# Chern classes in equivariant bordism

Stefan Schwede  Mathematisches Institut, Universität Bonn, Germany
<EMAIL_ADDRESS>
(Date: August 27, 2024; 2020 AMS Math. Subj. Class.: 55N22, 55N91, 55P91,
57R85)

###### Abstract.

We introduce Chern classes in $U(m)$-equivariant homotopical bordism that
refine the Conner-Floyd-Chern classes in the $\mathbf{MU}$-cohomology of
$BU(m)$. For products of unitary groups, our Chern classes form regular
sequences that generate the augmentation ideal of the equivariant bordism
rings. Consequently, the Greenlees-May local homology spectral sequence
collapses for products of unitary groups. We use the Chern classes to reprove
the $\mathbf{MU}$-completion theorem of Greenlees-May and La Vecchia.

## Introduction

Complex cobordism $\mathbf{MU}$ is arguably the most important cohomology
theory in algebraic topology. It represents the bordism theory of stably
almost complex manifolds, and it is the universal complex oriented cohomology
theory; via Quillen’s celebrated theorem [13], $\mathbf{MU}$ is the entry gate
for the theory of formal group laws into stable homotopy theory, and thus the
cornerstone of chromatic stable homotopy theory.

Tom Dieck’s homotopical equivariant bordism $\mathbf{MU}_{G}$ [17], defined
with the help of equivariant Thom spaces, strives to be the legitimate
equivariant refinement of complex cobordism, for compact Lie groups $G$. The
theory $\mathbf{MU}_{G}$ is the universal equivariantly complex oriented
theory; and for abelian compact Lie groups, the coefficient ring
$\mathbf{MU}_{G}^{*}$ carries the universal $G$-equivariant formal group law
[7]. Homotopical equivariant bordism receives a homomorphism from the
geometrically defined equivariant bordism theory; due to the lack of
equivariant transversality, this homomorphism is not an isomorphism for non-
trivial groups. In general, the equivariant bordism ring $\mathbf{MU}^{*}_{G}$
is still largely mysterious; the purpose of this paper is to elucidate its
structure for unitary groups, and for products of unitary groups.

Chern classes are important characteristic classes for complex vector bundles
that were originally introduced in singular cohomology. Conner and Floyd [4,
Corollary 8.3] constructed Chern classes for complex vector bundles in complex
cobordism; in the universal cases, these yield classes
$c_{k}\in\mathbf{MU}^{2k}(BU(m))$ that are nowadays referred to as Conner-
Floyd-Chern classes. Conner and Floyd’s construction works in much the same
way for any complex oriented cohomology theory, see [1, Part II, Lemma 4.3];
in singular cohomology, it reduces to the classical Chern classes. The purpose
of this note is to define and study Chern classes in $U(m)$-equivariant
homotopical bordism $\mathbf{MU}^{*}_{U(m)}$ that map to the Conner-Floyd-
Chern classes under tom Dieck’s bundling homomorphism [17, Proposition 1.2].
Our classes satisfy the analogous formal properties as their classical
counterparts, including the equivariant refinement of the Whitney sum formula,
see Theorem 1.4. Despite the many formal similarities, there are crucial
qualitative differences compared to Chern classes in complex oriented
cohomology theories: our Chern classes are not characterized by their
restriction to the maximal torus, and some of our Chern classes are zero-
divisors, see Remark 1.2.

We will use our Chern classes and the splitting of [15] to prove new structure
results about the equivariant bordism rings $\mathbf{MU}^{*}_{U(m)}$ for
unitary groups, or more generally for products of unitary groups. To put this
into context, we recall that in the special case when $G$ is an abelian
compact Lie group, the graded ring $\mathbf{MU}^{*}_{G}$ is concentrated in
even degrees and free as a module over the non-equivariant cobordism ring
$\mathbf{MU}^{*}$ [3, Theorem 5.3], [10], and the bundling homomorphism
$\mathbf{MU}^{*}_{G}\longrightarrow\mathbf{MU}^{*}(BG)$ is completion at the
augmentation ideal of $\mathbf{MU}^{*}_{G}$ [2, Theorem 1.1], [11]. For non-
abelian compact Lie groups $G$, however, the equivariant bordism rings
$\mathbf{MU}^{*}_{G}$ are still largely mysterious.

The main result of this note is the following:

Theorem. Let $m\geq 1$ be a natural number.

  1. (i)

The sequence of Chern classes $c_{m}^{(m)},c_{m-1}^{(m)},\dots,c_{1}^{(m)}$ is
a regular sequence that generates the augmentation ideal of the graded-
commutative ring $\mathbf{MU}^{*}_{U(m)}$.

  2. (ii)

The completion of $\mathbf{MU}_{U(m)}^{*}$ at the augmentation ideal is a
graded $\mathbf{MU}^{*}$-power series algebra in the above Chern classes.

  3. (iii)

The bundling homomorphism
$\mathbf{MU}_{U(m)}^{*}\longrightarrow\mathbf{MU}^{*}(BU(m))$ extends to an
isomorphism

$(\mathbf{MU}_{U(m)}^{*})^{\wedge}_{I}\ \longrightarrow\
\mathbf{MU}^{*}(BU(m))$

from the completion at the augmentation ideal.

We prove this result as a special case of Theorem 2.2 below; the more general
version applies to products of unitary groups. As we explain in Remark 2.4,
the regularity of the Chern classes also implies that the Greenlees-May local
homology spectral sequence converging to $\mathbf{MU}^{*}(BU(m))$ degenerates
because the relevant local homology groups vanish in positive degrees. As
another application we use the Chern classes in equivariant bordism to give a
reformulation and self-contained proof of work of Greenlees-May [6] and La
Vecchia [8] on the completion theorem for $\mathbf{MU}_{G}$, see Theorem 3.5.

##  1\. Equivariant $\mathbf{MU}$-Chern classes

In this section we introduce the Chern classes in $U(m)$-equivariant
homotopical bordism, see Definition 1.1. We establish their basic properties
in Theorem 1.4, including a Whitney sum formula and the fact that the bundling
homomorphism takes our Chern classes to the Conner-Floyd-Chern classes in
$\mathbf{MU}$-cohomology.

We begin by fixing our notation. For a compact Lie group $G$, we write
$\mathbf{MU}_{G}$ for the $G$-equivariant homotopical bordism spectrum
introduced by tom Dieck [17]. For our purposes, it is highly relevant that the
theories $\mathbf{MU}_{G}$ for varying compact Lie groups $G$ assemble into a
global stable homotopy type, see [14, Example 6.1.53]. For an integer $n$, we
write $\mathbf{MU}_{G}^{n}=\pi_{-n}^{G}(\mathbf{MU})$ for the $G$-equivariant
coefficient group in cohomological degree $n$.

Since $\mathbf{MU}$ comes with the structure of a global ring spectrum, it
supports graded-commutative multiplications on $\mathbf{MU}_{G}^{*}$, as well
as external multiplication pairings

$\times\ :\ \mathbf{MU}_{G}^{k}\times\mathbf{MU}_{K}^{l}\ \longrightarrow\
\mathbf{MU}_{G\times K}^{k+l}$

for all pairs of compact Lie groups $G$ and $K$. We write $\nu_{k}$ for the
tautological representation of the unitary group $U(k)$ on $\mathbb{C}^{k}$;
we denote its Euler class by

$e_{k}\ =\ e(\nu_{k})\ \in\ \mathbf{MU}^{2k}_{U(k)}\ ,$

compare [17, page 347]. We write $U(k,m-k)$ for the block subgroup of $U(m)$
consisting of matrices of the form $(\begin{smallmatrix}A&0\\\
0&B\end{smallmatrix})$ for $(A,B)\in U(k)\times U(m-k)$. We write
$\operatorname{tr}_{U(k,m-k)}^{U(m)}:\mathbf{MU}_{U(k,m-k)}^{*}\longrightarrow\mathbf{MU}_{U(m)}^{*}$
for the transfer associated to the inclusion $U(k,m-k)\longrightarrow U(m)$,
see for example [14, Construction 3.2.11].

######  Definition 1.1.

For $0\leq k\leq m$, the $k$-th Chern class in equivariant complex bordism is
the class

$c_{k}^{(m)}\ =\ \operatorname{tr}_{U(k,m-k)}^{U(m)}(e_{k}\times 1_{m-k})\
\in\ \mathbf{MU}^{2k}_{U(m)}\ ,$

where $1_{m-k}\in\mathbf{MU}_{U(m-k)}^{0}$ is the multiplicative unit. We also
set $c_{k}^{(m)}=0$ for $k>m$.

In the extreme cases $k=0$ and $k=m$, we recover familiar classes: since
$e_{0}$ is the multiplicative unit in the non-equivariant cobordism ring
$\mathbf{MU}^{*}$, the class $c_{0}^{(m)}=1_{m}$ is the multiplicative unit in
$\mathbf{MU}_{U(m)}^{0}$. In the other extreme, $c_{m}^{(m)}=e_{m}=e(\nu_{m})$
is the Euler class of the tautological $U(m)$-representation. As we will show
in Theorem 1.4 (ii), the classes $c_{k}^{(m)}$ are compatible in $m$ under
restriction to smaller unitary groups.

######  Remark 1.2.

We alert the reader that the restriction homomorphism
$\operatorname{res}^{U(m)}_{T^{m}}:\mathbf{MU}^{*}_{U(m)}\longrightarrow\mathbf{MU}^{*}_{T^{m}}$
is not injective for $m\geq 2$, where $T^{m}$ denotes a maximal torus in
$U(m)$. So the Chern classes in $\mathbf{MU}^{*}_{U(m)}$ are not characterized
by their restrictions to the maximal torus – in contrast to the non-
equivariant situation for complex oriented cohomology theories. To show this
we let $N$ denote the maximal torus normalizer inside $U(m)$. The class

$1-\operatorname{tr}_{N}^{U(m)}(1)\ \in\ \mathbf{MU}^{0}_{U(m)}$

has infinite order because the $U(m)$-geometric fixed point map takes it to
the multiplicative unit; in particular, this class is nonzero. The double
coset formula [9, IV Corollary 6.7 (i)]

$\operatorname{res}^{U(m)}_{T^{m}}(\operatorname{tr}_{N}^{U(m)}(1))\ =\
\operatorname{res}^{N}_{T^{m}}(1)\ =\ 1$

implies that the class $1-\operatorname{tr}_{N}^{U(m)}(1)$ lies in the kernel
of the restriction homomorphism
$\operatorname{res}^{U(m)}_{T^{m}}:\mathbf{MU}^{0}_{U(m)}\longrightarrow\mathbf{MU}^{0}_{T^{m}}$.

Moreover, the Chern class $c_{1}^{(2)}$ is a zero-divisor in the ring
$\mathbf{MU}^{*}_{U(2)}$, also in stark contrast to Chern classes in complex
oriented cohomology theories. Indeed, reciprocity for restriction and
transfers [14, Corollary 3.5.17 (v)] yields the relation

$\displaystyle c_{1}^{(2)}\cdot(1-\operatorname{tr}_{N}^{U(2)}(1))\ $
$\displaystyle=\ \operatorname{tr}_{U(1,1)}^{U(2)}(e_{1}\times
1)\cdot(1-\operatorname{tr}_{N}^{U(2)}(1))$ $\displaystyle=\
\operatorname{tr}_{U(1,1)}^{U(2)}((e_{1}\times
1)\cdot\operatorname{res}^{U(2)}_{U(1,1)}(1-\operatorname{tr}_{N}^{U(2)}(1)))\
=\ 0\ .$

One can also show that the class $1-\operatorname{tr}_{N}^{U(2)}(1)$ is
infinitely divisible by the Euler class $e_{2}=c_{2}^{(2)}$; so it is also in
the kernel of the completion map at the ideal $(e_{2})$.

The Chern class $c_{k}^{(m)}$ is defined as a transfer; so identifying its
restriction to a subgroup of $U(m)$ involves a double coset formula. The
following double coset formula will take care of all cases we need in this
paper; it ought to be well-known to experts, but I do not know a reference.
The case $l=1$ is established in [16, Lemma 4.2], see also [14, Example
3.4.13]. The double coset space $U(i,j)\backslash U(m)/U(k,l)$ is discussed at
various places in the literature, for example [12, Example 3], but I have not
seen the resulting double coset formula spelled out.

