# A stable splitting for spaces of commuting elements in unitary groups

Alejandro Adem  Department of Mathematics
University of British Columbia, Vancouver BC V6T 1Z2, Canada<EMAIL_ADDRESS>,  José Manuel Gómez  Departamento de Matemáticas
Universidad Nacional de Colombia sede Medellín, Medellín, Colombia
<EMAIL_ADDRESS>and  Simon Gritschacher  Department of Mathematics
University of Munich, Munich, Germany<EMAIL_ADDRESS>
###### Abstract.

We prove an analogue of Miller’s stable splitting of the unitary group $U(m)$
for spaces of commuting elements in $U(m)$. After inverting $m!$, the space
$\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ splits stably as a wedge of Thom-
like spaces of bundles of commuting varieties over certain partial flag
manifolds. Using Steenrod operations we prove that our splitting does not hold
integrally. Analogous decompositions for symplectic and orthogonal groups as
well as homological results for the one-point compactification of the
commuting variety in a Lie algebra are also provided.

##  1\. Introduction

###  1.1. Statement of results

In [13] Miller showed that the unitary group $U(m)$ admits a stable splitting

$U(m)_{+}\simeq\bigvee_{0\leq k\leq
m}\textnormal{Gr}_{k}(\mathbb{C}^{m})^{\mathfrak{u}_{k}}$

where

$\textnormal{Gr}_{k}(\mathbb{C}^{m})^{\mathfrak{u}_{k}}=(U(m)/U(m-k))_{+}\wedge_{U(k)}\mathfrak{u}_{k}^{+}$

is the Thom space of the adjoint bundle, i.e., of the vector bundle over the
Grassmannian $\textnormal{Gr}_{k}(\mathbb{C}^{m})$ associated with the adjoint
representation of $U(k)$. In this paper we prove an analogous stable splitting
for the space of commuting $n$-tuples in $U(m)$,

$\operatorname{Hom}(\mathbb{Z}^{n},U(m))=\\{(x_{1},\dots,x_{n})\in
U(m)^{n}\mid x_{i}x_{j}=x_{j}x_{i}\textnormal{ for all }1\leq i,j\leq n\\}\,,$

which holds after inverting $m!$. In our splitting the role of the adjoint
bundle is played by a bundle of commuting varieties over a generalised
Grassmannian. We denote by

$C_{n}(\mathfrak{u}_{k})=\\{(X_{1},\dots,X_{n})\in\mathfrak{u}_{k}^{n}~{}~{}|~{}~{}[X_{i},X_{j}]=0\text{
for all }1\leq i,j\leq n\\}$

the space of commuting $n$-tuples in the Lie algebra $\mathfrak{u}_{k}$. When
$k>1$ and $n>1$ the commuting variety $C_{n}(\mathfrak{u}_{k})$ is not a
vector space as it is not closed under addition. The group $U(k)$ acts on
$C_{n}(\mathfrak{u}_{k})$ diagonally by the adjoint representation, and this
action extends to one on the one-point compactification
$C_{n}(\mathfrak{u}_{k})^{+}$. Our splitting is indexed by a certain poset of
partitions of $m$ denoted $\mathcal{P}$. The elements
$\lambda=(\lambda_{a})_{a\in I}\in\mathcal{P}$ are partitions of $m$ into
$2^{n}$ parts indexed by the set of binary sequences $I=\\{0,1\\}^{n}$. Given
$a\in I$ we write $|a|=\sum_{i=1}^{n}a(i)$.

######  Theorem A.

After inverting $m!$ there is a stable splitting for all $n\in\mathbb{N}$,

$\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{+}\simeq\bigvee_{(\lambda_{a})_{a\in
I}\in\mathcal{P}}\left(U(m)_{+}\wedge_{\prod_{a\in
I}U(\lambda_{a})}\bigwedge_{a\in
I}C_{|a|}(\mathfrak{u}_{\lambda_{a}})^{+}\right)\,.$

The stable summands are Thom-like spaces for bundles of commuting varieties.
These commuting varieties are irreducible, and in the non-abelian case the
bundles that appear are not vector bundles, but rather bundles of infinite
subspace arrangements.

Our methods apply to prove an analogous stable splitting for spaces of
commuting elements in the compact symplectic group $Sp(m)$ (Theorem 4.8), but
not for $SU(m)$. This is because our proof uses a stable splitting of a
maximal torus which is equivariant for the Weyl group action, and we do not
know if such a splitting exists in the case of $SU(m)$. For orthogonal groups
our methods apply; but because the orthogonal groups have abelian subgroups
that are not contained in a maximal torus, we obtain a stable splitting only
of the path-component of $\operatorname{Hom}(\mathbb{Z}^{n},O(m))$ containing
the trivial homomorphism (Theorem 4.10).

To put our theorem in context we recall that the first author and F. Cohen [1]
have previously proved a stable splitting of the space
$\operatorname{Hom}(\mathbb{Z}^{n},G)$ for any compact Lie group $G$. Let
$S_{n}(G)\subseteq\operatorname{Hom}(\mathbb{Z}^{n},G)$ be the subspace of
those $n$-tuples of commuting elements of which at least one coordinate is the
neutral element $1_{G}\in G$. Then there is a stable splitting

$\operatorname{Hom}(\mathbb{Z}^{n},G)_{+}\simeq\bigvee_{0\leq r\leq
n}\left(\bigvee^{\binom{{n}}{{r}}}\operatorname{Hom}(\mathbb{Z}^{r},G)/S_{r}(G)\right).$

This splitting holds without inverting primes, but with the exception of the
trivial cases and the cases $G=SU(2)$ and $G=SO(3)$, the stable summands have
not been identified geometrically. For $G=U(m)$ and with $m!$ inverted, our
decomposition is finer than the above, in that there is a stable equivalence

$\operatorname{Hom}(\mathbb{Z}^{n},U(m))/S_{n}(U(m))\simeq\bigvee_{(\lambda_{a})_{a\in
I}\in\mathcal{S}}\left(U(m)_{+}\wedge_{\prod_{a\in
I}U(\lambda_{a})}\bigwedge_{a\in
I}C_{|a|}(\mathfrak{u}_{\lambda_{a}})^{+}\right)$

for a certain subset $\mathcal{S}\subseteq\mathcal{P}$ (Corollary 4.4).

Miller’s stable splitting is modelled on a filtration of $U(m)$ in which the
top filtration quotient is $\mathfrak{u}_{m}^{+}$. Similarly, our splitting is
obtained from a filtration with top filtration quotient
$C_{n}(\mathfrak{u}_{m})^{+}$. The next theorem shows that when $n\geq 2$,
certain primes _must_ be inverted for the top filtration quotient to split off
stably. Using Steenrod operations we prove:

######  Theorem B.

Let $p$ be a prime and let $n\geq 2$ be an integer. The projection

$\operatorname{Hom}(\mathbb{Z}^{n},U(p))_{+}\to C_{n}(\mathfrak{u}_{p})^{+}$

does not have a stable section up to homotopy at the prime $p$.

In contrast, it should be noted that Crabb [6] and Baird, Jeffrey and Selick
[4] have obtained a stable decomposition of
$\operatorname{Hom}(\mathbb{Z}^{n},SU(2))$ in which the space
$C_{n}(\mathfrak{su}_{2})^{+}$ appears as a stable summand.111In Crabb’s
notation this is $C(T,\emptyset)$ with $|T|=n$. In fact, this decomposition
coincides with the aforementioned one of Adem and Cohen applied to $G=SU(2)$.
Our splitting of $\operatorname{Hom}(\mathbb{Z}^{n},Sp(m))$ applied to
$Sp(1)=SU(2)$ reduces to theirs, but an ad-hoc argument is necessary to see
that this splitting holds without inverting $2$.

