(u^{1/p})]^{p}\left(\log({\rm
arth}(u^{1/p}))-\frac{u^{1/p}\log(u)}{p(1-u^{2/p}){\rm
arth}(u^{1/p})}\right)>0$ $\displaystyle\Leftrightarrow{\rm
arth}(u^{1/p})\log({\rm arth}(u^{1/p}))-\frac{u^{1/p}\log(u)}{p(1-u^{2/p})}>0$
$\displaystyle\Leftrightarrow{\rm arth}(y)\log({\rm
arth}(y))-\frac{y\log(y)}{1-y^{2}}>0,$

where $0<y<1$. Again, by differentiation,

(3.6) $\displaystyle\frac{d}{dy}\left({\rm arth}(y)\log({\rm
arth}(y))-\frac{y\log(y)}{1-y^{2}}\right)$
$\displaystyle=\frac{1}{(1-y^{2})^{2}}((1-y^{2})\log({\rm
arth}(y))-(1+y^{2})\log(y))>0$ $\displaystyle\Leftrightarrow(1-y^{2})\log({\rm
arth}(y))-(1+y^{2})\log(y)>0$ $\displaystyle\Leftrightarrow\frac{\log({\rm
arth}(y))}{\log(y)}-\frac{1+y^{2}}{1-y^{2}}<0,$

provided that $0<y<1$. Since $\log({\rm arth}(y))/\log(y)$ is strictly
decreasing on $(0,1)$ and $(1+y^{2})/(1-y^{2})$ is strictly increasing on
$(0,1)$, their difference is strictly decreasing on $(0,1)$ and the following
holds:

$\displaystyle\frac{\log({\rm
arth}(y))}{\log(y)}-\frac{1+y^{2}}{1-y^{2}}<\lim_{y\to
0+}\left(\frac{\log({\rm
arth}(y))}{\log(y)}-\frac{1+y^{2}}{1-y^{2}}\right)=1-1=0.$

Consequently, the derivative (3.6) is positive on $(0,1)$, the differentiated
expression in (3.6) is therefore strictly increasing on this interval, and we
have the following lower bound:

$\displaystyle{\rm arth}(y)\log({\rm
arth}(y))-\frac{y\log(y)}{1-y^{2}}>\lim_{y\to 0+}\left({\rm arth}(y)\log({\rm
arth}(y))-\frac{y\log(y)}{1-y^{2}}\right)=0.$

Thus, the derivative (3.5) is positive for $0<u<1$ and $p>0$ and the original
expression $\left[{\rm arth}(u^{1/p})\right]^{p}$ is strictly increasing with
respect to $p>0$, from which our lemma follows. ∎

######  Corollary 3.7.

For all $x,y\in D\in\\{{\mathbb{H}}^{2},\mathbb{B}^{2}\\}$, the inequality

$\displaystyle\mu_{D}(x,y)\geq\frac{8}{\pi\sqrt[4]{2}}\rho_{D}(x,y)^{1/4}$

holds.

###### Proof.

If $x=y$, the result holds trivially, so let us assume that $x\neq y$ below.
By Lemma 3.4, the expression $[{\rm arth}\left(({\rm
th}\,t)^{1/p}\right)]^{p}$ is strictly increasing with respect to $p$ for all
$t>0$ and $p>0$, so

$\displaystyle[{\rm arth}\left(({\rm th}\,t)^{1/4}\right)]^{4}>{\rm arth}({\rm
th}(t))=t\Leftrightarrow{\rm arth}({\rm th}(t)^{1/4})>t^{1/4}.$

Combining this result to Theorem 1.4, the formula (2.1) and [2, (5.29)], we
have

$\mu_{D}(x,y)=\frac{2\pi}{\mu({\rm th}(\rho_{D}(x,y)/2))}>\frac{8}{\pi}{\rm
arth}(({\rm th}(\rho_{D}(x,y)/2)^{1/4})>\frac{8}{\pi}(\rho_{D}(x,y)/2)^{1/4}.$

∎

######  Remark 3.8.

Corollary 3.7 gives a better lower bound for the modulus metric than the
inequality (3.3) if and only if the hyperbolic distance
$\rho_{\mathbb{B}^{2}}(x,y)$ is less than 2. See Figure 1 for more details. We
note that in Figure 1, $\rho_{\mathbb{B}^{2}}(x,0)<2$ if and only if $0<x\leq
0.75$. Also, in Figure 1, $\rho_{\mathbb{B}^{2}}(x,0)\geq 2$ if and only if
$0.75<x<1$.