######  Proposition 1.3 (Double coset formula).

Let $i,j,k,l$ be positive natural numbers such that $i+j=k+l$. Then

$\operatorname{res}^{U(i+j)}_{U(i,j)}\circ\operatorname{tr}_{U(k,l)}^{U(k+l)}\
=\ \sum_{0,k-j\leq d\leq
i,k}\,\operatorname{tr}_{U(d,i-d,k-d,j-k+d)}^{U(i,j)}\circ\gamma_{d}^{*}\circ\operatorname{res}^{U(k,l)}_{U(d,k-d,i-d,l-i+d)}\
,$

where $\gamma_{d}\in U(i+j)$ is the permutation matrix of the shuffle
permutation $\chi_{d}\in\Sigma_{i+j}$ given by

$\chi_{d}(a)\ =\ \begin{cases}a&\text{ for $1\leq a\leq d$,}\\\ a-d+i&\text{
for $d+1\leq a\leq k$,}\\\ a+d-k&\text{ for $k+1\leq a\leq k+i-d$, and}\\\
a&\text{ for $a>k+i-d$.}\end{cases}$

###### Proof.

We refer to [9, IV 6] or [14, Theorem 3.4.9] for the general double coset
formula for $\operatorname{res}^{G}_{K}\circ\operatorname{tr}_{H}^{G}$ for two
closed subgroups $H$ and $K$ of a compact Lie group $G$; we need to specialize
it to the situation at hand. We first consider a matrix $A\in U(m)$ such that
the center $Z$ of $U(i,j)$ is not contained in the $U(i,j)$-stabilizer

$S_{A}\ =\ U(i,j)\cap{{}^{A}U(k,l)}$

of the coset $A\cdot U(k,l)$. Then $S_{A}\cap Z$ is a proper subgroup of the
center $Z$ of $U(i,j)$, which is isomorphic to $U(1)\times U(1)$. So
$S_{A}\cap Z$ has strictly smaller dimension than $Z$. Since the center of
$U(i,j)$ is contained in the normalizer of $S_{A}$, we conclude that the group
$S_{A}$ has an infinite Weyl group inside $U(i,j)$. All summands in the double
coset formula indexed by such points then involve transfers with infinite Weyl
groups, and hence they vanish.

So all non-trivial contributions to the double coset formula stem from double
cosets $U(i,j)\cdot A\cdot U(k,l)$ such that $S_{A}$ contains the center of
$U(i,j)$. In particular the matrix $\left(\begin{smallmatrix}-E_{i}&0\\\
0&E_{j}\end{smallmatrix}\right)$ then belongs to $S_{A}$. We write
$L=A\cdot(\mathbb{C}^{k}\oplus 0^{l})$, a complex $k$-plane in
$\mathbb{C}^{k+l}$; we consider $x\in\mathbb{C}^{i}$ and $y\in\mathbb{C}^{j}$
such that $(x,y)\in L$. Because $\left(\begin{smallmatrix}-E_{i}&0\\\
0&E_{j}\end{smallmatrix}\right)\cdot L=L$, we deduce that $(-x,y)\in L$. Since
$(x,y)$ and $(-x,y)$ belong to $L$, so do the vectors $(x,0)$ and $(y,0)$. We
have thus shown that the $k$-plane $L=A\cdot(\mathbb{C}^{k}\oplus 0^{l})$ is
spanned by the intersections

$L\cap(\mathbb{C}^{i}\oplus 0^{j})\text{\qquad
and\qquad}L\cap(0^{i}\oplus\mathbb{C}^{j})\ .$

We organize the cosets with this property by the dimension of the first
intersection: we define $M_{d}$ as the closed subspace of $U(m)/U(k,l)$
consisting of those cosets $A\cdot U(k,l)$ such that

$\dim_{\mathbb{C}}(L\cap(\mathbb{C}^{i}\oplus 0^{j}))\ =\ d\text{\qquad
and\qquad}\dim_{\mathbb{C}}(L\cap(0^{i}\oplus\mathbb{C}^{j}))\ =\ k-d\ .$

If $M_{d}$ is non-empty, we must have $0,k-j\leq d\leq i,k$. The group
$U(i,j)$ acts transitively on $M_{d}$, and the coset $\gamma_{d}\cdot U(k,l)$
belongs to $M_{d}$; so $M_{d}$ is the $U(i,j)$-orbit type manifold of
$U(m)/U(k,l)$ for the conjugacy class of

$S_{\gamma_{d}}\ =\ U(i,j)\cap{{}^{\gamma_{d}}U(k,l)}\ =\ U(d,i-d,k-d,j-k+d)\
.$

The corresponding orbit space $U(i,j)\backslash
M_{d}=U(i,j)\cdot\gamma_{d}\cdot U(k,l)$ is a single point inside the double
coset space, so its internal Euler characteristic is 1. This orbit type thus
contributes the summand

$\operatorname{tr}_{U(d,i-d,k-d,j-k+d)}^{U(i,j)}\circ\gamma_{d}^{*}\circ\operatorname{res}^{U(k,l)}_{U(d,k-d,i-d,l-i+d)}$

to the double coset formula. ∎

In [4, Corollary 8.3], Conner and Floyd define Chern classes for complex
vector bundles in the non-equivariant $\mathbf{MU}$-cohomology rings. In the
universal cases, these yield classes $c_{k}\in\mathbf{MU}^{2k}(BU(m))$ that
are nowadays referred to as Conner-Floyd-Chern classes. The next theorem
spells out the key properties of our Chern classes $c_{k}^{(m)}$; parts (i),
(ii) and (iii) roughly say that all the familiar structural properties of the
Conner-Floyd-Chern classes in $\mathbf{MU}^{*}(BU(m))$ already hold for our
Chern classes in $U(m)$-equivariant $\mathbf{MU}$-theory. Part (iv) of the
theorem refers to the bundling maps
$\mathbf{MU}_{G}^{*}\longrightarrow\mathbf{MU}^{*}(BG)$ defined by tom Dieck
in [17, Proposition 1.2].

######  Theorem 1.4.

The Chern classes in homotopical equivariant bordism enjoy the following
properties.

  1. (i)

For all $0\leq k\leq m=i+j$, the relation

$\operatorname{res}^{U(m)}_{U(i,j)}(c_{k}^{(m)})\ =\
\sum_{d=0,\dots,k}c_{d}^{(i)}\times c_{k-d}^{(j)}$

holds in the group $\mathbf{MU}_{U(i,j)}^{2k}$.

  2. (ii)

The relation

$\operatorname{res}^{U(m)}_{U(m-1)}(c_{k}^{(m)})\ =\
\begin{cases}c_{k}^{(m-1)}&\text{ for $0\leq k\leq m-1$, and}\\\ \ 0&\text{
for $k=m$}\end{cases}$

holds in the group $\mathbf{MU}_{U(m-1)}^{2k}$.

  3. (iii)

Let $T^{m}$ denote the diagonal maximal torus of $U(m)$. Then the restriction
homomorphism

$\operatorname{res}^{U(m)}_{T^{m}}\ :\ \mathbf{MU}_{U(m)}^{2k}\
\longrightarrow\ \mathbf{MU}^{2k}_{T^{m}}$

takes the class $c_{k}^{(m)}$ to the $k$-th elementary symmetric polynomial in
the classes $p_{1}^{*}(e_{1}),\dots,p_{m}^{*}(e_{1})$, where
$p_{i}:T^{m}\longrightarrow T=U(1)$ is the projection to the $i$-th factor.

  4. (iv)

The bundling map

$\mathbf{MU}_{U(m)}^{*}\ \longrightarrow\ \mathbf{MU}^{*}(BU(m))$

takes the class $c_{k}^{(m)}$ to the $k$-th Conner-Floyd-Chern class.

###### Proof.

(i) This property exploits the double coset formula for
$\operatorname{res}^{U(m)}_{U(i,j)}\circ\operatorname{tr}_{U(k,m-k)}^{U(m)}$
recorded in Proposition 1.3, which is the second equation in the following
list:

$\displaystyle\operatorname{res}^{U(m)}_{U(i,j)}(c_{k}^{(m)})\ $
$\displaystyle=\
\operatorname{res}^{U(m)}_{U(i,j)}(\operatorname{tr}_{U(k,m-k)}^{U(m)}(e_{k}\times
1_{m-k}))$ $\displaystyle=\
\sum_{d=0,\dots,k}\operatorname{tr}_{U(d,i-d,k-d,j-k+d)}^{U(i,j)}(\gamma_{d}^{*}(\operatorname{res}^{U(k,m-k)}_{U(d,k-d,i-d,j-k+d)}(e_{k}\times
1_{m-k})))$ $\displaystyle=\
\sum_{d=0,\dots,k}\operatorname{tr}_{U(d,i-d,k-d,j-k+d)}^{U(i,j)}(\gamma_{d}^{*}(e_{d}\times
e_{k-d}\times 1_{i-d}\times 1_{j-k+d}))$ $\displaystyle=\
\sum_{d=0,\dots,k}\operatorname{tr}_{U(d,i-d,k-d,j-k+d)}^{U(i,j)}(e_{d}\times
1_{i-d}\times e_{k-d}\times 1_{j-k+d})$ $\displaystyle=\
\sum_{d=0,\dots,k}\operatorname{tr}_{U(d,i-d)}^{U(i)}(e_{d}\times
1_{i-d})\times\operatorname{tr}_{U(k-d,j-k+d)}^{U(j)}(e_{k-d}\times
1_{j-k+d})$ $\displaystyle=\ \sum_{d=0,\dots,k}c_{d}^{(i)}\times
c_{k-d}^{(j)}$

Part (ii) for $k<m$ follows from part (i) by restriction from $U(m-1,1)$ to
$U(m-1)$:

$\displaystyle\operatorname{res}^{U(m)}_{U(m-1)}(c_{k}^{(m)})\ $
$\displaystyle=\
\operatorname{res}^{U(m-1,1)}_{U(m-1)}(\operatorname{res}^{U(m)}_{U(m-1,1)}(c_{k}^{(m)}))$
$\displaystyle=\ \operatorname{res}^{U(m-1,1)}_{U(m-1)}(c_{k-1}^{(m-1)}\times
c_{1}^{(1)}\ +\ c_{k}^{(m-1)}\times c_{0}^{(1)})$ $\displaystyle=\
c_{k-1}^{(m-1)}\times\operatorname{res}^{U(1)}_{1}(c_{1}^{(1)})\ +\
c_{k}^{(m-1)}\times\operatorname{res}^{U(1)}_{1}(c_{0}^{(1)})\ =\
c_{k}^{(m-1)}\ .$

We have used that the class $c_{1}^{(1)}=e_{1}$ is in the kernel of the
augmentation
$\operatorname{res}^{U(1)}_{1}:\mathbf{MU}_{U(1)}^{*}\longrightarrow\mathbf{MU}^{*}$.
The Euler class $c_{m}^{(m)}=e(\nu_{m})$ restricts to 0 in
$\mathbf{MU}^{*}_{U(m-1)}$ because the restriction of the tautological
$U(m)$-representation to $U(m-1)$ splits off a trivial 1-dimensional summand.

(iii) An inductive argument based on property (i) shows the desired relation:

$\displaystyle\operatorname{res}^{U(m)}_{T^{m}}(c_{k}^{(m)})\ $
$\displaystyle=\ \operatorname{res}^{U(m)}_{U(1,\dots,1)}(c_{k}^{(m)})$
$\displaystyle=\ \sum_{A\subset\\{1,\dots,m\\},|A|=k}\quad\prod_{a\in
A}p_{a}^{*}(c_{1}^{(1)})\cdot\prod_{b\not\in A}p_{b}^{*}(c_{0}^{(1)})$
$\displaystyle=\ \sum_{A\subset\\{1,\dots,m\\},|A|=k}\quad\prod_{a\in
A}p_{a}^{*}(e_{1})\ .$

(iv) As before we let $T^{m}$ denote the diagonal maximal torus in $U(m)$. The
splitting principle holds for non-equivariant complex oriented cohomology
theories, see for example [5, Proposition 8.10]. In other words, the right
vertical map in the commutative square of graded rings is injective:

$\textstyle{\mathbf{MU}^{*}_{U(m)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{res}^{U(m)}_{T^{m}}}$$\textstyle{\mathbf{MU}^{*}(BU(m))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(Bi)^{*}}$$\textstyle{\mathbf{MU}^{*}_{T^{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{MU}^{*}(BT^{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{MU}^{*}[[p_{1}^{*}(e_{1}),\dots,p_{m}^{*}(e_{1})]]}$

The $k$-th Conner-Floyd-Chern class is characterized as the unique element of
$\mathbf{MU}^{2k}(BU(m))$ that maps to the $k$-th elementary symmetric
polynomial in the classes $p_{1}^{*}(e_{1}),\dots,p_{m}^{*}(e_{1})$. Together
with part (iii), this proves the claim. ∎

##  2\. Regularity results

In this section we use the Chern classes to formulate new structural
properties of the equivariant bordism ring $\mathbf{MU}_{U(m)}^{*}$. In
particular, we can say what $\mathbf{MU}_{U(m)}^{*}$ looks like after dividing
out some of the Chern classes, and after completing at the Chern classes. The
following theorem states these facts more generally for $U(m)\times G$ instead
of $U(m)$; by induction on the number of factors, we can then deduce
corresponding results for products of unitary groups, see Theorem 2.2. The
results in this section make crucial use of the splitting theorem for global
functors established in [15].