A motivation for proving the stable splitting lies in the effort of computing
the homology of $\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ which is still
unknown. The homology of $C_{n}(\mathfrak{u}_{m})^{+}$ appears much more
tractable than that of $\operatorname{Hom}(\mathbb{Z}^{n},U(m))$, although
explicit computations are difficult even when $m$ is small. Unfortunately,
inverting $m!$ has the effect of killing all torsion in
$\operatorname{Hom}(\mathbb{Z}^{n},U(m))$. But even if Theorem A fails to hold
without inverting primes, it remains to be determined if it holds at the level
of integral homology. Preliminary calculations indicate that this is the case
when $n=2$ and $m=2,3$.

######  Question C.

Is there an isomorphism

$\tilde{H}_{*}(\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{+};\mathbb{Z})\cong\bigoplus_{(\lambda_{a})_{a\in
I}\in\mathcal{P}}\tilde{H}_{*}\left(U(m)_{+}\wedge_{\prod_{a\in
I}U(\lambda_{a})}\bigwedge_{a\in
I}C_{|a|}(\mathfrak{u}_{\lambda_{a}})^{+};\mathbb{Z}\right)?$

This question may be related to the Steenrod splitting of the homology of
symmetric products [19, Section 22]; indeed, the quotient of
$C_{n}(\mathfrak{u}_{m})^{+}$ by the adjoint action of $U(m)$ is a symmetric
smash product of spheres, and a key step in the proof of Theorem B is the
comparison of $\operatorname{Hom}(\mathbb{Z}^{n},U(p))$ with a symmetric
product of spheres. In fact, the moduli space of representations, i.e., the
quotient of $\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ by the conjugation
action of $U(m)$ is the $m$-fold symmetric product of the $n$-torus,
$\textnormal{SP}^{m}((S^{1})^{n})$. Rationally then, the stable factors in our
splitting correspond precisely to those in a rational stable splitting of the
symmetric product.

This article has an appendix in which we record basic results about the
compactified commuting variety $C_{n}(\mathfrak{g})^{+}$ where $\mathfrak{g}$
is the Lie algebra of a compact connected Lie group $G$. These results are
used frequently throughout the earlier sections, but we have decided to
concentrate them in the appendix which is self-contained and may be of
independent interest. Amongst other results we will prove the following:

######  Theorem D.

Let $G$ be a compact connected Lie group of rank $r\geq 1$ and dimension $d$,
and let $W$ denote the Weyl group of $G$. For every $n\geq 1$,
$C_{n}(\mathfrak{g})^{+}$ is a $\mathbb{Z}[1/|W|]$-homology sphere of
dimension $nr$ if $n$ is even, and of dimension $d+(n-1)r$ if $n$ is odd.

A computation of the mod-$p$ cohomology of $C_{n}(\mathfrak{u}_{p})^{+}$ when
$n$ is even is also included.

###  1.2. Organisation of the paper

In Section 2 we review Miller’s stable splitting for the unitary groups
$U(m)$. In Section 3 we introduce the poset $\mathcal{P}$ and construct the
filtration of $\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ which will lead to the
stable splitting of Theorem A. The theorem is proved in Section 4. The non-
splitting result, Theorem B, is proved in Section 5. The article concludes
with Appendix A which contains in particular the proof of Theorem D.

##  2\. The stable splitting for the unitary group

In this section we briefly review Miller’s stable splitting for the group
$U(m)$. We follow the presentation of Crabb [7].

We start by constructing a suitable filtration of $U(m)$. For every $0\leq
k\leq m$, let

$F^{k}(U(m))=\\{A\in U(m)\mid\dim(\ker(A-\mathds{1}))\geq m-k\\}\,.$

In other words, the elements in $F^{k}(U(m))$ are the matrices that have $1$
as an eigenvalue with multiplicity at least $m-k$. This way we obtain an
increasing filtration of $U(m)$

(1) $\\{\mathds{1}\\}=F^{0}(U(m))\subseteq F^{1}(U(m))\subseteq\cdots\subseteq
F^{m}(U(m))=U(m).$

Alternatively, this filtration is obtained in the following way. Let
$T\subseteq U(m)$ be the maximal torus consisting of diagonal matrices and let
$F^{k}(T)$ be the subspace of $T\cong(S^{1})^{m}$ with at least $m-k$
coordinates equal to $1$. Consider the map

$\displaystyle\tilde{\varphi}\colon\thinspace U(m)\times T$ $\displaystyle\to
U(m)$ $\displaystyle(g,t)$ $\displaystyle\mapsto gtg^{-1}\,.$

Then $F^{k}(U(m))=\tilde{\varphi}(U(m)\times F^{k}(T))$. We will take this
point of view in the next section to construct filtrations of
$\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ from filtrations of $T^{n}$.

Let $V_{k}(\mathbb{C}^{m})=U(m)/U(m-k)$ denote the Stiefel manifold of
orthonormal $k$-frames in $\mathbb{C}^{m}$ and let
$\textnormal{Gr}_{k}(\mathbb{C}^{m})=V_{k}(\mathbb{C}^{m})/U(k)$ be the
Grassmannian. Let $\mathfrak{u}_{k}$ denote the Lie algebra of $U(k)$. It is
the space of $k\times k$ skew-Hermitian matrices with the commutator bracket.
We next explain how the stratum

$F_{k}(U(m)):=F^{k}(U(m))\backslash F^{k-1}(U(m))$

can be identified with the total space of the vector bundle

(2) $\mathfrak{u}_{k}\to
V_{k}(\mathbb{C}^{m})\times_{U(k)}\mathfrak{u}_{k}\to\textnormal{Gr}_{k}(\mathbb{C}^{m})$

associated via the adjoint representation.

######  Definition 2.1.

The _Cayley transform_ is the map

$\displaystyle\psi_{m}\colon\thinspace$ $\displaystyle\mathfrak{u}_{m}$
$\displaystyle\to F_{m}(U(m))$ $\displaystyle X$
$\displaystyle\mapsto(X-\mathds{1})(X+\mathds{1})^{-1}\,.$

Observe that the space $F_{m}(U(m))$ is an open dense subspace of $U(m)$ on
which $U(m)$ acts by conjugation. The following lemma is basic:

######  Lemma 2.2.

The map $\psi_{m}\colon\thinspace\mathfrak{u}_{m}\to F_{m}(U(m))$ is a
$U(m)$–equivariant diffeomorphism.