Figure 1. 1: The graph of $\mu_{\mathbb{B}^{2}}(x,0)$, its lower bound in
Corollary 3.7 (dashed), and its lower bound in (3.3) (dotted) when
$\rho_{\mathbb{B}^{2}}(x,0)<2$, where $0<x\leq 0.75$. 1: The graph of
$\mu_{\mathbb{B}^{2}}(x,0)$, its lower bound in (3.3) (dotted), and its lower
bound in Corollary 3.7 (dashed) when $\rho_{\mathbb{B}^{2}}(x,0)\geq 2$, where
$0.75<x<1$.

In article [14], the Euclidean midpoint rotation was introduced as a new way
to find upper and lower bounds for the triangular ratio metric. This metric
was originally introduced in 2002 by P. Hästö [8] and recently studied in [12,
13, 15]. In the midpoint rotation, two distinct points $x,y\in\mathbb{B}^{2}$
are rotated around their Euclidean midpoint $(x+y)/2$ so that the smaller
angle $\nu$ between lines $L(x,y)$ and $L(0,(x+y)/2)$ varies on the interval
$[0,\pi/2]$. See Figure 2. Here, we assume that $x\neq-y$, because otherwise
the midpoint $(x+y)/2$ is the origin and the hyperbolic distance
$\rho_{\mathbb{B}^{2}}(x,y)$ is invariant under rotations around the origin.
As explained in Theorem 3.9, the hyperbolic distance of the rotated points is
decreasing with respect to $\nu$ and, since $\nu=0$ when the rotated points
are collinear with the origin and $\nu=\pi/2$ when their absolute values are
equivalent, the Euclidean midpoint rotation yields upper and lower bounds for
the hyperbolic metric.

••••$0$$x$$y$$\nu$ Figure 2. In the Euclidean midpoint rotation, two distinct
points $x,y$ in the unit disk $\mathbb{B}^{2}$ are rotated around their
midpoint $(x+y)/2$ so that the smaller angle $\nu$ between the lines $L(x,y)$
and $L(0,(x+y)/2)$ increases from $0$ to $\pi/2$.

######  Theorem 3.9.

For all distinct points $x,y\in\mathbb{B}^{2}$ such that $x\neq-y$, the
hyperbolic distance $\rho_{\mathbb{B}^{2}}(x,y)$ decreases as $x$ and $y$ are
rotated around their Euclidean midpoint so that the smaller angle between the
lines $L(x,y)$ and $L(0,(x+y)/2)$ increases from $0$ to $\pi/2$. Furthermore,

$\displaystyle\frac{2|x-y|}{\sqrt{4-8x\cdot y+(|x|^{2}+|y|^{2})^{2}}}\leq{\rm
th}\frac{\rho_{\mathbb{B}^{2}}(x,y)}{2}\leq\frac{|x-y|}{1-x\cdot y},$

where equality holds in the first inequality if and only if $|x|=|y|$ and in
the second inequality if and only if $x,y$ are collinear with the origin. Note
also that ${|x-y|}/({1-x\cdot y})<1$ if and only if $|x+y|+|x-y|<2$.

###### Proof.

Fix distinct points $x,y\in\mathbb{B}^{2}$ such that $x\neq-y$. Denote
$d=|x-y|/2$ and $k=|x+y|/2$, and note that $k,d\in(0,1)$. Let
$\nu\in[0,\pi/2]$ be the magnitude of the smaller angle between the lines
$L(x,y)$ and $L(0,(x+y)/2)$. Now, we have,

$\displaystyle{\rm th}(\rho_{\mathbb{B}^{2}}(x,y)/2)$
$\displaystyle=\frac{|x-y|}{|1-x\overline{y}|}$
$\displaystyle=\frac{2d}{|1-(k+d(\cos(\nu)+\sin(\nu)i))(k+d(-\cos(\nu)+\sin(\nu)i)|}$
$\displaystyle=\frac{2d}{|1+d^{2}-k^{2}-2kd\sin(\nu)i|}=\frac{2d}{\sqrt{(1+d^{2}-k^{2})^{2}+4k^{2}d^{2}\sin^{2}(\nu)}}.$

Trivially, the quotient above is decreasing with respect to $\nu$ and it
attains its minimum $2d/\sqrt{(1+d^{2}-k^{2})^{2}+4k^{2}d^{2}}$ at $\nu=\pi/2$
and its maximum $2d/(1+d^{2}-k^{2})$ with respect to $\nu$ at $\nu=0$. Given
$|x-y|^{2}=|x|^{2}+|y|^{2}-2x\cdot y$, we can easily show that