######  Theorem 2.1.

For every compact Lie group $G$ and all $0\leq k\leq m$, the sequence of Chern
classes

$(c_{m}^{(m)}\times 1_{G},\ c_{m-1}^{(m)}\times 1_{G},\dots,\
c_{k+1}^{(m)}\times 1_{G})$

is a regular sequence in the graded-commutative ring
$\mathbf{MU}^{*}_{U(m)\times G}$ that generates the kernel of the surjective
restriction homomorphism

$\operatorname{res}^{U(m)\times G}_{U(k)\times G}\ :\ \mathbf{MU}_{U(m)\times
G}^{*}\ \longrightarrow\mathbf{MU}_{U(k)\times G}^{*}\ .$

In particular, the sequence of Chern classes
$(c_{m}^{(m)},c_{m-1}^{(m)},\dots,c_{1}^{(m)})$ is a regular sequence that
generates the augmentation ideal of the graded-commutative ring
$\mathbf{MU}^{*}_{U(m)}$.

###### Proof.

We argue by downward induction on $k$. The induction starts with $k=m$, where
there is nothing to show. Now we assume the claim for some $k\leq m$, and we
deduce it for $k-1$. The inductive hypothesis shows that $c_{m}^{(m)}\times
1_{G},\dots,c_{k+1}^{(m)}\times 1_{G}$ is a regular sequence in the graded-
commutative ring $\mathbf{MU}^{*}_{U(m)\times G}$, and that the restriction
homomorphism $\operatorname{res}^{U(m)\times G}_{U(k)\times G}$ factors
through an isomorphism

$\mathbf{MU}_{U(m)\times G}^{*}/(c_{m}^{(m)}\times
1_{G},\dots,c_{k+1}^{(m)}\times 1_{G})\ \cong\mathbf{MU}_{G\times U(k)}^{*}\
.$

We exploit that the various equivariant bordism spectra $\mathbf{MU}_{G}$
underlie a global spectrum, see [14, Example 6.1.53]; thus the restriction
homomorphism $\operatorname{res}^{U(k)\times G}_{U(k-1)\times G}$ is
surjective by Theorem 1.4 and Proposition 2.2 of [15]. Hence the standard long
exact sequence unsplices into a short exact sequence of graded
$\mathbf{MU}^{*}$-modules:

$0\ \longrightarrow\ \mathbf{MU}_{U(k)\times G}^{*-2k}\
\xrightarrow{(e_{k}\times 1_{G})\cdot-\ }\ \mathbf{MU}_{U(k)\times
G}^{*}\xrightarrow{\operatorname{res}^{U(k)\times G}_{U(k-1)\times G}}\
\mathbf{MU}_{U(k-1)\times G}^{*}\ \longrightarrow\ 0$

Because

$\operatorname{res}^{U(m)\times G}_{U(k)\times G}(c_{k}^{(m)}\times 1_{G})\ =\
c_{k}^{(k)}\times 1_{G}\ =\ e_{k}\times 1_{G}\ ,$

we conclude that $c_{k}^{(m)}\times 1_{G}$ is a non zero-divisor in
$\mathbf{MU}_{U(m)\times G}^{*}/(c_{m}^{(m)}\times 1_{G},c_{m-1}^{(m)}\times
1_{G},\dots,c_{k+1}^{(m)}\times 1_{G})$, and that additionally dividing out
$c_{k}^{(m)}\times 1_{G}$ yields $\mathbf{MU}_{U(k-1)\times G}^{*}$. This
completes the inductive step. ∎

We can now identify the completion of $\mathbf{MU}^{*}_{U(m)}$ at the
augmentation ideal as an $\mathbf{MU}^{*}$-power series algebra on the Chern
classes. We state this somewhat more generally for products of unitary groups,
which we write as

$U(m_{1},\dots,m_{l})\ =\ U(m_{1})\times\dots\times U(m_{l})\ ,$

for natural numbers $m_{1},\dots,m_{l}\geq 1$. For $1\leq i\leq l$, we write
$p_{i}:U(m_{1},\dots,m_{l})\longrightarrow U(m_{i})$ for the projection to the
$i$-th factor, and we set

$c^{[i]}_{k}\ =\ p_{i}^{*}(c_{k}^{(m_{i})})\ =\
1_{U(m_{1},\dots,m_{i-1})}\times c_{k}^{(m_{i})}\times
1_{U(m_{i+1},\dots,m_{l})}\ \in\ \mathbf{MU}_{U(m_{1},\dots,m_{l})}^{2k}\ .$

The following theorem was previously known for tori, i.e., for
$m_{1}=\dots=m_{l}=1$.

######  Theorem 2.2.

Let $m_{1},\dots,m_{l}\geq 1$ be positive integers.

  1. (i)

The sequence of Chern classes

(2.3)
$c_{m_{1}}^{[1]},\dots,c_{1}^{[1]},c_{m_{2}}^{[2]},\dots,c_{1}^{[2]},\dots,c_{m_{l}}^{[l]},\dots,c_{1}^{[l]}$

is a regular sequence that generates the augmentation ideal of the graded-
commutative ring $\mathbf{MU}^{*}_{U(m_{1},\dots,m_{l})}$.

  2. (ii)

The completion of $\mathbf{MU}_{U(m_{1},\dots,m_{l})}^{*}$ at the augmentation
ideal is a graded $\mathbf{MU}^{*}$-power series algebra in the Chern classes
(2.3).

  3. (iii)

The bundling map
$\mathbf{MU}_{U(m_{1},\dots,m_{l})}^{*}\longrightarrow\mathbf{MU}^{*}(BU(m_{1},\dots,m_{l}))$
extends to an isomorphism

$(\mathbf{MU}_{U(m_{1},\dots,m_{l})}^{*})^{\wedge}_{I}\ \longrightarrow\
\mathbf{MU}^{*}(BU(m_{1},\dots,m_{l}))$

from the completion at the augmentation ideal.

###### Proof.

Part (i) follows from Theorem 2.1 by induction on the number $l$ of factors.

We prove parts (ii) and (iii) together. We must show that for every $n\geq 1$,
$\mathbf{MU}_{U(m_{1},\dots,m_{l})}^{*}/I^{n}$ is free as an
$\mathbf{MU}^{*}$-module on the monomials of degree less than $n$ in the Chern
classes (2.3). There is nothing to show for $n=1$. The short exact sequence

$0\ \longrightarrow\ I^{n}/I^{n+1}\ \longrightarrow\
\mathbf{MU}_{U(m_{1},\dots,m_{l})}^{*}/I^{n+1}\ \longrightarrow\
\mathbf{MU}_{U(m_{1},\dots,m_{l})}^{*}/I^{n}\ \longrightarrow\ 0$

and the inductive hypothesis reduce the claim to showing that $I^{n}/I^{n+1}$
is free as an $\mathbf{MU}^{*}$-module on the monomials of degree exactly $n$
in the Chern classes (2.3). Since the augmentation ideal $I$ is generated by
these Chern classes, the $n$-th power $I^{n}$ is generated, as a module over
$\mathbf{MU}_{U(m_{1},\dots,m_{l})}^{*}$, by the monomials of degree $n$. So
$I^{n}/I^{n+1}$ is generated by these monomials as a module over
$\mathbf{MU}^{*}$.

The bundling map
$\mathbf{MU}_{U(m_{1},\dots,m_{l})}^{*}\longrightarrow\mathbf{MU}^{*}(BU(m_{1},\dots,m_{l}))$
is a homomorphism of augmented $\mathbf{MU}^{*}$-algebras, and it takes the
Chern class $c_{k}^{[i]}$ to the inflation of the $k$-th Conner-Floyd-Chern
class along the projection to the $i$-th factor. By the theory of complex
orientations, the collection of these Conner-Floyd-Chern classes are
$\mathbf{MU}^{*}$-power series generators of
$\mathbf{MU}^{*}(BU(m_{1},\dots,m_{l}))$; in particular, the images of the
Chern class monomials are $\mathbf{MU}^{*}$-linearly independent in
$\mathbf{MU}^{*}(BU(m_{1},\dots,m_{l}))$. Hence these classes are themselves
linearly independent in $I^{n}/I^{n+1}$. ∎

######  Remark 2.4.

Greenlees and May [6, Corollary 1.6] construct a local homology spectral
sequence

$E_{2}^{p,q}\ =\ H^{I}_{-p,-p}(\mathbf{MU}_{G}^{*})\ \Longrightarrow\
\mathbf{MU}^{p+q}(BG)\ .$

The regularity results about Chern classes from Theorem 2.2 imply that
whenever $G=U(m_{1},\dots,m_{l})$ is a product of unitary groups, the
$E_{2}^{p,q}$-term vanishes for all $p\neq 0$, and the spectral sequence
degenerates into the isomorphism

$E_{2}^{0,*}\ \cong\ (\mathbf{MU}_{U(m_{1},\dots,m_{l})}^{*})^{\wedge}_{I}\
\cong\ \mathbf{MU}^{*}(BU(m_{1},\dots,m_{l}))$

of Theorem 2.2 (iii).

######  Remark 2.5.

The previous regularity theorems are special cases of the following more
general results that hold for every global $\mathbf{MU}$-module $E$:

  * •

For every compact Lie group $G$, the sequence of Chern classes
$c_{m}^{(m)}\times 1_{G},\dots,c_{1}^{(m)}\times 1_{G}$ acts regularly on the
graded $\mathbf{MU}^{*}_{U(m)\times G}$-module $E^{*}_{U(m)\times G}$.

  * •

The restriction homomorphism

$\operatorname{res}^{U(m)\times G}_{G}\ :\ E_{U(m)\times G}^{*}\
\longrightarrow\ E_{G}^{*}$

factors through an isomorphism

$E_{U(m)\times G}^{*}/(c_{m}^{(m)}\times 1_{G},\dots,c_{1}^{(m)}\times 1_{G})\
\cong\ E_{G}^{*}\ .$

  * •

For all $m_{1},\dots,m_{l}\geq 1$, the sequence of Chern classes (2.3) acts
regularly on the graded $\mathbf{MU}^{*}_{U(m_{1},\dots,m_{l})}$-module
$E^{*}_{U(m_{1},\dots,m_{l})}$.

As in Remark 2.4, the regularity properties also imply the degeneracy of the
Greenlees-May local homology spectral sequence converging to
$E^{*}(BU(m_{1},\dots,m_{l}))$.

##  3\. The $\mathbf{MU}$-completion theorem via Chern classes

In this section we use the Chern classes to reformulate the
$\mathbf{MU}_{G}$-completion theorem of Greenlees-May [6] and La Vecchia [8],
for any compact Lie group $G$, and we give a short and self-contained proof.
We emphasize that the essential arguments of this section are all contained in
[6] and [8]; the Chern classes let us arrange them in a more conceptual and
concise way. The references [6, 8] ask for a finitely generated ideal of
$\mathbf{MU}_{G}^{*}$ that is ‘sufficiently large’ in the sense of [6,
Definition 2.4]; while we have no need to explicitly mention sufficiently
large ideals, the new insight is that the ideal generated by the Chern classes
of any faithful $G$-representation is ‘sufficiently large’.

######  Construction 3.1 (Chern classes of representations).

We let $V$ be a complex representation of a compact Lie group $G$. We let
$\rho:G\longrightarrow U(m)$ be a continuous homomorphism that classifies $V$,
i.e., such that $\rho^{*}(\nu_{m})$ is isomorphic to $V$; here
$m=\dim_{\mathbb{C}}(V)$. The $k$-th Chern class of $V$ is

$c_{k}(V)\ =\ \rho^{*}(c_{k}^{(m)})\ \in\ \mathbf{MU}_{G}^{2k}\ .$

In particular, $c_{0}(V)=1$, $c_{m}(V)=e(V)$ is the Euler class, and
$c_{k}(V)=0$ for $k>m$.