###### Proof.

The inverse of $\psi_{m}$ is given by the formula
$\psi_{m}^{-1}(A)=(\mathds{1}-A)^{-1}(\mathds{1}+A)$. ∎

Thus, when $k=m$ the Cayley transform identifies $F_{k}(U(m))$ with the total
space of the adjoint bundle (2). More generally, for $0\leq k\leq m$ the
stratum $F_{k}(U(m))$ fibres over $\textnormal{Gr}_{k}(\mathbb{C}^{m})$ by
considering the map

$\displaystyle F_{k}(U(m))$
$\displaystyle\to\textnormal{Gr}_{k}(\mathbb{C}^{m})$ $\displaystyle A$
$\displaystyle\mapsto\ker(A-\mathds{1})^{\perp}\,.$

The point is that a transformation $A\in F_{k}(U(m))$ can be written in the
form

$\mathds{1}\oplus A|_{V_{1}}:V_{0}\oplus V_{1}\to V_{0}\oplus V_{1}$

where $V_{0}=\ker(A-\mathds{1})$ and $V_{1}=V_{0}^{\perp}$. So $A$ is uniquely
determined by $V_{1}\in\textnormal{Gr}_{k}(\mathbb{C}^{m})$ and the
restriction $A|_{V_{1}}$. Note that $A|_{V_{1}}$ is an element of
$F_{k}(U(V_{1}))\cong\mathfrak{u}_{k}$. More precisely, the map

$\displaystyle V_{k}(\mathbb{C}^{m})\times_{U(k)}\mathfrak{u}_{k}$
$\displaystyle\to F_{k}(U(m))$ $\displaystyle[(\bar{g},X)]$
$\displaystyle\mapsto g(\mathds{1}\oplus\psi_{k}(X))g^{-1}$

is a diffeomorphism over $\textnormal{Gr}_{k}(\mathbb{C}^{m})$ (here $g\in
U(m)$ is a representative of an element $\bar{g}\in
U(m)/U(m-k)=V_{k}(\mathbb{C}^{m})$). It follows that the subquotient
$F^{k}(U(m))/F^{k-1}(U(m))$ can be identified with the Thom space of the
adjoint bundle:

$\textnormal{Gr}_{k}(\mathbb{C}^{m})^{\mathfrak{u}_{k}}=(U(m)/U(m-k))_{+}\wedge_{U(k)}\mathfrak{u}_{k}^{+}\,.$

In particular, the top filtration quotient $U(m)/F^{m-1}(U(m))$ is
$\mathfrak{u}_{m}^{+}$.

Finally, one shows that the filtration (1) splits stably. The top quotient is
split off using a Pontryagin-Thom construction: Let $\mathcal{H}(m)$ be the
vector space of hermitian matrices of size $m\times m$. The embedding

$\displaystyle U(m)\times\mathcal{H}(m)$
$\displaystyle\to\textnormal{Mat}_{m}(\mathbb{C})$ $\displaystyle(A,Z)$
$\displaystyle\mapsto A\exp(-Z)$

is onto $GL_{m}(\mathbb{C})$, and so $GL_{m}(\mathbb{C})$ is a tubular
neighbourhood of $U(m)$ inside
$\textnormal{Mat}_{m}(\mathbb{C})\cong\mathfrak{u}_{m}\times\mathcal{H}(m)$.
Collapsing the complement to a point we obtain a map

$\mathfrak{u}_{m}^{+}\wedge\mathcal{H}(m)^{+}\to
U(m)_{+}\wedge\mathcal{H}(m)^{+}.$

Note that $\mathcal{H}(m)^{+}$ is a sphere, and so this is a stable map
$\mathfrak{u}_{m}^{+}\to U(m)_{+}$. One checks that this is a stable section
up to homotopy of the projection

(3) $U(m)_{+}\to U(m)/F^{m-1}(U(m))\cong\mathfrak{u}_{m}^{+}\,.$

The general case, which shows that
$\textnormal{Gr}_{k}(\mathbb{C}^{m})^{\mathfrak{u}_{k}}$ splits off stably
from $F^{k}(U(m))_{+}$, is handled similarly using the Pontryagin-Thom
construction fibrewise. For this one notes that the stable section constructed
above is $U(m)$-equivariant. Taking the wedge over the various sections
$\textnormal{Gr}_{k}(\mathbb{C}^{m})^{\mathfrak{u}_{k}}\to
F^{k}(U(m))_{+}\hookrightarrow U(m)_{+}$ leads to a stable map

$\bigvee_{0\leq k\leq
m}\textnormal{Gr}_{k}(\mathbb{C}^{m})^{\mathfrak{u}_{k}}\to U(m)_{+}$

which is a homotopy equivalence. The details are in [7].

One may attempt the same approach to construct a stable splitting of
$\operatorname{Hom}(\mathbb{Z}^{n},U(m))$. However, the Pontryagin-Thom
collapse does not apply naively in that situation, because the open embedding
$U(m)\times\mathcal{H}(m)\hookrightarrow\textnormal{Mat}_{m}(\mathbb{C})$
considered above does not produce a tubular neighbourhood of any sort for
$\operatorname{Hom}(\mathbb{Z}^{n},U(m))$.

##  3\. The filtration of $\operatorname{Hom}(\mathbb{Z}^{n},U(m))$

The aim of this section is to describe the filtration of
$\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ on which our stable splitting is
based. Throughout this section let $n\geq 1$ and $m\geq 0$ be fixed.

###  3.1. Commuting elements

The space $\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ of $n$-tuples of pairwise
commuting elements in $U(m)$ (defined in Section 1) is topologized as a
subspace of $U(m)^{n}$. Let $T\subseteq U(m)$ be the maximal torus consisting
of all diagonal matrices. Consider the map

$\displaystyle\tilde{\varphi}\colon\thinspace U(m)\times T^{n}$
$\displaystyle\to\operatorname{Hom}(\mathbb{Z}^{n},U(m))$
$\displaystyle(g,t_{1},\dots,t_{n})$
$\displaystyle\mapsto(gt_{1}g^{-1},\dots,gt_{n}g^{-1})\,.$

It is easily seen to be surjective, as any abelian subgroup of $U(m)$ is
contained in a maximal torus. There is an action of the normaliser
$N_{U(m)}(T)$ on $U(m)\times T^{n}$ defined by

$h\cdot(g,t_{1},\dots,t_{n})=(gh^{-1},ht_{1}h^{-1},\dots,ht_{n}h^{-1})\,,\quad
h\in N_{U(m)}(T)\,.$

The map $\tilde{\varphi}$ is invariant under this action and so it descends to
a map

(4) $\varphi\colon\thinspace
U(m)\times_{N_{U(m)}(T)}T^{n}\to\operatorname{Hom}(\mathbb{Z}^{n},U(m))\,.$

Notice that

$U(m)\times_{N_{U(m)}(T)}T^{n}=U(m)/T\times_{\Sigma_{m}}T^{n}\,$

where the symmetric group $\Sigma_{m}$ appears as the Weyl group
$N_{U(m)}(T)/T$. It acts on $T^{n}$ by permuting the $m$ factors of
$T\cong(S^{1})^{m}$.

###  3.2. Indexing poset

A filtration of $\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ can be constructed
by starting with a $\Sigma_{m}$-invariant filtration of $T^{n}$ and pushing it
forward via $\varphi$. The filtration of $T^{n}$ that we shall consider is
indexed over a poset $\mathcal{P}$ of partitions of $m$ which we will now
describe. Let $I=\\{0,1\\}^{n}$ be the set of binary sequences of length $n$,
and let $\mathcal{P}$ be the set of $I$-indexed ordered integer partitions of
$m$, i.e.,

$\mathcal{P}=\left\\{(\lambda_{a})_{a\in
I}\in\mathbb{Z}^{I}~{}\left|~{}\lambda_{a}\geq 0\textnormal{ for all }a\in
I\textnormal{ and }\sum_{a\in I}\lambda_{a}=m\right\\}\right.\,.$

For example, when $n=2$, elements in $\mathcal{P}$ are of the form
$\lambda=(\lambda_{(0,0)},\lambda_{(0,1)},\lambda_{(1,0)},\lambda_{(1,1)})$,
where each $\lambda_{a}$ is a non-negative integer and

$\lambda_{(0,0)}+\lambda_{(0,1)}+\lambda_{(1,0)}+\lambda_{(1,1)}=m.$

The set $I=\\{0,1\\}^{n}$ is totally ordered by the lexicographic order in
which $(0,\dots,0)$ is the smallest element and $(1,\dots,1)$ is the largest
element. With this in mind, to each partition $\lambda=(\lambda_{a})_{a\in I}$
we can associate an $m\times n$ matrix $M(\lambda)$ with entries in
$\\{0,1\\}$ defined by blocks in the following way. In the first
$\lambda_{(0,\dots,0)}$ rows of $M(\lambda)$ we put the row $(0,\dots,0)$.
Subsequently, we define the next $\lambda_{(0,\dots,0,1)}$ rows in
$M(\lambda)$ to be equal to $(0,\dots,0,1)$. We continue this way, using the
order in $I$, until we reach the last $\lambda_{(1,\dots,1)}$ rows in
$M(\lambda)$ and these rows are defined to be equal to $(1,\dots,1)$. Note
that the assignment $\lambda\mapsto M(\lambda)$ is one-to-one.