$\displaystyle\frac{2d}{\sqrt{(1+d^{2}-k^{2})^{2}+4k^{2}d^{2}}}$
$\displaystyle=\frac{4|x-y|}{\sqrt{(4+|x-y|^{2}-|x+y|^{2})^{2}+4|x+y|^{2}|x-y|^{2}}}$
$\displaystyle=\frac{2|x-y|}{\sqrt{4-8x\cdot y+(|x|^{2}+|y|^{2})^{2}}}$

and

$\frac{2d}{1+d^{2}-k^{2}}=\frac{4|x-y|}{4+|x-y|^{2}-|x+y|^{2}}=\frac{|x-y|}{1-x\cdot
y}.$

Note that the points always stay in $\mathbb{B}^{2}$ if the rotation is done
so that $\nu$ increases on the interval $[0,\pi/2]$ whereas a rotation
decreasing $v$ might move one of the points outside of $\mathbb{B}^{2}$ if
$k+d>1$. ∎

By [2, 7.64(26-27), p. 156],

(3.10)
$\displaystyle\frac{|x-y|}{\min\\{|x-y|+\sqrt{1-|x|^{2}}\sqrt{1-|y|^{2}},1+|x||y|\\}}$
$\displaystyle\leq{\rm th}\frac{\rho_{\mathbb{B}^{2}}(x,y)}{2}$
$\displaystyle\leq\frac{|x-y|}{\max\\{|x-y|+(1-|x|)(1-|y|),1-|x||y|\\}}.$

Compared to these bounds, the bounds of Theorem 3.9 are better in some cases
but not always. For instance, if $x=0.6+0.3i$ and $y=0.1+0.1i$, both the upper
and lower bound of Theorem 3.9 are better than those of inequality (3.10), but
the bounds of Theorem 3.9 are worse for $x=-0.7+0.7i$ and $y=0.65-0.6i$. We
summarize our observations in the following table.

$x$ | $y$ | L.H.S Thm 3.9 | L.H.S (3.10) | R.H.S Thm 3.9 | R.H.S (3.10)
---|---|---|---|---|---
$0.6+0.3i$ | $0.1+0.1i$ | 0.575624 | 0.491855 | 0.591776 | 0.594959
$-0.7+0.7i$ | $0.65-0.6i$ | 0.997999 | 0.999183 | 0.999555 | 0.999381

######  Corollary 3.11.

For all distinct points $x,y\in\mathbb{B}^{2}$ such that $x\neq-y$, the
distance $\mu_{\mathbb{B}^{2}}(x,y)$ decreases as $x$ and $y$ are rotated
around their Euclidean midpoint so that the smaller angle between lines
$L(x,y)$ and $L(0,(x+y)/2)$ increases from $0$ to $\pi/2$. Furthermore, if
$|x+y|+|x-y|<2$,

$\displaystyle\gamma_{2}\left(\frac{\sqrt{4-8x\cdot
y+(|x|^{2}+|y|^{2})^{2}}}{2|x-y|}\right)\leq\mu_{\mathbb{B}^{2}}(x,y)\leq\gamma_{2}\left(\frac{1-x\cdot
y}{|x-y|}\right),$

where equality holds in the first inequality if and only if $|x|=|y|$ and in
the second inequality if and only if $x,y$ are collinear with the origin. If
$|x+y|+|x-y|\geq 2$, only the first inequality above holds.

######  Lemma 3.12.

The modulus metric defined in the unit disk is not Hölder continuous with
respect to the Euclidean metric.

###### Proof.

First, fix $w>0$ and $x\in(0,1)\subset\mathbb{B}^{2}$. By Theorem 1.4, the
formula (2.1), and the inequality [6, (7.21), p. 122],

$\frac{\mu_{\mathbb{B}^{2}}(x,0)}{|x|^{w}}=\frac{\gamma_{2}(1/{\rm
th}(\rho_{\mathbb{B}^{2}}(x,0)/2))}{|x|^{w}}=\frac{\gamma_{2}(1/|x|)}{|x|^{w}}=\frac{2\pi}{|x|^{w}\mu(|x|)}\geq\frac{2\pi}{|x|^{w}U(|x|)},$

where $U(r)=\log(2(1+\sqrt{1-r^{2}})/r)$. Since $\lim_{r\to
0+}U(r)=\infty=\lim_{r\to 0+}r^{-w}$ and $(\partial/\partial
r)r^{-w}=-wr^{-w-1}>0$ for all $r>0$, it follows from L’Hôpital’s rule that