######  Example 3.2.

As an example, we consider the tautological representation $\nu_{2}$ of
$SU(2)$ on $\mathbb{C}^{2}$. By the general properties of Chern classes we
have $c_{0}(\nu_{2})=1$, $c_{2}(\nu_{2})=e(\nu_{2})$ is the Euler class, and
$c_{k}(\nu_{2})=0$ for $k\geq 3$. The first Chern class of $\nu_{2}$ can be
rewritten by using a double coset formula as follows:

$\displaystyle c_{1}(\nu_{2})\ $ $\displaystyle=\
\operatorname{res}^{U(2)}_{SU(2)}(c_{1}^{(2)})\ =\
\operatorname{res}^{U(2)}_{SU(2)}(\operatorname{tr}_{U(1,1)}^{U(2)}(e_{1}\times
1))$ $\displaystyle=\
\operatorname{tr}_{T}^{SU(2)}(\operatorname{res}^{U(1,1)}_{T}(e_{1}\times 1))\
=\ \operatorname{tr}_{T}^{SU(2)}(e(\chi))\ .$

Here $T=\\{(\begin{smallmatrix}\lambda&0\\\ 0&\lambda^{-1}\end{smallmatrix})\
:\ \lambda\in U(1)\\}$ is the diagonal maximal torus of $SU(2)$, $\chi:T\cong
U(1)$ is the character that projects onto the upper left diagonal entry, and
$e(\chi)\in\mathbf{MU}^{2}_{T}$ is its Euler class.

######  Construction 3.3.

We construct a $G$-equivariant $\mathbf{MU}_{G}$-module $K(G,V)$ associated to
a complex representation $V$ of a compact Lie group $G$. The construction is a
special case of one used by Greenlees and May [6, Section 1], based on the
sequence of Chern classes $c_{1}(V),\dots,c_{m}(V)$, where
$m=\dim_{\mathbb{C}}(V)$.

For any equivariant homotopy class $x\in\mathbf{MU}_{G}^{l}$, we write
$\mathbf{MU}_{G}[1/x]$ for the $\mathbf{MU}_{G}$-module localization of
$\mathbf{MU}_{G}$ with $x$ inverted; in other words, $\mathbf{MU}_{G}[1/x]$ is
a homotopy colimit (mapping telescope) in the triangulated category of the
sequence

$\mathbf{MU}_{G}\ \xrightarrow{-\cdot x}\ \Sigma^{l}\mathbf{MU}_{G}\
\xrightarrow{-\cdot x}\Sigma^{2l}\mathbf{MU}_{G}\ \xrightarrow{-\cdot x}\
\Sigma^{3l}\mathbf{MU}_{G}\ \xrightarrow{-\cdot x}\ \dots\ \ .$

We write $K(x)$ for the fiber of the morphism
$\mathbf{MU}_{G}\longrightarrow\mathbf{MU}_{G}[1/x]$. Then we define

$K(G,V)\ =\
K(c_{1}(V))\wedge_{\mathbf{MU}_{G}}\dots\wedge_{\mathbf{MU}_{G}}K(c_{m}(V))\
.$

The smash product of the morphisms $K(c_{i}(V))\longrightarrow\mathbf{MU}_{G}$
provides a morphism of $G$-equivariant $\mathbf{MU}_{G}$-module spectra

$\epsilon_{V}\ :\ K(G,V)\ \longrightarrow\ \mathbf{MU}_{G}.$

By general principles, the module $K(G,V)$ only depends on the radical of the
ideal generated by the classes $c_{1}(V),\dots,c_{m}(V)$. But more is true: as
a consequence of Theorem 3.5 below, $K(G,V)$ is entirely independent, as a
$G$-equivariant $\mathbf{MU}_{G}$-module, of the faithful representation $V$.

######  Proposition 3.4.

Let $V$ be a faithful complex representation of a compact Lie group $G$.

  1. (i)

The morphism $\epsilon_{V}:K(G,V)\longrightarrow\mathbf{MU}_{G}$ is an
equivalence of underlying non-equivariant spectra.

  2. (ii)

For every non-trivial closed subgroup $H$ of $G$, the $H$-geometric fixed
point spectrum $\Phi^{H}(K(G,V))$ is trivial.

###### Proof.

(i) We set $m=\dim_{\mathbb{C}}(V)$. The Chern classes
$c_{1}(V),\dots,c_{m}(V)$ belong to the augmentation ideal of
$\mathbf{MU}_{G}^{*}$, so they restrict to 0 in $\mathbf{MU}_{\\{1\\}}^{*}$,
and hence the underlying non-equivariant spectrum of
$\mathbf{MU}_{G}[1/c_{i}(V)]$ is trivial for each $i=1,\dots,m$. Hence the
morphisms $K(c_{i}(V))\longrightarrow\mathbf{MU}_{G}$ are underlying non-
equivariant equivalences for $i=1,\dots,m$. So also the morphism
$\epsilon_{V}$ is an underlying non-equivariant equivalence.

(ii) We let $H$ be a non-trivial closed subgroup of $G$. We set $W=V-V^{H}$,
the orthogonal complement of the $H$-fixed points. This is a complex
$H$-representation with $W^{H}=0$; moreover, $W$ is nonzero because $H$ acts
faithfully on $V$ and $H\neq\\{1\\}$. For $k=\dim_{\mathbb{C}}(W)$ we then
have

$e(W)\ =\ c_{k}(W)\ =\ c_{k}(W\oplus V^{H})\ =\
c_{k}(\operatorname{res}^{G}_{H}(V))\ =\ \operatorname{res}^{G}_{H}(c_{k}(V))\
;$

the second equation uses the fact that adding a trivial representation leaves
Chern classes unchanged, by part (ii) of Theorem 1.4.

Since $W^{H}=0$, the geometric fixed point homomorphism
$\Phi^{H}:\mathbf{MU}_{H}^{*}\longrightarrow\Phi_{H}^{*}(\mathbf{MU})$ sends
the Euler class $e(W)=\operatorname{res}^{G}_{H}(c_{k}(V))$ to an invertible
element. The functor $\Phi^{H}\circ\operatorname{res}^{G}_{H}$ commutes with
inverting elements. Since the class
$\Phi^{H}(\operatorname{res}^{G}_{H}(c_{k}(V)))$ is already invertible, the
localization morphism
$\mathbf{MU}_{G}\longrightarrow\mathbf{MU}_{G}[1/c_{k}(V)]$ induces an
equivalence on $H$-geometric fixed points. Since the functor
$\Phi^{H}\circ\operatorname{res}^{G}_{H}$ is exact, it annihilates the fiber
$K(c_{k}(V))$ of the localization
$\mathbf{MU}_{G}\longrightarrow\mathbf{MU}_{G}[1/c_{k}(V)]$. The functor
$\Phi^{H}\circ\operatorname{res}^{G}_{H}$ is also strong monoidal, in the
sense of a natural equivalence of non-equivariant spectra

$\Phi^{H}(X\wedge_{\mathbf{MU}_{G}}Y)\ \simeq\
\Phi^{H}(X)\wedge_{\Phi^{H}(\mathbf{MU}_{G})}\Phi^{H}(Y)\ ,$

for all $G$-equivariant $\mathbf{MU}_{G}$-modules $X$ and $Y$. Since $K(G,V)$
contains $K(c_{k}(V))$ as a factor (with respect to
$\wedge_{\mathbf{MU}_{G}}$), we conclude that the spectrum $\Phi^{H}(K(G,V))$
is trivial. ∎

The following ’completion theorem’ is a reformulation of the combined work of
Greenlees-May [6, Theorem 1.3] and La Vecchia [8]. It is somewhat more precise
in that an unspecified ‘sufficiently large’ finitely generated ideal of
$\mathbf{MU}_{G}^{*}$ is replaced by the ideal generated by the Chern classes
of a faithful $G$-representation. The proof is immediate from the properties
of $K(G,V)$ listed in Proposition 3.4. We emphasize, however, that our proof
is just a different way of arranging some arguments from [6] and [8] while
taking advantage of the Chern class formalism.

Since the morphism $\epsilon_{V}:K(G,V)\longrightarrow\mathbf{MU}_{G}$ is a
non-equivariant equivalence of underlying spectra, the morphism
$EG_{+}\wedge\mathbf{MU}_{G}\longrightarrow\mathbf{MU}_{G}$ that collapses the
universal space $EG$ to a point admits a unique lift to a morphism of
$G$-equivariant $\mathbf{MU}_{G}$-modules
$\psi:EG_{+}\wedge\mathbf{MU}_{G}\longrightarrow K(G,V)$ across
$\epsilon_{V}$.

######  Theorem 3.5.

Let $V$ be a faithful complex representation of a compact Lie group $G$. Then
the morphism

$\psi\ :\ EG_{+}\wedge\mathbf{MU}_{G}\ \longrightarrow\ K(G,V)$

is an equivalence of $G$-equivariant $\mathbf{MU}_{G}$-module spectra.

###### Proof.

Because the underlying space of $EG$ is contractible, the composite

$EG_{+}\wedge\mathbf{MU}_{G}\ \xrightarrow{\ \psi\ }\ K(G,V)\ \xrightarrow{\
\epsilon_{V}\ }\ \mathbf{MU}_{G}$

is an equivariant equivalence of underlying non-equivariant spectra. Since
$\epsilon_{V}$ is an equivariant equivalence of underlying non-equivariant
spectra by Proposition 3.4, so is $\psi$. For all non-trivial closed subgroups
$H$ of $G$, source and target of $\psi$ have trivial $H$-geometric fixed
points spectra, again by Proposition 3.4. So the morphism $\psi$ induces an
equivalence on geometric fixed point spectra for all closed subgroup of $G$,
and it is thus an equivariant equivalence. ∎

## References

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# Cascading and Direct Approaches to Unsupervised
Constituency Parsing on Spoken Sentences

###### Abstract

Past work on unsupervised parsing is constrained to written form. In this
paper, we present the first study on _unsupervised spoken constituency
parsing_ given unlabeled spoken sentences and unpaired textual data. The goal
is to determine the spoken sentences’ hierarchical syntactic structure in the
form of constituency parse trees, such that each node is a span of audio that
corresponds to a constituent. We compare two approaches: (1) cascading an
unsupervised automatic speech recognition (ASR) model and an unsupervised
parser to obtain parse trees on ASR transcripts, and (2) direct training an
unsupervised parser on continuous word-level speech representations. This is
done by first splitting utterances into sequences of word-level segments, and
aggregating self-supervised speech representations within segments to obtain
segment embeddings. We find that separately training a parser on the unpaired
text and directly applying it on ASR transcripts for inference produces better
results for unsupervised parsing. Additionally, our results suggest that
accurate segmentation alone may be sufficient to parse spoken sentences
accurately. Finally, we show the direct approach may learn head-directionality
correctly for both head-initial and head-final languages without any explicit
inductive bias.

Index Terms—  Unsupervised constituency parsing, unsupervised word
segmentation, self-supervised speech representations

##  1 Introduction

Unsupervised constituency parsing is a long-standing research challenge in
natural language processing [1, 2, 3, 4] that aims to automatically determine
the syntactic constituent structure of sentences without access to any
training labels. It sheds light on how children are able to learn high-level
linguistic information, such as syntax and grammar, without expert
supervision. Additionally, constituency parse trees have also been shown to
improve and provide greater interpretability to a variety of downstream tasks
such as semantic role labeling [5], word representation learning [6], and
speech synthesis [7, 8, 9].

Fig. 1:  Diagram of our proposed direct approach to unsupervised spoken
constituency parsing, using only raw speech and unpaired text. Textual
transcripts of the input sentence are only shown for illustrative purposes.

To the best of our knowledge, syntactic parsing on speech was done exclusively
in a supervised manner, using paired text transcripts and syntactic labels for
training. However, these approaches cannot be applied to low-resource
languages without paired data. This motivates us to explore a more realistic
setting where only raw speech and limited unpaired textual data are available.

With no speech-text pairs or tree-text pairs, the first approach to
unsupervised spoken constituency parsing is to cascade unsupervised ASR with
an unsupervised parser. We compare training the parser on limited unpaired
text and ASR transcripts and find that training on ASR transcripts does not
help the model parse ASR transcripts of the same domain.