######  Example 3.1.

Consider the particular case $n=2$ and $m=6$, and let $\lambda$ be the
$I$-indexed partition of $m=6$ given by

$\lambda_{(0,0)}=1,\ \lambda_{(0,1)}=1,\ \lambda_{(1,0)}=2,\
\lambda_{(1,1)}=2.$

In this case, the matrix $M(\lambda)$ associated to $\lambda$ is the $6\times
2$ matrix given by

$M(\lambda)=\begin{pmatrix}0&0\\\ \hline\cr 0&1\\\ \hline\cr 1&0\\\ 1&0\\\
\hline\cr 1&1\\\ 1&1\end{pmatrix}.$

Using these matrices $\\{M(\lambda)~{}|~{}\lambda\in\mathcal{P}\\}$ we can
define a partial order on $\mathcal{P}$ in the following way: Given
$\lambda,\mu\in\mathcal{P}$ we declare $\mu\leq\lambda$ if, after suitable
permutations of rows, the matrix $M(\mu)$ is obtained from $M(\lambda)$ by
replacing some entries that are equal to $1$ with a $0$.

######  Example 3.2.

Consider the case $n=2$ and $m=6$, and let $\mu,\lambda$ be the $I$-indexed
partitions of $m=6$ defined by

$\displaystyle\mu_{(0,0)}$ $\displaystyle=1,\ \mu_{(0,1)}=2,\ \mu_{(1,0)}=2,\
\mu_{(1,1)}=1,$ $\displaystyle\lambda_{(0,0)}$ $\displaystyle=1,\
\lambda_{(0,1)}=1,\ \lambda_{(1,0)}=2,\ \lambda_{(1,1)}=2.$

In this example the matrices $M(\mu)$ and $M(\lambda)$ are as follows

$M(\mu)=\begin{pmatrix}0&0\\\ \hline\cr 0&1\\\ 0&1\\\ \hline\cr 1&0\\\ 1&0\\\
\hline\cr 1&1\end{pmatrix},\ \ \ \ \ M(\lambda)=\begin{pmatrix}0&0\\\
\hline\cr 0&1\\\ \hline\cr 1&0\\\ 1&0\\\ \hline\cr 1&1\\\ 1&1\end{pmatrix}.$

In this case, $M(\mu)$ is obtained from $M(\lambda)$ by replacing the entry
$(5,1)$ with a $0$ and then permuting rows $3$ and $5$. This shows that
$\mu\leq\lambda$.

We endow the poset $\mathcal{P}$ with the upper topology in which the closed
sets are generated by the subbasis
$\\{\mathcal{P}_{\leq\lambda}\mid\lambda\in\mathcal{P}\\}$ where
$\mathcal{P}_{\leq\lambda}=\\{\mu\in\mathcal{P}\mid\mu\leq\lambda\\}$. If $X$
is a space and $f\colon\thinspace X\to\mathcal{P}$ is a continuous map, then
for all $\lambda\in\mathcal{P}$ the subspace

$F^{\lambda}(X):=f^{-1}(\mathcal{P}_{\leq\lambda})\subseteq X$

is closed, and there is an inclusion $F^{\mu}(X)\subseteq F^{\lambda}(X)$
whenever $\mu\leq\lambda$. Thus, $f$ induces a filtration of $X$ by closed
subspaces indexed by $\mathcal{P}$.

###  3.3. Construction of the filtration

To filter $T^{n}$ we define a map $\Psi_{T}\colon\thinspace
T^{n}\to\mathcal{P}$ in the following way. An element $x\in T^{n}$ can be
evidently viewed as a matrix of size $m\times n$ with entries in $S^{1}$. In
this picture the Weyl group $\Sigma_{m}$ permutes the rows of $x$. To each row
$x_{i}=(x_{i1},\dots,x_{in})\in(S^{1})^{n}$, $1\leq i\leq m$ of $x$ we can
assign a binary sequence $\textnormal{seq}(x_{i})\in I$ by setting

$\textnormal{seq}(x_{i})(j)=\begin{cases}0&\textnormal{if }x_{ij}=1,\\\
1&\textnormal{if }x_{ij}\neq 1\end{cases}$

for all $1\leq j\leq n$. For $a\in I$ we let $\lambda_{a}$ be the number of
$1\leq i\leq m$ such that $\textnormal{seq}(x_{i})=a$ and set
$\Psi_{T}(x):=(\lambda_{a})_{a\in I}$.

######  Example 3.3.

Suppose that $n=2$ and $m=6$, and let $x\in T^{2}$ be the element
corresponding to the matrix

$x=\begin{pmatrix}*&*\\\ 1&1\\\ *&1\\\ *&1\\\ *&*\\\ 1&*\end{pmatrix},$

where the $*$‘s can take any value in $S^{1}$ different from $1$. Then
$\lambda=\Psi_{T}(x)\vdash 6$ is given by

$\lambda_{(0,0)}=1,\,\lambda_{(0,1)}=1,\,\lambda_{(1,0)}=2,\,\lambda_{(1,1)}=2\,.$

It is easily checked that for each $\lambda\in\mathcal{P}$ the subspace

$F^{\lambda}(T^{n}):=\Psi_{T}^{-1}(\mathcal{P}_{\leq\lambda})\subseteq T^{n}$

is closed, and thus $\Psi_{T}\colon\thinspace T^{n}\to\mathcal{P}$ is
continuous. The filtration defined by $\Psi_{T}$ refines the fat wedge
filtration of $T^{n}$.

######  Example 3.4.

Suppose that $n=1$. Then $I=\\{0,1\\}$ and thus all partitions
$\lambda\in\mathcal{P}$ are of the form $\lambda=(m-k,k)$ for some $0\leq
k\leq m$. Projection onto the second component defines a homeomorphism of
posets $\mathcal{P}\cong\mathbb{N}_{\leq m}$. Then
$\Psi_{T}^{-1}(\mathbb{N}_{\leq k})$ is precisely the subspace of
$T\cong(S^{1})^{m}$ consisting of those $m$-tuples that have at least $m-k$
entries equal to $1$. Therefore, the filtration associated with
$\Psi_{T}\colon\thinspace T\to\mathbb{N}_{\leq m}$ is the fat wedge filtration
of $T$.

######  Example 3.5.

More generally, for any $n\geq 1$ we can define a map of posets
$f\colon\thinspace\mathcal{P}\to\mathbb{N}$ by sending $(\lambda_{a})_{a\in
I}$ to $\sum_{a\in I}\lambda_{a}|a|$, where $|a|:=\sum_{i=1}^{n}a(i)$. For
$x\in T^{n}$, $f(\Psi_{T}(x))$ is precisely the number of entries of $x$ that
are different from $1$. Therefore, the filtration defined by $f\Psi_{T}$ is
the fat wedge filtration of $T^{n}\cong(S^{1})^{nm}$.