$\lim_{r\to 0+}r^{w}U(r)=\lim_{r\to 0+}\frac{U(r)}{r^{-w}}=\lim_{r\to
0+}\frac{U^{\prime}(r)}{\frac{\partial}{\partial r}r^{-w}}=\lim_{r\to
0+}\frac{-1/(r\sqrt{1-r^{2}})}{-wr^{-w-1}}=\lim_{r\to
0+}\frac{r^{w}}{w\sqrt{1-r^{2}}}=0.$

Consequently,

$\displaystyle\lim_{x\to
0+}\frac{\mu_{\mathbb{B}^{2}}(x,0)}{|x|^{w}}\geq\lim_{x\to
0+}\frac{2\pi}{|x|^{w}U(|x|)}=\frac{2\pi}{\lim_{x\to
0+}|x|^{w}U(|x|)}=\infty.$

Thus, the quotient $\mu_{\mathbb{B}^{2}}(x,0)/|x|^{w}$ approaches infinity as
$x\to 0+$ for all $w>0$, from which our result follows. ∎

######  3.13.

Proof of Theorem 1.6. The proof follows from Lemma 3.12. $\square$

######  Corollary 3.14.

The metric $\lambda^{-1}_{\mathbb{B}^{2}}(x,y)$ defined in the unit disk is
not Hölder continuous with respect to the hyperbolic metric.

###### Proof.

The result follows from Lemma 3.12 and the identity
$\mu_{\mathbb{B}^{2}}(x,y)\lambda_{\mathbb{B}^{2}}(x,y)=4$, see [2, 16.7(1)].
∎

Acknowledgments. We are indebted to Prof. Matti Vuorinen for introducing this
topic to us and offering useful suggestions during the writing process. We
also wish to extend our appreciation to the reviewers for the priceless
feedback they provided.

Funding. The first author was financially supported by the University of Turku
Graduate School, Doctoral Programme in Exact Sciences (EXACTUS). The research
of the second author was supported by the Magnus Ehrnrooth Foundation.

### Conflict of interest.

The authors declare that they have no conflict of interest.

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Asymptotic Analysis of Synchronous Signal Processing

Marc Vilà-Insa 0000-0002-7032-1411,
and Jaume Riba 0000-0002-5515-8169, Senior Member, IEEE
This work was (partially) funded by project RODIN (PID2019-105717RB-C22) by MCIU/AEI/10.13039/501100011033, project MAYTE (PID2022-136512OB-C21) by MCIU/AEI/10.13039/501100011033 and ERDF “A way of making Europe”, grant 2021 SGR 01033 and grant 2022 FI SDUR 00164 by Departament de Recerca i Universitats de la Generalitat de Catalunya.
The authors are with the Signal Processing and Communications Group (SPCOM), Departament de Teoria del Senyal i Comunicacions, Universitat Politècnica de Catalunya (UPC), 08034 Barcelona, Spain (e-mail: {marc.vila.insa, jaume.riba}@upc.edu).
This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.

This paper extends various theoretical results from stationary data processing to cs processes under a unified framework.
We first derive their asymptotic eigenbasis, which provides a link between their Fourier and kl expansions, through a unitary transformation dictated by the cyclic spectrum.
By exploiting this connection and the optimalities offered by the kl representation, we study the asymptotic performance of smoothing, filtering and prediction of cs processes, without the need for deriving explicit implementations.
We obtain mmse expressions that depend on the cyclic spectrum and include classical limits based on the power spectral density as particular cases.
We conclude this work by applying the results to a practical scenario, in order to quantify the achievable gains of synchronous signal processing.

Cyclostationary processes, cyclic Wiener filter, Karhunen-Loève expansion, asymptotic bounds.

§ INTRODUCTION

§ PRELIMINARY DEFINITIONS AND BACKGROUND

§ KL REPRESENTATION OF CS PROCESSES

§ CYCLIC WIENER FILTERING: PROBLEM STATEMENT

§ ASYMPTOTIC SMOOTHING PERFORMANCE

§ ASYMPTOTIC FILTERING PERFORMANCE

§ ASYMPTOTIC PREDICTION PERFORMANCE

§ NUMERICAL ILLUSTRATION

§ CONCLUSIONS