We also propose a framework to directly parse spoken input without any
intermediate textual form. First, we split an utterance into word-level
segments, and transform each segment into a continuous embedding. Then we
directly use this sequence of segment embeddings as input for our unsupervised
parser. We refer to this as the direct approach.

Contributions. (1) We perform the first investigation of unsupervised
constituency parsing on spoken sentences using only raw speech and unpaired
text. (2) We demonstrate that for parsing ASR transcripts, training on limited
unpaired text is still better than training on ASR transcripts, and we
quantify the effects of ASR errors on unsupervised parsing. (3) We propose a
framework to directly parse continuous speech without intermediate lexical
unit discovery. (4) We show that our direct approach may induce parse trees
with the correct branching direction for different spoken languages.

##  2 RELATED WORK

###  2.1 Unsupervised Constituency Parsing

Previous studies in unsupervised constituency parsing focus on obtaining
constituency tree structures from large unannotated text corpora, usually by
encouraging neural models to follow syntactic structure [4, 10, 11], or
parameterizing linguistic models with neural networks [12, 13, 14].

A recent line of work in visually-grounded grammar induction leverages paired
images to improve unsupervised constituency parsing [15, 16, 17]. We consider
AV-NSL [18] in particular to be most relevant to our work, as they extend this
approach to audio-visual learning and attempt to learn constituency parse
trees from raw speech and image pairs. Unlike AV-NSL, our work does not rely
on paired visual grounding data.

###  2.2 Parsing Speech with Supervision

Past works on syntactic parsing of speech address topics such as disfluency
detection [19, 20], or incorporating prosodic cues [21, 22]. However, most
previous works require oracle transcripts, which is an unrealistic setting.
Yoshikawa et al. [23] shows that it is possible to build a supervised
dependency parser that jointly detect disfluencies and ASR errors, and Pupier
et al. [24] build an end-to-end supervised dependency parser for French that
jointly predicts transcription and dependency tree from raw speech signals.
Both works show an improvement over cascading baseline systems.

Additionally, prosodic features are shown to be closely related to syntax [25,
26] and beneficial for both constituency parsing [27, 28] and dependency
parsing [29], even under the presence of ASR errors [30]. These works motivate
us to explore ways of using speech features to improve unsupervised
constituency parsing of spoken data.

###  2.3 Unsupervised Spoken Language Modeling

Unsupervised spoken language modeling [31] aims to learn a spoken language
model that simultaneously learns different levels of linguistic structure from
raw speech signals with little or no textual data. The ZeroSpeech Challenge
2021 [32, 33] proposes to evaluate such a model at the acoustic, lexical,
syntactic, and semantic levels. They find that while current spoken language
models excel at the acoustic and lexical levels, higher levels of linguistic
structure are much more difficult to model. They only require their models to
be able to determine how grammatical a sentence is, while our work aims to
solve the more challenging problem of producing the exact constituency
structure of a sentence.

##  3 METHOD

###  3.1 Background

Constituency parsing is usually formulated under the binary setting in order
to reduce computation complexity. [34] This setting entails that for a
sentence with $n$ words $\\{{x}_{1},{x}_{2},...,{x}_{n}\\}$, each constituent
spanning $x_{i:j}$ is composed of two constituents spanning $x_{i:k}$ and
$x_{k+1:j}$ for some $k$ such that $i\leq k<j$. Our unsupervised parser
follows the chart-based Deep Inside-Outside Recursive Autoencoder (DIORA)
framework proposed by [12] to produce binary parse trees without using any
syntactically labeled data.

####  3.1.1 Chart-based Constituency Parsing

Chart-based parsers find the optimal tree out of all valid binary parse trees
by filling the upper-triangular portion of an $n\times n$ chart with a score
$s_{i,j}$ for each cell. For $1\leq i<j\leq n$, the score in cell $(i,j)$
represents how likely the span $x_{i:j}$ is a constituent. The CKY dynamic
programming algorithm [35, 36] is then used to determine the parse tree with
the highest total score.

####  3.1.2 Unsupervised Parser Architecture: DIORA

DIORA consists of an encoder and a decoder, and operates similarly to masked
language models. The framework recursively encodes all but one of the words
from the input sentence as a vector, and optimizes that vector to reconstruct
the missing word. The core assumption is that the most efficient weights to
produce such an encoding can be used as scores for chart-based constituency
parsing. DIORA initially represents the input sentence with pretrained ELMo
character embeddings, but subsequent work [17] shows that the framework can
also be used with randomly initialized word embeddings.

###  3.2 Cascading Parsing with Unsupervised ASR

A straightforward approach to unsupervised spoken constituency parsing is to
obtain word-level transcripts from unsupervised ASR, then represent each word
with a randomly initialized vector. An unsupervised parser can then produce
parse trees using these vector sequences as input.

Our unsupervised ASR system adopts the wav2vec-U framework [37]. wav2vec-U
first phonemizes unpaired text data, then performs a series of preprocessing
steps on unlabeled speech to produce higher-level features with length similar
to phoneme sequences. They use adversarial training to train a model to
predict phoneme sequences from speech features. A weighted finite-state
transducer (WFST) trained on the unpaired text data is then used to decode the
output into words. Further improvements are be obtained through Hidden Markov
Model (HMM) self-training. The phoneme output of the HMM achieves a lower
phone error rate and can be decoded into more accurate word-level transcripts.

###  3.3 Direct Parsing on Speech Segments

The direct parsing approach extends the DIORA framework by training on
continuous word-level speech embeddings instead of ELMo embeddings. Using
continuous segment representations allows our parser to benefit from
continuous information in speech, in comparison to the discretization approach
recently proposed in spoken language modeling [31]. This design choice is
supported by AV-NSL [18], which finds that continuous segment representations
outperforms discrete representations for audio-visual parsing.

From each spoken utterance, we prepare (1) frame-level features, and (2) word-
level segments. Frame-level features can extracted from a pretrained self-
supervised speech model such as XLSR-53 [37], and word-level segments can be
determined with unsupervised word segmentation models [38, 39]. Mirroring AV-
NSL, we represent each segment with a continuous embedding parameterized by a
simple weighted average of frame-level features. Weights are determined by a
learnable two-layer MLP that is jointly optimized with the parser. This
sequence of word-level segment embeddings is then directly used as input for
our parser, then jointly optimized with the reconstruction loss proposed in
DIORA [12].

##  4 Experiments

###  4.1 Datasets, Preprocessing, and Hyperparameters

Experiments are mainly conducted on the SpokenCOCO dataset [40], a 742h
English read-speech dataset produced by 2.3k speakers reading the captions in
MSCOCO [41]. Each image in MSCOCO corresponds to 5 captions on average.
Following [16], we use the spoken captions of the 83k/5k/5k image split for
training, validation, and testing respectively. The textual captions of the
remaining 31k images are used as unpaired text data for unsupervised ASR. We
focus on the more practical setting of unsupervised spoken constituency
parsing using speech and unpaired text data only, hence we do not utilize the
image data.

Additional experiments in Korean are done on the Zeroth-Korean
corpus111https://github.com/goodatlas/zeroth, which contains 51.6hrs of audio
spoken by 105 speakers for training, and 1.6hrs by 10 different speakers for
testing. We use the utterances of 10 speakers in the original training set for
validation.

We note that due to a lack of labelled speech data, we are limited to
experimenting on high-resource languages. Following [15, 16], ground-truth
parse trees are obtained from the outputs of an off-the-shelf parser [42] on
the normalized text captions. Punctuation is removed from the trees, and we
run forced alignment using the Montreal Forced Aligner [43] to obtain oracle
word boundaries.

For all experiments, we use the same hyperparameters as the randomly
initialized DIORA experiment in [17], with a batch size of 32 and learning
rate of $5e-3$. We perform unsupervised model selection with the
reconstruction loss of DIORA on the validation set. Our cascading and direct
systems are trained for 10 epochs and 2000 batches respectively, as we found
our direct systems to converge much faster. Further details are available in
our training code222https://github.com/roger-tseng/speech-parsing.

###  4.2 Evaluation

Unsupervised constituency parsing on text is typically evaluated with $F_{1}$
score of constituents, where a match is only counted if a predicted
constituent and a oracle constituent consist of the exact same words. However,
erroneous word segmentation or ASR may introduce mismatch in the number of
word-level leaves between model predictions and ground truth parse trees.
Therefore, we match the constituents first by calculating an alignment between
our word-level segments and oracle text, similar to SParseval [44].

We use forced alignment to determine the spans of audio that correspond to
each word in the oracle sentence. We then compute the optimal one-to-one
mapping that maximizes total span overlap between oracle segmentation and our
proposed word segmentations333This mapping is determined via bipartite weight
mapping, where the nodes are speech segments and the weights are given by the
overlap duration across nodes.. This allows us to first match nodes between
predicted and ground truth parse trees, and calculate an $F_{1}$ score that
jointly considers segmentation and parsing performance. We include whole
sentence spans in our evaluation, in order to compare to AV-NSL [18]. For all
experiments, we evaluate fully unsupervised parsing with this proposed $F_{1}$
score, and report the average and standard deviation of corpus-level $F_{1}$
of the best model before convergence over five different random seeds.

###  4.3 Results of Cascading Systems

We train two unsupervised ASR models to observe how varying accuracy of ASR
may affect parsing performance. The two models are trained with and without
self-training following the original setup of wav2vec-U. They are denoted as
ASR-ST and ASR respectively. We use a 100-hour subset of speech from the
training set, and 150k unpaired text sentences as our training data. Word-
level transcriptions for the entire SpokenCOCO dataset are decoded from the
phoneme output sequences of the ASR models. Word error rate of training set
transcripts is 13.15% and 28.25% for AST-ST and ASR.

The training set transcripts are then used to train our parser. We follow [16]
and use the 10,000 most commonly occurring words in their respective training
set transcripts as the vocabulary set for the parser. Since we assume the
availability of unpaired text data in the cascade scenario, we also consider
training a parser from the unpaired text data alone.

| Training split | Training | Testing | $F_{1}$
---|---|---|---|---
(A) | Entire train set | oracle | oracle |  57.15 $\pm$ 2.09
(B) | Unpaired text | oracle | ASR-ST |  44.08 $\pm$ 1.64
(C) | Entire train set | ASR-ST | ASR-ST |  40.53 $\pm$ 1.65
(D) | Unpaired text | oracle | ASR |  34.97 $\pm$ 1.32
(E) | Entire train set | ASR | ASR |  31.01 $\pm$ 1.17
Table 1: $F_{1}$ score of our casacading systems. The leftmost column lists
whether training data comes from the unpaired text split or the training
split. We also list the $F_{1}$ score obtained by training and testing our
parser on oracle transcripts in row (A) as a topline.

Effect of ASR errors on parsing. When comparing results across different
blocks, we can see that parsing accuracy is heavily degraded when ASR errors
are present. Additionally, one might expect that training a parser on ASR
transcripts would allow it to better handle text with ASR errors during
inference. However, by comparing rows (B)/(C) and rows (D)/(E), we see that
training our parser on unpaired oracle text is consistently better than
training on ASR transcripts. We note that this occurs in spite of the training
set being nearly 3x larger than the unpaired text set. We hypothesize that
this is partially caused by the decoding process of wav2vec-U. Uncommon words
rarely get decoded by the WFST language model. As a result, the ASR
transcripts contain fewer types of words compared to the original vocabulary,
and parser trained on imperfect transcripts deal with more out-of-vocabulary
words. The training set transcripts from the AST-ST model only use 8.2k words
out of the original 16k words present in the ground truth captions.

###  4.4 Results of Direct Systems

For direct systems, only speech features and word boundaries are required. We
extract frame-level speech features from the 14th layer of XLSR-53 [45], a
publicly available wav2vec 2.0 model pretrained on 53 languages. For word
boundaries, we naively split all utterances into 0.5-second segments, to
encompass approximately one word in each segment444As a reference, average
word length in SpokenCOCO is about 0.4 seconds, see Appendix of [40]..

Since this method of segmentation is very inaccurate, the parsing results are
similarly poor (Table 2 row (D)). However, when provided with ground truth
segmentation during testing (Table 2 row (C)), the parser trained on fixed
length segments is able to achieve a performance similar to the parser trained
on ground truth. This suggests that our direct system is limited by
segmentation accuracy during inference.