Since every commuting $n$-tuple in $U(m)$ is conjugate to an $n$-tuple in $T$,
the quotient of $\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ by the conjugation
action of $U(m)$ is homeomorphic to $T^{n}/\Sigma_{m}$. Clearly, $\Psi_{T}$
factors through a continuous map $\overline{\Psi}_{T}\colon\thinspace
T^{n}/\Sigma_{m}\to\mathcal{P}$.

######  Construction 3.6.

We let

$\Psi\colon\thinspace\operatorname{Hom}(\mathbb{Z}^{n},U(m))\to\mathcal{P}$

be the composition of the quotient map
$\operatorname{Hom}(\mathbb{Z}^{n},U(m))\to T^{n}/\Sigma_{m}$ with
$\overline{\Psi}_{T}$, and let

$F^{\lambda}(\operatorname{Hom}(\mathbb{Z}^{n},U(m))):=\Psi^{-1}(\mathcal{P}_{\leq\lambda})$

be the filtration of $\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ induced by
$\Psi$.

Of course,

$F^{\lambda}(\operatorname{Hom}(\mathbb{Z}^{n},U(m)))=\varphi((U(m)/T\times_{\Sigma_{m}}F^{\lambda}(T^{n})))\,,$

and so the filtration of $\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ is the
filtration of $T^{n}$ pushed forward by $\varphi$.

We will frequently adjoin a basepoint to
$\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ in which case we extend the
filtration to include the basepoint at every filtration stage.

The filtration of $\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ generalises the
one used in Miller’s splitting and it refines the filtration used in the
splitting of Adem and Cohen.

######  Example 3.7.

Suppose that $n=1$. The filtration of
$\operatorname{Hom}(\mathbb{Z},U(m))\cong U(m)$ induced by
$\Psi\colon\thinspace U(m)\to\mathcal{P}\cong\mathbb{N}_{\leq m}$ is given by

$\Psi^{-1}(\mathbb{N}_{\leq k})=\left\\{g\in
U(m)\mid\dim\ker(g-\mathds{1})\geq m-k\right\\}\,.$

This is the filtration on which the stable splitting of $U(m)$ is based (see
Section 2).

######  Example 3.8.

Let $f\colon\thinspace\mathcal{P}\to\mathbb{N}$ be the map of posets defined
by

$f(\lambda)=n-|\\{i\in\\{1,\dots,n\\}\mid\sum_{a\in
I}\lambda_{a}a(i)=0\\}|\,.$

Then the filtration of $\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ induced by
$f\Psi\colon\thinspace\operatorname{Hom}(\mathbb{Z}^{n},U(m))\to\mathbb{N}$
coincides with the one induced by the fat wedge filtration of $U(m)^{n}$,

$(f\Psi)^{-1}(\mathbb{N}_{\leq
k})=\\{(g_{1},\dots,g_{n})\in\operatorname{Hom}(\mathbb{Z}^{n},U(m))\mid\textnormal{at
least }n-k\textnormal{ of the }g_{i}\textnormal{ are equal to
}\mathds{1}\\}\,.$

This is the filtration that is used in the stable splitting of Adem and Cohen.

###  3.4. Subquotients of the filtration

The map
$\Psi\colon\thinspace\operatorname{Hom}(\mathbb{Z}^{n},U(m))\to\mathcal{P}$
gives a partition of $\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ into “strata”
(these are not manifolds in general)

$\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{\lambda}:=\Psi^{-1}(\lambda),\;\lambda\in\mathcal{P}$

that we will now describe.

From the introduction, recall the definition of the variety
$C_{n}(\mathfrak{u}_{m})$ of commuting elements in the Lie algebra
$\mathfrak{u}_{m}$. We topologize $C_{n}(\mathfrak{u}_{m})$ as a subspace of
the Euclidean space $\mathfrak{u}_{m}^{n}$. Also recall that $I=\\{0,1\\}^{n}$
is totally ordered via the lexicographic order. By fixing this total order on
$I$ we can view, for every $\lambda\in\mathcal{P}$, the direct product

$U(\lambda):=\prod_{a\in I}U(\lambda_{a})$

as a subgroup of $U(m)$ in the obvious way. For $a\in I$ we write
$|a|:=\sum_{i=1}^{n}a(i)$.

######  Proposition 3.9.

For every $\lambda\in\mathcal{P}$ there is a $U(m)$-equivariant homeomorphism

$\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{\lambda}\cong
U(m)\times_{U(\lambda)}\prod_{a\in I}C_{|a|}(\mathfrak{u}_{\lambda_{a}})\,.$

###### Proof.

According to [5, Proposition II.3.2] it is enough to construct a
$U(m)$-equivariant map

$p_{\lambda}\colon\thinspace\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{\lambda}\to
U(m)/U(\lambda)$

and to identify the fibre over the neutral coset, as a $U(\lambda)$-space,
with $\prod_{a\in I}C_{|a|}(\mathfrak{u}_{\lambda_{a}})$.

Recall that elements of $T^{n}$ can be viewed as $m\times n$ matrices with
entries in $S^{1}$. Given $x\in T^{n}$, we can assign to each row
$x_{i}=(x_{i1},\dots,x_{in})\in(S^{1})^{n}$, $1\leq i\leq m$ a binary sequence
$\textnormal{seq}(x_{i})\in I$ as in Section 3.3. Let $R_{\lambda}$ be the
subspace of $\Psi^{-1}_{T}(\lambda)\subseteq T^{n}$ defined by

$R_{\lambda}=\\{x\in\Psi^{-1}_{T}(\lambda)\mid\textnormal{seq}(x_{1})\leq\cdots\leq\textnormal{seq}(x_{m})\\}\,.$

As the $\Sigma_{m}$-orbit of any $x\in\Psi_{T}^{-1}(\lambda)$ intersects
$R_{\lambda}$ non-trivially, the restriction of the action map
$\tilde{\varphi}\colon\thinspace U(m)\times
T^{n}\to\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ to $U(m)\times R_{\lambda}$,

$\tilde{\varphi}|\colon\thinspace U(m)\times
R_{\lambda}\to\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{\lambda}\,,$

is surjective. Thus, for every
$(g_{1},\dots,g_{n})\in\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{\lambda}$
there is a $h=h(g_{1},\dots,g_{n})\in U(m)$ such that
$(h^{-1}g_{1}h,\dots,h^{-1}g_{n}h)\in R_{\lambda}$. It is easy to see that $h$
is uniquely determined modulo $U(\lambda)$ by $(g_{1},\dots,g_{n})$, and thus
the assignment

$\displaystyle
p_{\lambda}\colon\thinspace\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{\lambda}$
$\displaystyle\to U(m)/U(\lambda)$ $\displaystyle(g_{1},\dots,g_{n})$
$\displaystyle\mapsto h(g_{1},\dots,g_{n})U(\lambda)$

is well-defined. As $\tilde{\varphi}|$ is a closed surjection, hence a
quotient map, and $p_{\lambda}\tilde{\varphi}|\colon\thinspace U(m)\times
R_{\lambda}\to U(m)/U(\lambda)$ is continuous being the obvious projection,
$p_{\lambda}$ is continuous as well.