###  4.5 A Hybrid Approach: Segmenting speech with word boundaries determined
by unsupervised ASR

| Approach | Segmentation | $F_{1}$
---|---|---|---
Training | Testing
(A) | AV-NSL | ground truth | ground truth | 55.51
(B) | Ours | ground truth | ground truth |  57.11 $\pm$ 0.00
(C) | Direct | every 0.5 sec. | ground truth |  57.10 $\pm$ 0.01
(D) | Direct | every 0.5 sec. | every 0.5 sec. |  3.88 $\pm$ 0.00
(E) | Hybrid | AST-ST | AST-ST |  40.44 $\pm$ 1.72
(F) | Hybrid | ASR | ASR |  28.49 $\pm$ 0.57
Table 2: $F_{1}$ score of direct and hybrid systems. We include $F_{1}$ scores
obtained by training and testing using oracle segmentation in rows (A) and (B)
as toplines.

We compare our systems with AV-NSL under oracle segmentation settings in rows
(A) and (B). We find that our parser outperforms AV-NSL despite not using any
visual grounding information. This suggests that the DIORA framework may be
better suited for unsupervised spoken constituency parsing.

We experimented with a speech-only unsupervised word segmentation method [46],
but found it to be suboptimal. Therefore, we consider a hybrid approach that
uses forced alignment to obtain word boundaries from unsupervised ASR
transcripts. We find that when word boundaries are sufficiently accurate,
using word boundaries alone can achieve similar accuracy to cascading systems,
as shown in Table 1 row (C) and Table 2 row (E). This implies that accurate
word segmentation is necessary for unsupervised constituency parsing from
speech, which aligns with the findings in AV-NSL.

###  4.6 On the Inductive Bias and Trivial Tree Structure

Due to the head-initial property of English [47], constituency parse trees
tend to be right-branching, especially if punctuation is removed. On the other
hand, for head-final languages such as Japanese and Korean, trees are left-
branching instead.

In our direct and hybrid systems, we observe that our models tend to converge
to producing right-branching trees on SpokenCOCO555We note that right-
branching is a difficult baseline even for parsers trained on oracle text. As
shown in Table 1 row (A) and [17], unsupervised text parsers only marginally
improve on right-branching trees.. It is worthwhile to note that our framework
does not apply any inductive bias that encourages the model to favor right-
branching trees; hence, it is non-trivial for such a phenomenon to emerge. We
hypothesize that our systems learn a language’s branching direction from
continuous spoken input without supervision.

We empirically verify this claim by conducting experiments on Korean, a
primarily left-branching language. Over 5 runs with different random seeds, 3
runs converge towards producing some left-branching structures (Fig. 2),
supporting our hypothesis.

| English | Korean
---|---|---
Rule-based |  |
Left branching | 24.68 | 27.15
Right branching | 57.11 | 7.60
Speech only |  |
0.5 sec. segmentation |  57.10 $\pm$ 0.01 |  18.53 $\pm$ 8.99
Table 3: Results of direct systems trained on English (right-branching) and
Korean (left-branching), respectively, using the same setting as Table 2 row
(C).

(a) Generated parse tree

(b) Oracle parse tree

Fig. 2: A sample pair of oracle and generated Korean parse trees. Only textual
transcripts are shown for ease of visualization.

##  5 CONCLUSION

The work investigates cascading and direct approaches to perform constituency
parsing on speech input, while only requiring raw speech and unpaired data.
For cascading systems, we empirically show that parsers trained on ASR
transcripts do not parse ASR transcripts better than parsers trained on
unpaired text. For direct and hybrid systems, our results suggest that using
segmentation alone may be sufficient to produce unsupervised parse trees.

For future work, we expect to extend our system to end-to-end training to
jointly optimize word segmentation and parsing. Additionally, we also plan to
investigate whether unsupervised spoken constituency parsing can be improve
other speech processing tasks under low-resource scenarios, such as text-to-
speech, spoken question answering, or spoken content retrieval.

Acknowledgement. We thank the helpful discussions with Freda Shi, Shang-Wen
Li, Ali Elkahky, and Abdelrahman Mohamed. We also thank the National Center
for High-performance Computing (NCHC) of National Applied Research
Laboratories (NARLabs) in Taiwan for providing computational and storage
resources.

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#  The Interplay of Regularizing Factors in the Model of Upper Hybrid
Oscillations of Cold Plasma

Maria I. Delova  Mathematics and Mechanics Department, Lomonosov Moscow State
University, Leninskie Gory, Moscow, 119991, Russian Federation,
<EMAIL_ADDRESS>and  Olga S. Rozanova  Mathematics and Mechanics
Department, Lomonosov Moscow State University, Leninskie Gory, Moscow, 119991,
Russian Federation<EMAIL_ADDRESS>
###### Abstract.

A one-dimensional nonlinear model of the so-called upper hybrid oscillations
in a magnetoactive plasma is investigated taking into account electron-ion
collisions. It is known that both the presence of an external magnetic field
of strength $B_{0}$ and a sufficiently large collisional factor $\nu$ help
suppress the formation of a finite-dimensional singularity in a solution
(breaking of oscillations). Nevertheless, the suppression mechanism is
different: an external magnetic field increases the oscillation frequency, and
collisions tend to stabilize the medium and suppress oscillations. In terms of
the initial data and the coefficients $B_{0}$ and $\nu$, we establish a
criterion for maintaining the global smoothness of the solution. Namely, for
fixed $B_{0}$ and $\nu\geq 0$ one can precisely divide the initial data into
two classes: one leads to stabilization to the equilibrium, and the other
leads to the destruction of the solution in a finite time.

Next, we examine the nature of the stabilization. We show that for small
$|B_{0}|$ an increase in the intensity factor first leads to a change in the
oscillatory behavior of the solution to monotonic damping, which is then again
replaced by oscillatory damping. At large values of $|B_{0}|$, the solution is
characterized by oscillatory damping regardless of the value of the intensity
factor $\nu$.

###### Key words and phrases:

Quasilinear hyperbolic system, plasma oscillations, electron-ion collisions,
magnetic effect, blow up

###### 1991 Mathematics Subject Classification:

Primary 35Q60; Secondary 35L60, 35L67, 34M10

## Introduction

The equations of hydrodynamics of ”cold” plasma in the non-relativistic
approximation take the form

(1) $\ \begin{array}[]{l}\displaystyle{\partial n\over\partial
t}+\mbox{div}\,(n{\bf V})=0\,,\quad\displaystyle{\partial{\bf V}\over\partial
t}+\left({\bf V}\cdot\nabla\right){\bf V}=\frac{e}{m}\,\left({\bf
E}+\displaystyle{1\over c}\left[{\bf V}\times{\bf B}\right]\right)-\nu_{e}{\bf
V},\vspace{0.5em}\\\ \displaystyle{1\over c}\frac{\partial{\bf E}}{\partial
t}=-\displaystyle{4\pi\over c}en{\bf V}+{\rm rot}\,{\bf
B}\,,\quad\displaystyle{1\over c}\frac{\partial{\bf B}}{\partial t}=-{\rm
rot}\,{\bf E}\,,\quad\mbox{div}\,{\bf B}=0\,,\end{array}$

where $e,m$ — charge and mass of the electron (here the electron charge has a
negative sign: $e<0$), $c$ — speed of light; $n,{\bf V}$ — density and
velocity of electrons; ${\bf E},{\bf B}$ — vectors of electric and magnetic
fields, the term $-\nu_{e}{\bf V}$ describes electron-ion collisions, which
can be interpreted as a friction between particles, see, for example, [1].
System (1) is described in sufficient detail in textbooks and monographs (see,
for example, [9]). Currently, the great attention is paid to a study of the
cold plasma due to a possibility of acceleration of electrons in the wake wave
of a powerful laser pulse [7], nevertheless theoretical results in this field
are very scarce.

It is commonly known that the plasma oscillations described by (1), tend to
break. Mathematically, the breaking process means a blow up of solution, and
the appearance of a delta-shape singularity of the electron density [6]. Among
main interests is a study of possibility of existence of a smooth solution as
long as possible.

To obtain exact mathematical results we use the model of upper hybrid
oscillations of cold magnetoactive plasma (see, for example, [6]), a
simplification that the physicists traditionally use to study the effect of an
external magnetic field. The model was appeared, apparently, for the first
time, in [5], further it was used in various contexts, e.g. [11], [12], [10].
To derive the model they expand the solution to (1) in a series with respect a
small parameter and retain physically significant terms. In the Cartesian
coordinates $x,y,z$ the solution has the following structure:

${\bf V}=(V_{x}(x,t),V_{y}(x,t),0),\,\,{\bf E}(x,t)=(E_{x}(x,t),0,0),\,\,{\bf
B}(x,t)=(0,0,b_{0}),\,b_{0}\equiv{\rm const}.$

In the dimensionless quantities

$\begin{array}[]{c}\rho=k_{p}x,\quad\theta=\omega_{p}t,\quad{V_{1}}=\displaystyle{V_{x}\over
c},\quad{V_{2}}=\displaystyle{V_{y}\over c},\vspace{1 ex}\\\
{E}=-\,\displaystyle{e\,E_{x}\over
m\,c\,\omega_{p}},\quad{N}=\displaystyle{n\over
n_{0}},\quad{B}_{0}=-\,\displaystyle{e\,b_{0}\over
m\,c\,\omega_{p}},\quad\nu=\displaystyle{\nu_{e}\over\omega_{p}},\end{array}$

where $\omega_{p}=\left(4\pi e^{2}n_{0}/m\right)^{1/2}$ is the plasma
frequency, $n_{0}$ is the value of the unperturbed electron density,
$k_{p}=\omega_{p}/c$, the result takes the form

(2) $\begin{array}[]{c}\displaystyle{\partial
V_{1}\over\partial\theta}+V_{1}\displaystyle{\partial
V_{1}\over\partial\rho}=-E-B_{0}V_{2}-\nu V_{1},\\\ \displaystyle{\partial
V_{2}\over\partial\theta}+V_{1}\displaystyle{\partial
V_{2}\over\partial\rho}=B_{0}V_{1}-\nu V_{2},\\\ \displaystyle{\partial
E\over\partial\theta}+V_{1}\displaystyle{\partial
E\over\partial\rho}=V_{1}.\end{array}$

The solutions of this system are called in [5] nonlinear zero-temperature
Bernstein modes.

We consider (2) with the initial conditions

(3) $V_{1}(\rho,0)=V_{1}^{0}(\rho),\quad V_{2}(\rho,0)=V_{2}^{0}(\rho),\quad
E(\rho,0)=E^{0}(\rho),\quad\rho\in{\mathbb{R}}.$

Due to the Gauss theorem the electron density can be found as

(4)
${N}(\rho,\theta)=1-\displaystyle{\partial{E}(\rho,\theta)\over\partial\rho}.$

System (2) is of hyperbolic type. It is well known that for such systems,
there exists, locally in time, a unique solution to the Cauchy problem of the
same class as the initial data. The blow up of a solution is associated with
unbounded derivatives [4].

The Cauchy problem (2), (3) was studied in the following cases:

  * •

$\nu=B_{0}=0$ in [15];

  * •

$B_{0}=0$, $\nu>0$ in [14], see also [16] for the relativistic case;

  * •

$\nu=0$, $B_{0}\neq 0$ in [18].

In all these case we obtained the criterion of the singularity formation for
the solution in terms of the initial data and parameters $\nu$, $B_{0}$. In
other words, for any specific initial data we can say in advance whether the
solution keeps the initial smoothness for all $\theta>0$ or blows up in a
finite time $\theta_{*}>0$. Moreover, for the case $\nu>0$ the global
smoothness of solution means a stabilization to the trivial equilibrium state
[16].

Both parameters $\nu$ and $B_{0}$ act as regularizers in the following sense:
if we fix an arbitrary initial data (3), we can obtain the global smoothness
of solution by increasing $\nu$ or $|B_{0}|$. For example, the criterion of
the singularity formation for the case $\nu=0$ is the following.

######  Theorem 1.

[18]. For the existence of a $C^{1}$ – smooth
$\frac{2\pi}{\sqrt{1+B_{0}^{2}}}$ \- periodic solution
$(V_{1}(\theta,\rho),\,V_{2}(\theta,\rho),\,E(\theta,\rho))$ of the problem
(2) with $\nu=0$, (3), with $(V_{1}^{0},V_{2}^{0},E^{0})\in
C^{2}({\mathbb{R}})$, it is necessary and sufficient that at any point
$\rho\in\mathbb{R}$

(5)
$\left((V_{1}^{0})^{\prime}\right)^{2}+2\,(E^{0})^{\prime}+2B_{0}\,(V_{2}^{0})^{\prime}-B_{0}^{2}-1<0.$

If the opposite inequality (5) holds at least at one point $\rho_{0}$, then
the derivatives of the solution turn to infinity in a finite time.