It remains to identify the fibre over $\mathds{1}U(\lambda)$. Using $\oplus$
to denote the direct sum of matrices or representations, the elements of
$p_{\lambda}^{-1}(\mathds{1}U(\lambda))$ are precisely those $n$-tuples of the
form

$(g_{1},\dots,g_{n})=\bigoplus_{a\in I}(g_{1}^{(a)},\dots,g_{n}^{(a)})$

where
$(g_{1}^{(a)},\dots,g_{n}^{(a)})\in\operatorname{Hom}(\mathbb{Z}^{n},U(\lambda_{a}))$
satisfies for all $i\in\\{1,\dots,n\\}$

$g_{i}^{(a)}-\mathds{1}=\begin{cases}0&\textnormal{if }a(i)=0\\\
\textnormal{non-singular}&\textnormal{if }a(i)=1\,.\end{cases}$

The Cayley transform

$\displaystyle\psi_{k}\colon\thinspace\mathfrak{u}_{k}$
$\displaystyle\to\\{g\in U(k)\mid g-\mathds{1}\textnormal{ is non-
singular}\\}$

from Definition 2.1 respects commutativity, i.e., for all
$X,Y\in\mathfrak{u}_{k}$ we have $[X,Y]=0$ if and only if
$[\psi_{k}(X),\psi_{k}(Y)]=\mathds{1}$. Therefore, we can define for every
$a\in I$ a $U(\lambda_{a})$-equivariant map

$\psi^{\prime}_{\lambda_{a}}\colon\thinspace
C_{|a|}(\mathfrak{u}_{\lambda_{a}})\to\operatorname{Hom}(\mathbb{Z}^{n},U(\lambda_{a}))$

in the following fashion. Let
$\iota_{a}\colon\thinspace\\{1,\dots,|a|\\}\to\\{1,\dots,n\\}$ be the unique
monotone injection with $\textnormal{im}(\iota_{a})=\\{i\in\\{1,\dots,n\\}\mid
a(i)=1\\}$. Let

(5)
$\textnormal{sh}_{a}\colon\thinspace\operatorname{Hom}(\mathbb{Z}^{|a|},U(\lambda_{a}))\to\operatorname{Hom}(\mathbb{Z}^{n},U(\lambda_{a}))$

be the map defined by
$\textnormal{sh}_{a}(g_{1},\dots,g_{|a|})=(g_{1}^{\prime},\dots,g_{n}^{\prime})$
with

$g_{i}^{\prime}=\begin{cases}\mathds{1}&\textnormal{if
}\iota_{a}^{-1}(i)=\emptyset\\\ g_{j}&\textnormal{if
}\iota_{a}(j)=i\,.\end{cases}$

Then $\psi^{\prime}_{\lambda_{a}}$ is defined by

$\psi^{\prime}_{\lambda_{a}}(X_{1},\dots,X_{|a|})=\textnormal{sh}_{a}(\psi_{\lambda_{a}}(X_{1}),\dots,\psi_{\lambda_{a}}(X_{|a|}))\,.$

These maps assemble into a $U(\lambda)$-equivariant map

$\prod_{a\in I}C_{|a|}(\mathfrak{u}_{\lambda_{a}})\xrightarrow{\prod_{a\in
I}\psi^{\prime}_{\lambda_{a}}}\prod_{a\in
I}\operatorname{Hom}(\mathbb{Z}^{n},U(\lambda_{a}))\xrightarrow{\oplus}\operatorname{Hom}(\mathbb{Z}^{n},U(m))$

which is easily seen to be a homeomorphism onto
$p_{\lambda}^{-1}(\mathds{1}U(\lambda))\subseteq\operatorname{Hom}(\mathbb{Z}^{n},U(m))$.
∎

Observe that the homogeneous space $U(m)/U(\lambda)$ is the flag manifold
consisting of all the orthogonal decompositions of $\mathbb{C}^{m}$ into
vector subspaces of dimensions $\lambda_{a}$, $a\in I$. By Proposition 3.9,
the space $\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{\lambda}$ is the total
space of a fibre bundle

$\prod_{a\in
I}C_{|a|}(\mathfrak{u}_{\lambda_{a}})\to\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{\lambda}\to
U(m)/U(\lambda)$

whose fibres are products of commuting varieties. This generalises naturally
the case $n=1$ where, for $\lambda=(m-k,k)$, the space $U(m)_{\lambda}$ is the
total space of the adjoint bundle

$V_{k}(\mathbb{C}^{m})\times_{U(k)}\mathfrak{u}_{k}\to\textnormal{Gr}_{k}(\mathbb{C}^{m})$

(see Section 2).

Given a $\mathcal{P}$-indexed filtration
$(F^{\lambda}(X))_{\lambda\in\mathcal{P}}$ of a space $X$, we write
$F^{<\lambda}(X):=\bigcup_{\mu<\lambda}F^{\mu}(X)$.

######  Corollary 3.10.

For each $\lambda\in\mathcal{P}$ there is a $U(m)$-equivariant homeomorphism

$F^{\lambda}\operatorname{Hom}(\mathbb{Z}^{n},U(m))/F^{<\lambda}\operatorname{Hom}(\mathbb{Z}^{n},U(m))\cong
U(m)_{+}\wedge_{U(\lambda)}\bigwedge_{a\in
I}C_{|a|}(\mathfrak{u}_{\lambda_{a}})^{+}\,.$

###### Proof.

The space $F^{<\lambda}\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ is a closed
subspace of the compact Hausdorff space
$F^{\lambda}\operatorname{Hom}(\mathbb{Z}^{n},U(m))$, hence

$F^{\lambda}\operatorname{Hom}(\mathbb{Z}^{n},U(m))/F^{<\lambda}\operatorname{Hom}(\mathbb{Z}^{n},U(m))\cong\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{\lambda}^{+}\,.$

By Proposition 3.9 this is homeomorphic to

$\left(U(m)\times_{U(\lambda)}\prod_{a\in
I}C_{|a|}(\mathfrak{u}_{\lambda_{a}})\right)^{+}\cong
U(m)_{+}\wedge_{U(\lambda)}\bigwedge_{a\in
I}C_{|a|}(\mathfrak{u}_{\lambda_{a}})^{+}\,,$

and these identifications are evidently $U(m)$-equivariant. ∎

Let $f\colon\thinspace\mathcal{P}\to\mathbb{N}$ be the map of posets sending
$\lambda\mapsto\sum_{a\in I}\lambda_{a}|a|$ and let

$F^{k}\operatorname{Hom}(\mathbb{Z}^{n},U(m))=(f\Psi)^{-1}(\mathbb{N}_{\leq
k})$

be the induced filtration of $\operatorname{Hom}(\mathbb{Z}^{n},U(m))$. For
$(g_{1},\dots,g_{n})\in\operatorname{Hom}(\mathbb{Z}^{n},U(m))$, the number
$(f\Psi)(g_{1},\dots,g_{n})$ is the total number of eigenvalues of
$g_{1},\dots,g_{n}$ that are different from $1$.

######  Lemma 3.11.

For each $k\geq 0$, the space $F^{k}\operatorname{Hom}(\mathbb{Z}^{n},U(m))$
admits a finite CW-structure in such a way that the inclusion

$F^{k-1}\operatorname{Hom}(\mathbb{Z}^{n},U(m))\to
F^{k}\operatorname{Hom}(\mathbb{Z}^{n},U(m))$

is the inclusion of a subcomplex. In particular, the inclusion is a
cofibration. Moreover, the same statement holds for $U(m)$-equivariant CW-
structures instead of ordinary ones.

###### Proof.

The space $\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ is evidently an affine
real algebraic variety.

An element $(g_{1},\dots,g_{n})\in\operatorname{Hom}(\mathbb{Z}^{n},U(m))$
belongs to $F^{k}\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ if and only if the
matrix $\bigoplus_{i=1}^{n}g_{i}\in U(mn)$, i.e., the block sum of
$g_{1},\dots,g_{n}$, has at least $m-k$ eigenvalues equal to $1$. It follows
that $F^{k}\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ is precisely the subset of
$\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ cut out by the vanishing of all
$(m-k+1)\times(m-k+1)$ minors of $\mathds{1}-\bigoplus_{i=1}^{n}g_{i}$. These
minors are polynomial expressions in the coordinates of the $g_{i}$, and so
this is an algebraic subset of $\operatorname{Hom}(\mathbb{Z}^{n},U(m))$. In
the same way we see that $F^{k-1}\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ is
an algebraic subset of $F^{k}\operatorname{Hom}(\mathbb{Z}^{n},U(m))$.

The existence of CW-structures is now guaranteed by the semialgebraic
triangulation theorems. In the non-equivariant case we apply [12, Theorem 3].
To obtain an equivariant CW-structure we observe that
$F^{k}\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ is a semialgebraic $U(m)$-space
in the sense of [15, Definition 3.1]. Then [15, Theorem 3.5] yields a
semialgebraic triangulation of the orbit space
$F^{k}\operatorname{Hom}(\mathbb{Z}^{n},U(m))/U(m)$ in such a way that the
$U(m)$-isotropy type is constant across each open simplex in the
triangulation. Moreover, using the non-equivariant triangulation theorem, we
may assume that $F^{k-1}\operatorname{Hom}(\mathbb{Z}^{n},U(m))/U(m)$ is a
union of open simplices in this triangulation. By passing to the barycentric
subdivision these triangulations give compatible equivariant triangulations of
the orbit spaces (see [10, Theorem 5.5]) which then lift to compatible
equivariant CW-structures on $F^{k}\operatorname{Hom}(\mathbb{Z}^{n},U(m))$
and $F^{k-1}\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ by [10, Proposition 6.1].
∎

In the subposet $f^{-1}(\mathbb{N}_{\leq k})\subseteq\mathcal{P}$ through
which $f\Psi$ factors, the points $\lambda\in\mathcal{P}$ with $f(\lambda)=k$
are open. This implies a homeomorphism

$F^{k}\operatorname{Hom}(\mathbb{Z}^{n},U(m))\backslash
F^{k-1}\operatorname{Hom}(\mathbb{Z}^{n},U(m))\cong\bigsqcup_{\lambda\in
f^{-1}(k)}\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{\lambda}\,,$

and consequently a homeomorphism

$F^{k}\operatorname{Hom}(\mathbb{Z}^{n},U(m))/F^{k-1}\operatorname{Hom}(\mathbb{Z}^{n},U(m))\cong\bigvee_{\lambda\in
f^{-1}(k)}\left(U(m)_{+}\wedge_{U(\lambda)}\bigwedge_{a\in
I}C_{|a|}(\mathfrak{u}_{\lambda_{a}})^{+}\right)\,.$

By Lemma 3.11 this quotient admits both a non-equivariant as well as a
$U(m)$-equivariant CW-structure in which the wedge point is a $0$-cell. These
induce CW-structures on each wedge summand.

######  Corollary 3.12.

For every $\lambda\in\mathcal{P}$ the space
$U(m)_{+}\wedge_{U(\lambda)}\bigwedge_{a\in
I}C_{|a|}(\mathfrak{u}_{\lambda_{a}})^{+}$ admits the structure of a finite
CW-complex, and also that of a finite $U(m)$-equivariant CW-complex.

##  4\. The stable splitting

We now turn to the stable splitting for
$\operatorname{Hom}(\mathbb{Z}^{n},U(m))$. We begin by briefly explaining the
general idea.

###  4.1. Sketch of idea

It is well-known [3] that the action map (4) is an isomorphism in homology
with coefficients in $\mathbb{Z}[1/m!]$. After inverting $m!$, this implies a
stable equivalence

$(U(m)/T\times_{\Sigma_{m}}T^{n})_{+}\simeq\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{+}\,.$

The fat wedge filtration of the torus $T^{n}_{+}\cong(S^{1})^{mn}_{+}$ with
its permutation action by $\Sigma_{mn}$ admits a $\Sigma_{mn}$-equivariant
stable splitting (see e.g. [8, Proposition 5.2]). By restriction to the
diagonal subgroup $\Sigma_{m}\leq\Sigma_{mn}$ one obtains a
$\Sigma_{m}$-equivariant stable splitting of the filtration
$(F^{\lambda}(T^{n}_{+}))_{\lambda\in\mathcal{P}}$ defined in Section 3.3.
This implies a stable equivalence

$(U(m)/T\times_{\Sigma_{m}}T^{n})_{+}\simeq\bigvee_{\lambda\in\mathcal{P}}(U(m)/T)_{+}\wedge_{\Sigma_{m}}F^{\lambda}(T^{n}_{+})/F^{<\lambda}(T^{n}_{+})\,.$

Finally, via the Cayley transform there are maps for all
$\lambda\in\mathcal{P}$,

$(U(m)/T)_{+}\wedge_{\Sigma_{m}}F^{\lambda}(T^{n}_{+})/F^{<\lambda}(T^{n}_{+})\to
U(m)_{+}\wedge_{U(\lambda)}\bigwedge_{a\in
I}C_{|a|}(\mathfrak{u}_{\lambda_{a}})^{+}\,,$

and these are isomorphisms in homology with coefficients in $\mathbb{Z}[1/m!]$
(see Section A).

###  4.2. Statement and corollaries

What we shall actually prove is the following slightly refined statement from
which the above will follow:

######  Theorem 4.1.

Let $\lambda\in\mathcal{P}$. After inverting all primes
$p\leq\max\\{\lambda_{a}\mid a\in I,\,|a|>1\\}$ the quotient map

$\zeta_{\lambda}\colon\thinspace
F^{\lambda}\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{+}\to
U(m)_{+}\wedge_{U(\lambda)}\bigwedge_{a\in
I}C_{|a|}(\mathfrak{u}_{\lambda_{a}})^{+}$

collapsing $F^{<\lambda}\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{+}$ admits a
stable section up to homotopy.

We will prove the theorem in Section 4.3. Let us deduce Theorem A as a
corollary:

######  Corollary 4.2.

After inverting $m!$ there is a stable equivalence for all $n\in\mathbb{N}$

$\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{+}\simeq\bigvee_{\lambda\in\mathcal{P}}\left(U(m)_{+}\wedge_{U(\lambda)}\bigwedge_{a\in
I}C_{|a|}(\mathfrak{u}_{\lambda_{a}})^{+}\right)\,.$

###### Proof.

We consider the filtration of $\operatorname{Hom}(\mathbb{Z}^{n},U(m))$
defined in Lemma 3.11. For each $k\leq mn$, the lemma and the subsequent
discussion yield a cofibre sequence

$F^{k-1}\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{+}\to
F^{k}\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{+}\to\bigvee_{\lambda\in
f^{-1}(k)}\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{\lambda}^{+}\,.$

Now assume that all spaces are localised so as to invert $m!$. By Theorem 4.1
we can choose, for each $\lambda\in f^{-1}(k)$, a stable section up to
homotopy