We can see that the frequency of oscillations increases with $B_{0}$ and the
smooth solution never stabilazes to a constant state.

For $B_{0}=0$ we can draw on the plane
$((E^{0})^{\prime}(\rho_{0}),(V_{1}^{0})^{\prime}(\rho_{0}))$ a smooth curve
dividing the plane into two domains: one of them corresponds to a globally
smooth solution that stabilizes to the zero equilibrium if the point
$((E^{0})^{\prime}(\rho_{0}),(V_{1}^{0})^{\prime}(\rho_{0}))$ gets there for
all $\rho_{0}\in\mathbb{R}$. If, for certain initial data, the point
$((E^{0})^{\prime}(\rho_{0}),(V_{1}^{0})^{\prime}(\rho_{0}))$ does not belong
to this domain for at least one $\rho_{0}$, the corresponding solution blows
up. The domain of smoothness expands with increasing $\nu$, and the character
of stabilization changes with $\nu$: for $\nu\in(0,2)$ the damping is
oscillatory, while for $\nu\geq 2$ damping is monotonic starting from a
certain moment of time.

The purpose of this article is to study the relationship between the
regularizing factors $\nu$ and $B_{0}$. It is natural to expect that a smooth
solution is stabilized in the presence of both factors, and the high intensity
of electron-ion collisions suppresses oscillations. However, as we are going
to show, the nature of the stabilization is drastically different. Namely, for
any fixed $B_{0}$ there is a limit of the oscillation frequency
$\Omega(B_{0},\nu)$ as $\nu\to\infty$, equal to $B_{0}$.

Moreover, for every fixed $B_{0}$, $|B_{0}|\leq\bar{B}_{0}=\frac{\sqrt{2}}{4}$
there are two threshold values of the intensity factor of electron-ion
collisions $\nu_{1}(B_{0})$ and $\nu_{2}(B_{0})$,
$\,0<\nu_{1}(B_{0})<\nu_{2}(B_{0})<\infty$ such that for
$\nu\in[\nu_{1},\nu_{2}]$ the oscillations of the medium are completely
suppressed. For other relationships between the parameters $B_{0}$ and $\nu$,
including $|B_{0}|\geq\bar{B}_{0}$, stabilization is accompanied by
oscillations.

As for the criterion for the formation of singularities, similar to the
previous cases, it can also be obtained. However, it is technically much more
complex due to the fact that there are five independent parameters. The
analytical formulas obtained here are very cumbersome, which hampers the
analysis of the solution. However, the use of numerical methods allows one to
study the domain of smoothness of the solution when some of the parameters are
fixed, and using the methods of computer algebra, one can study the limiting
behavior of the oscillation frequency.

The paper is organized as follows. In Sec.1 we obtain a matrix Riccati
equation to describe the behavior of the space derivatives of the solution.
Then we use the Radon lemma to linearize it and obtain an implicit criterion
of the singularity formation. In Sec.2 we propose an algorithm to find the
domain of smoothness of the solution numerically and test it for the initial
data corresponding to a standard laser pulse, traditionally used in numerical
computations. In Sec.3 we study the character of stabilization of a smooth
solution, in particular, oscillations, induced by a large intensity of
collisions. In Sec.4 we discuss the results and their applicability in
physics.

##  1\. A criterion of the singularity formation

System (2) along characteristics can be written as follows:

(6) $\displaystyle\frac{dV_{1}}{d\theta}=-E-B_{0}V_{2}-\nu
V_{1},\quad\frac{dV_{2}}{d\theta}=B_{0}V_{1}-\nu
V_{2},\quad\frac{dE}{d\theta}=V_{1},\quad\frac{d\rho}{d\theta}=V_{1}.$

It immediately implies that along every characteristic
$V_{1}^{2}(t)+V_{2}^{2}(t)+E^{2}(t)\leq V_{1}^{2}(0)+V_{2}^{2}(0)+E^{2}(0)$,
therefore components of the solution remain bounded till the moment of a
possible singularity formation.

Let us denote $q_{1}=\frac{\partial V_{1}}{\partial\rho},q_{2}=\frac{\partial
V_{2}}{\partial\rho},s=\frac{\partial E}{\partial\rho}$ and (2) with respect
to $\rho$ and obtain along characteristics $\frac{d\rho}{d\theta}=V_{1}$ the
following Cauchy problem:

(7) $\frac{dq_{1}}{d\theta}=-q_{1}^{2}-s-B_{0}q_{2}-\nu
q_{1},\quad\frac{dq_{2}}{\partial\theta}=-q_{1}q_{2}+B_{0}q_{1}-\nu
q_{2},\quad\frac{ds}{d\theta}=q_{1}(1-s),$

(8) $q_{1}(\rho,0)=q_{1}^{0}(\rho),\quad q_{2}(\rho,0)=q_{2}^{0}(\rho),\quad
s(\rho,0)=s^{0}(\rho).$

Note that (7) is split off from the (6) system and can be written as the
matrix Riccati equation.

In what follows we need the following version of the Radon lemma (1927) [8],
Theorem 3.1, see also [13].

######  Theorem 2 (The Radon lemma).

A matrix Riccati equation

(9) $\dot{W}=M_{21}(t)+M_{22}(t)W-WM_{11}(t)-WM_{12}(t)W,$

($W=W(t)$ is a matrix $(n\times m)$, $M_{21}$ is a matrix $(n\times m)$,
$M_{22}$ is a matrix $(m\times m)$, $M_{11}$ is a matrix $(n\times n)$,
$M_{12}$ is a matrix $(m\times n)$) is equivalent to the homogeneous linear
matrix equation

(10) $\dot{Y}=M(t)Y,\quad M=\left(\begin{array}[]{cc}M_{11}&M_{12}\\\
M_{21}&M_{22}\end{array}\right),$

($Y=Y(t)$ is a matrix $(n\times(n+m))$, $M$ is a matrix $((n+m)\times(n+m))$ )
in the following sense.

Let on some interval ${\mathcal{J}}\in\mathbb{R}$ the matrix-function
$\,Y(t)=\left(\begin{array}[]{c}Q(t)\\\ P(t)\end{array}\right)$ ($Q$ is a
matrix $(n\times n)$, $P$ is a matrix $(n\times m)$)  be a solution of (10)
with the initial data

(11) $Y(0)=\left(\begin{array}[]{c}I\\\ W_{0}\end{array}\right)$

($I$ is the identity matrix $(n\times n)$, $W_{0}$ is a constant matrix
$(n\times m)$)  and $\det Q\neq 0$ on ${\mathcal{J}}$. Then
$W(t)=P(t)Q^{-1}(t)$ is the solution of (9) with $W(0)=W_{0}$ on
${\mathcal{J}}$.

In our case (7)

(12) $\displaystyle W=\begin{pmatrix}q_{1}\\\ q_{2}\\\
s\end{pmatrix},M_{21}=\begin{pmatrix}0\\\ 0\\\
0\end{pmatrix},M_{22}=\begin{pmatrix}-\nu&-B_{0}&-1\\\ B_{0}&-\nu&0\\\
1&0&0\\\ \end{pmatrix},$ $\displaystyle M_{11}=\begin{pmatrix}0\\\
\end{pmatrix},M_{21}=\begin{pmatrix}1&0&0\\\ \end{pmatrix}.$

######  Lemma 1.

The eigenvalues of matrix $M$, constructed by the rules of Theorem 2 for the
blocks (12) are the following: $\lambda_{1}=0$, $\lambda_{i}$, $i=2,3,4$ are
roots of a 3rd order algebraic equation

(13) $\lambda^{3}+2\nu\lambda^{2}+(1+B_{0}^{2}+\nu^{2})\lambda+\nu=0.$

The real parts of $\lambda_{k}$, $k=2,3,4$ are negative for $\nu>0$.

###### Proof.

The method for obtaining (13) is standard. We use the Routh table for
polynomials to show that all roots of (13) are in the left half-plane. $\Box$

######  Lemma 2.

The solution to the Cauchy problem (10), (11) for the matrix $M$, constructed
by the rules of Theorem 2 for the blocks (12), is

$W(\theta)=\sum\limits_{k=1}^{4}C_{k}{\bf v}_{k}e^{\lambda_{k}\theta},$

where ${\bf v}_{k}=v_{kj}$, $k,j=1,\dots,4$ are the eigenvectors of $M$ and
$C_{k}$, $k=1,\dots,4$ are solutions to the algebraic linear system

$W(0)=\sum\limits_{k=1}^{4}C_{k}{\bf v}_{k}.$

The function $Q(\theta)$, a part of the solution, has the form

(14) $Q(\theta)=\sum\limits_{k=1}^{4}C_{k}v_{k1}e^{\lambda_{k}\theta}.$

The proof is standard. The function $Q(\theta)$ depends on the initial data
$q_{1}^{0},q_{2}^{0},s^{0}$ and parameters $B_{0},\nu$.

######  Theorem 3.

For the existence of a $C^{1}$ – smooth solution
$(V_{1}(\theta,\rho),\,V_{2}(\theta,\rho),\,E(\theta,\rho))$ of the problem
(2), (3), where $(V_{1}^{0},V_{2}^{0},E^{0})\in C^{2}({\mathbb{R}})$, it is
necessary and sufficient that at any point $\rho\in\mathbb{R}$ the function
$Q(\theta)$, given as (14), does not have roots at the half-axis $\theta>0$.
Otherwise, the solution blows up at the time

(15)
$T_{c}=\inf\limits_{\theta_{*}>0}\,\\{\rho_{0}\in{\mathbb{R}}\,|\,Q(\theta_{*},\rho_{0})=0\\}.$

###### Proof.

If we solve the respective system (10), then we find the solution to (7)
according to Theorem 2. The relation $W(\theta)={P(\theta)}Q^{-1}(\theta)$
implies that the derivatives of the solution to the Cauchy problem (7), (8) go
to infinity in a finite time along the characteristic starting from the point
$\rho_{0}\in\mathbb{R}$ if and only if there is $\theta_{*}>0$ such that
$Q(\theta_{*},\rho_{0})=0$. The smallest root $\theta_{*}>0$ over all
$\rho_{0}\in\mathbb{R}$ corresponds to the blow up time (15). $\Box$

In what follows we need explicit expressions for $\lambda_{k}$ and ${\bf
v}_{k}$. To write them we introduce the following constants:

(16) $\displaystyle K$ $\displaystyle=$ $\displaystyle
B_{0}^{6}+(2\nu^{2}+3)B_{0}^{4}+(\nu^{4}-5\nu^{2}+3)B_{0}^{2}+\left(1-\frac{3}{4}\nu^{2}\right),$
(17) $\displaystyle K_{1}$ $\displaystyle=$
$\displaystyle\left(8\nu^{3}+36(2B_{0}^{2}-1)\nu+24\sqrt{3K}\right)^{\frac{1}{3}},\quad
K_{2}=\frac{1}{9}(3+3B_{0}^{2}-\nu^{2}).$

Then the eigenvalues can be written as

$\displaystyle\lambda_{2}=\frac{1}{6}K_{1}-6\frac{K_{2}}{K_{1}}-\frac{2}{3}\nu,\quad\lambda_{3,4}=-\frac{1}{12}K_{1}+3\frac{K_{2}}{K_{1}}-\frac{2}{3}\nu\pm{\,\rm
i\,}\frac{\sqrt{3}}{2}\left(\frac{1}{6}K_{1}+6\frac{K_{2}}{K_{1}}\right).$

The respective eigenvectors are

$\displaystyle{\bf v}_{1}=\begin{pmatrix}1\\\ 0\\\ 0\\\
0\end{pmatrix},\quad{\bf
v}_{k}=\begin{pmatrix}-\frac{\nu}{((\lambda_{k}+\nu)^{2}+1+B_{0}^{2})\lambda_{k}}\\\
-\frac{\nu}{(\lambda_{k}+\nu)^{2}+1+B_{0}^{2}}\\\
-\frac{\lambda_{k}^{2}+\lambda_{k}\nu+1+B_{0}^{2}}{B_{0}((\lambda_{k}+\nu)^{2}+1+B_{0}^{2})}\\\
1\end{pmatrix},\quad k=2,3,4.$

##  2\. Global smoothness induced by the regularizing factors for a standard
pulse

We call the ”standard pulse” a set of initial data that looks like:

(18) $V_{1}^{0}(\rho)=V_{2}^{0}(\rho)=0,\quad
E_{0}(\rho)=k\,\rho\,\exp\left\\{-\displaystyle{\rho^{2}\over\sigma}\right\\},\quad
k=\left(\displaystyle{a_{*}\over\rho_{*}}\right)^{2},\;\sigma=\frac{\rho_{*}^{2}}{2}.$

The data (18) was chosen in the assumption that the oscillations are excited
by a laser pulse with a frequency of $\omega_{l}$ ($\omega_{l}\gg\omega_{p}$)
when it is focused in a line, which can be achieved by using a cylindrical
lens [19]. The parameters $\rho_{*}$ and $a_{*}$ characterize the scale of the
localization region and the maximum value
$E_{\max}=a_{*}^{2}/(\rho_{*}2\sqrt{{\rm e}})\approx 0.3a_{*}^{2}/\rho_{*}$ of
the electric field, respectively. The specific values of $\rho_{*}$ and
$a_{*}$ are not important here. This kind of data are traditionally used for
numerical simulation of different phenomena in a rarefied plasma, see e.g.
[3].

Although the way of applying Theorem 3 is formally clear, the technical side
is fraught with great difficulties. Indeed, we need to solve the
transcendental equation $Q(\theta)=0$ with very cumbersome coefficients, which
depends on the eigenvectors of $M$. Nevertheless, for any fixed set of
parameters $q_{1}^{0},q_{2}^{0},s^{0},B_{0},\nu$ we use a numerical procedure
to study the effect of each particular parameter.

To study the blow-up for the data (18) we choose the most ”dangerous” point
$\rho_{0}=0$, where the derivative $(E_{0})^{\prime}$ has a maximum. There
$q_{1}^{0}=q_{2}^{0}=0,\,s^{0}=k$, this reduces the number of free parameters
to three and simplifies the analysis.

First, we picture on the plane $(\nu,B_{0})$ the set of parameters that leads
to a finite time blow-up for a fixed $s^{0}=k$. If $k<\frac{1}{2}$, then this
set is empty. This follows, for example, from Theorem 1 for the case $\nu=0$
and from the estimate of the critical $k$ from [14], $\nu\in[0,2)$, $B_{0}=0$,

(19) $k_{cr}=\frac{1}{1+\exp(-\frac{\nu\pi}{\sqrt{4-\nu^{2}}})}.$

We note also that the nonnegativity of density dictates the requirement $k<1$
(see (4)). Thus, we choose for the numerical illustrations the interval
$k=s^{0}\in(\frac{1}{2},1)$. To estimate the sign of minimum of $Q(\theta)$
for $\theta>0$ we used Newton’s method built into the Optimization package of
MAPLE. Fig.1, presents the ”blow-up” domains for $k=0.6$, $k=0.7$ and $k=0.8$
and shows that this domain expands with increasing $k$. The limits of $\nu$ as
$B_{0}\to 0$ on the boundary of the domain correspond to (19).

Figure 1.  The ”blow up” domains on the plane $\nu$ (horizontal) and $B_{0}$
(vertical) for different values of $k$, the step is 0.01 for both variables.
The picture is symmetrical about the axis $B_{0}=0$.

The next numerical experiment that we can perform is to fix $\nu$ and $B_{0}$
and to find a domain $\mathcal{D}$, consisting of points
$q_{1}^{0},q_{2}^{0},s^{0}$, corresponding to a globally in time smooth
solution (2), (18), provided at every point $\rho\in\mathbb{R}$ the
derivatives of the initial data fall in this domain. In other words, we try to
picture an analog of the plain domain found in [14] for a fixed $\nu$.
However, the considered domain $\mathcal{D}$ is three-dimensional, and the
result of calculations is difficult to present in a convenient form. Therefore
one can draw a section of this domain, for example, with $q_{2}^{0}=\rm
const$. We do not present the result of this computation here, just note that
the ”blow up” domain, the complement of the domain $\mathcal{D}$ to the entire
space, expands with increasing $q_{2}^{0}$.

##  3\. Character of stabilization

In this section, we establish certain properties of smooth solutions.

######  Theorem 4.

Let the initial data (3) are such that the solution of the Cauchy problem (2),
(3), is globally $C^{1}$ – smooth.

1\. For any fixed $B_{0}$ there exists
$\lim\limits_{\nu\to\infty}\Im\,\lambda_{3}(B_{0},\nu)=B_{0}$;

2\. The motion is oscillatory if and only if $K>0$, given as (16), is
positive. If, in addition, $\nu>0$, then the solution decays to the trivial
steady state. If $K\leq 0$, then the solution decays to the trivial steady
state monotonically starting from a sufficiently large $\theta$.

###### Proof.

1\. The first properties follows from the expansion

$\lambda_{3}(B_{0},\nu)=-\nu+{\,\rm
i\,}B_{0}\,+O\left(\frac{1}{\nu}\right),\quad\nu\to\infty.$

It implies that the oscillations cannot be suppressed by a large frequency of
collisions.

2\. If the function $Q(\theta)$ does not have roots for $\theta>0$ for all
$\rho_{0}\in\mathbb{R}$, then taking into account Lemma 2 we conclude that the
solution to (2), (3) stabilizes to to the trivial state provided $\nu>0$ since
$e^{\lambda_{k}\theta}\to 0$, $\theta\to\infty$, $k=2,3,4$.

The stabilization is monotonic starting from a certain moment of time if and
only if $\Im\lambda_{k}=0$ for all $k=2,3,4.$

If $K$ is nonnegative, then $\Im\,\lambda_{2}=0$, and $\Im\,\lambda_{3,4}\neq
0$ provided $K_{1}^{2}+36K_{2}\neq 0$. It can be readily checked that
$K_{1}^{2}+36K_{2}=0$ implies $K=0$. We take into account that $K_{1}$ and
$K_{2}$, given as (17), both vanish at the hyperbola $3B_{0}^{2}-\nu^{2}+9=0$,
so the ratio $\frac{K_{2}}{K_{1}}$ is finite.

If $K<0$, then it suffices to prove that $\Im\,\lambda_{3}=0$. Since
$|\Im\,\lambda_{3}|=|\Im\,\lambda_{4}|$, Lemma 1 implies that $\lambda_{k}$,
$k=2,3,4$, are real and negative. First of all we notice that if $K<0$, then
necessarily $K_{1}=a+{\,\rm i\,}b\,$, $a,b\in\mathbb{R}$, $ab\neq 0$. Then it
is easy to compute that

(20)
$\displaystyle\Im\,\lambda_{3}=\frac{\sqrt{3}a-b}{12(a^{2}+b^{2})}(36K_{2}+a^{2}+b^{2}).$

Since $K_{1}=(A\pm{\,\rm i\,}B)^{\frac{1}{3}},$
$A=8\nu^{3}+36(2B_{0}^{2}-1)\nu,$ $B=24\sqrt{-3K}$, we can find $a$ and $b$,
then substitute the result in (20) and check directly that
$\Im\,\lambda_{3}=0$. This is a rather cumbersome computation, however, it can
be done using computer algebra.

Note that since we are extracting the 3rd order roots of complex numbers, we
must account for the corresponding branch of this multivalued function. $\Box$

######  Corollary 1.

Let the solution of (2), (3) is globally smooth in time. For every fixed
$B_{0}$, $|B_{0}|\leq\bar{B}_{0}=\frac{\sqrt{2}}{4}$ there are two threshold
values of the factor of the intensity of collisions, $\nu_{1}(B_{0})$ and
$\nu_{2}(B_{0})$,
$\,\frac{3}{4}\sqrt{6}<\nu_{1}(B_{0})<\nu_{2}(B_{0})<+\infty$ such that for
$\nu\in[\nu_{1},\nu_{2}]$ the oscillations of the medium are completely
suppressed. For other relations between the parameters $|B_{0}|$ and $\nu>0$,
including $B_{0}\geq\bar{B}_{0}$, the stabilization is accompanied by
oscillations.

For the proof  we have to study the curve $K(B_{0},\nu)=0$ on the plane
$(B_{0},\nu)$. We resolve $K=0$ with respect to $\nu$ and get two branches
$\nu=\nu_{\pm}(B_{0})$ with the following properties:

$\displaystyle\nu_{\pm}=\frac{1}{4}\frac{\sqrt{2+40B_{0}^{2}-16B_{0}^{4}\pm
2\sqrt{1-24B_{0}^{2}+192B_{0}^{4}-512B_{0}^{6}}}}{B0}.$

Thus,

$\displaystyle\quad
B_{0}\in\left[-\bar{B}_{0},\bar{B}_{0}\right],\quad\bar{B}_{0}=\frac{\sqrt{2}}{4}\approx
0.353\dots,\quad\nu_{\pm}\left(\bar{B}_{0}\right)=\frac{3}{4}\sqrt{6}\approx
1.837\dots,$

the limit values are

$\displaystyle\lim\limits_{B_{0}\to 0}\nu_{-}=2,\quad\lim\limits_{B_{0}\to
0}\nu_{+}=+\infty.$

The points $K(B_{0},\nu)=0$ for $B_{0}\geq 0$ are presented in Fig.2, the
picture is symmetric with respect to $B_{0}=0$. Thus, if we fix any
$B_{0}\in\left[-\bar{B}_{0},\bar{B}_{0}\right]$ and begin to increase $\nu$,
we start in the domain $K>0$, then at $\nu_{1}=\nu_{-}(B_{0})$ we fall in the
domain $K\leq 0$ and then leave this domain at $\nu_{2}=\nu_{+}(B_{0})$. This
observation proves the corollary. We note that the results is in compliance
with the case $B_{0}=0$, studied in [14], where it was shown that the
monotonic decay takes place for $\nu\geq 2$. $\Box$

Figure 2. Values $B_{0},\,\nu$ corresponding to oscillatory regime (I) and
monotonic regime (II) for $B_{0}\geq 0$; the picture is symmetrical about the
axis $B_{0}=0$. The ”blow up” domain for the standard pulse (Sec.2, Fig.1) is
located near the origin of coordinates and belongs to the oscillatory regime.

######  Remark 1.

The following expansions also make it possible to trace the effect of low
collision intensity on the character of oscillations:

$\displaystyle\lambda_{2}(B_{0},\nu)$ $\displaystyle=$
$\displaystyle-\frac{1}{B_{0}^{2}+1}\nu+O\left(\nu^{2}\right),\quad\nu\to 0,$
$\displaystyle\lambda_{3,4}(B_{0},\nu)$ $\displaystyle=$
$\displaystyle\pm{\,\rm
i\,}\sqrt{B_{0}^{2}+1}-\frac{1}{2}\frac{2B_{0}^{2}+1}{B_{0}^{2}+1}\nu+O\left(\nu^{2}\right).$

We see that in the first approximation, the low intensity of collisions does
not change the frequency of the oscillations, but causes their exponential
decay. Generally speaking, the rate of this damping does not decrease at large
values of $|B_{0}|$.

######  Remark 2.

It should be noticed that the oscillations induced by friction are known in
mechanical systems (see, e.g. [2]).

##  4\. Discussion

We performed a mathematical analysis of equations of upper hybrid oscillations
(2) for all allowed values of the constant parameters $\nu$ and $B_{0}$.
Nevertheless, it should be noted that the most interesting phenomena, such as
the persistence of oscillations at large values of $\nu$, can hardly be
observed experimentally, since the physical values of $\nu$ are of the order
$10^{-2}$ (see [14] for details). For small values of $\nu$, the effect of the
external magnetic field is very close to the case $\nu=0$ considered in [18]:
basically, increasing the magnetic field prevents blow-up (except for
specially adapted initial data). It should also be noted that the most
powerful regularizing factor is the dependence of the intensity of electron-
ion collisions on the density. An analysis of this situation in the absence of
a magnetic field was carried out in [17]. It is natural to expect that, in the
presence of a magnetic field, the effect of suppressing the formation of
singularities for all smooth initial data will be retained.

## Acknowledgment

Supported by the Ministry of Education and Science of the Russian Federation
as part of the program of the Moscow Center for Fundamental and Applied
Mathematics under the agreement 075-15-2019-1621.

The authors are grateful to E.V.Chizhonkov for drawing attention to the
problem, stimulating discussions and constant interest.

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