$\sigma_{\lambda}\colon\thinspace\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{\lambda}^{+}\to
F^{\lambda}\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{+}$

of the projection $\zeta_{\lambda}$. By taking wedge sum we have a stable map

$\bigvee_{\lambda\in
f^{-1}(k)}\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{\lambda}^{+}\xrightarrow{\bigvee_{\lambda}\sigma_{\lambda}}\bigvee_{\lambda\in
f^{-1}(k)}F^{\lambda}\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{+}\to
F^{k}\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{+}$

in which the second map is the sum over the various inclusions. By
construction, this is a stable splitting, up to homotopy, of the cofibre
sequence above. Induction on $k\leq mn$ yields the asserted equivalence. ∎

######  Example 4.3.

Consider $\operatorname{Hom}(\mathbb{Z}^{2},U(2))$. The stable summands that
appear in Corollary 4.2 are listed in Table 1. All the stable summands are
wedges of spheres, even integrally, except for $C_{2}(\mathfrak{u}_{2})^{+}$.
Away from $2$,

$C_{2}(\mathfrak{u}_{2})^{+}\simeq S^{4}$

by Theorem A.4 and the fact that $C_{2}(\mathfrak{u}_{2})^{+}$ is simply-
connected. This may be explained in more geometric terms: First, one has
$C_{2}(\mathfrak{u}_{2})^{+}\cong\Sigma^{2}C_{2}(\mathfrak{su}_{2})^{+}$ by
Proposition A.9. By observing that two elements of $\mathfrak{su}_{2}$ commute
if and only if they are linearly dependent, one sees that
$C_{2}(\mathfrak{su}_{2})$ is the quotient of twice the tautological line
bundle $L\to\mathbb{R}P^{2}$ by its zero section. This implies

$C_{2}(\mathfrak{su}_{2})^{+}\cong S(2L)^{\lozenge}\,,$

the unreduced suspension of the sphere bundle in $2L$. It is explained in [6,
Section 4] that this space is stably equivalent to
$S^{2}\vee\Sigma^{2}\mathbb{R}P^{2}$, and so
$C_{2}(\mathfrak{su}_{2})^{+}\simeq S^{2}$ away from $2$. As a result, there
is a stable equivalence away from $2$

$\operatorname{Hom}(\mathbb{Z}^{2},U(2))_{+}\simeq
S^{0}\vee(\bigvee^{2}S^{1})\vee(\bigvee^{2}S^{2})\vee(\bigvee^{4}S^{3})\vee(\bigvee^{5}S^{4})\vee(\bigvee^{2}S^{5})\,.$

(0,0) | (0,1) | (1,0) | (1,1) | Stable summand
---|---|---|---|---
$2$ | $0$ | $0$ | $0$ | $S^{0}$
$0$ | $2$ | $0$ | $0$ | $C_{1}(\mathfrak{u}_{2})^{+}\cong S^{4}$
$0$ | $0$ | $2$ | $0$ | $C_{1}(\mathfrak{u}_{2})^{+}\cong S^{4}$
$0$ | $0$ | $0$ | $2$ | $C_{2}(\mathfrak{u}_{2})^{+}$
$1$ | $1$ | $0$ | $0$ | $(U(2)/U(1)\times_{U(1)}C_{1}(\mathfrak{u}_{1}))^{+}\cong S^{1}\vee S^{3}$
$1$ | $0$ | $1$ | $0$ | $(U(2)/U(1)\times_{U(1)}C_{1}(\mathfrak{u}_{1}))^{+}\cong S^{1}\vee S^{3}$
$1$ | $0$ | $0$ | $1$ | $(U(2)/U(1)\times_{U(1)}C_{2}(\mathfrak{u}_{1}))^{+}\cong S^{2}\vee S^{4}$
$0$ | $1$ | $1$ | $0$ | $(U(2)/U(1)\times_{U(1)}C_{2}(\mathfrak{u}_{1}))^{+}\cong S^{2}\vee S^{4}$
$0$ | $1$ | $0$ | $1$ | $(U(2)\times_{U(1)\times U(1)}C_{1}(\mathfrak{u}_{1})\times C_{1}(\mathfrak{u}_{1}))^{+}\cong S^{3}\vee S^{5}$
$0$ | $0$ | $1$ | $1$ | $(U(2)\times_{U(1)\times U(1)}C_{1}(\mathfrak{u}_{1})\times C_{1}(\mathfrak{u}_{1}))^{+}\cong S^{3}\vee S^{5}$
Table 1. The table shows the stable summands of
$\operatorname{Hom}(\mathbb{Z}^{2},U(2))_{+}$. The partitions are indexed by
$\\{0,1\\}^{2}$ and their parts are listed in the corresponding columns.

We can use Theorem 4.1 to identify the summands in the stable splitting of
$\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ proved by Adem and Cohen [1, Theorem
6.5]. These are of the form

$\operatorname{Hom}(\mathbb{Z}^{n},U(m))/S_{n}(U(m))$

where $S_{n}(U(m))\subseteq\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ is the
subspace for which at least one coordinate is the identity.

Let $\mathcal{S}\subseteq\mathcal{P}$ be the subset defined by

$\mathcal{S}=\\{\lambda\in\mathcal{P}\mid\sum_{a\in I}\lambda_{a}a(i)\neq
0\textnormal{ for all }1\leq i\leq n\\}\,.$

Observe that if $\lambda\in\mathcal{S}$, then
$\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{\lambda}\subseteq\operatorname{Hom}(\mathbb{Z}^{n},U(m))\backslash
S_{n}(U(m))$.

######  Corollary 4.4.

After inverting $m!$ there is a stable equivalence

$\operatorname{Hom}(\mathbb{Z}^{n},U(m))/S_{n}(U(m))\simeq\bigvee_{\lambda\in\mathcal{S}}\left(U(m)_{+}\wedge_{U(\lambda)}\bigwedge_{a\in
I}C_{|a|}(\mathfrak{u}_{\lambda_{a}})^{+}\right)\,.$

###### Proof.

By definition of $\mathcal{S}$ we have that

$S_{n}(U(m))=\bigcup_{\lambda\in\mathcal{P}\backslash\mathcal{S}}F^{\lambda}\operatorname{Hom}(\mathbb{Z}^{n},U(m))\,.$

Let $F^{k}\operatorname{Hom}(\mathbb{Z}^{n},U(m))$ denote the filtration used
in the proof of Corollary 4.2. The induced filtration of the subspace
$S_{n}(U(m))$ satisfies

$F^{k}S_{n}(U(m))=\bigcup_{\lambda\in f^{-1}(\mathbb{N}_{\leq
k})\cap(\mathcal{P}\backslash\mathcal{S})}F^{\lambda}\operatorname{Hom}(\mathbb{Z}^{n},U(m))\,.$

An argument analogous to that of Lemma 3.11 shows that the inclusions
$F^{k-1}S_{n}(U(m))\to F^{k}S_{n}(U(m))$ are inclusions of subcomplexes in a
CW-structure. Thus, there is a cofibre sequence of the form

$F^{k-1}S_{n}(U(m))_{+}\to F^{k}S_{n}(U(m))_{+}\to\bigvee_{\lambda\in
f^{-1}(k)\cap(\mathcal{P}\backslash\mathcal{S})}\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{\lambda}^{+}\,.$

The maps $F^{k}S_{n}(U(m))\to F^{k}\operatorname{Hom}(\mathbb{Z}^{n},U(m))$
are likewise inclusions of subcomplexes. Therefore, the space
$\operatorname{Hom}(\mathbb{Z}^{n},U(m))/S_{n}(U(m))$ is filtered by the
subspaces

$\bar{F}^{k}:=F^{k}\operatorname{Hom}(\mathbb{Z}^{n},U(m))/F^{k}S_{n}(U(m))$

which are in an induced cofibre sequence

$\bar{F}^{k-1}\to\bar{F}^{k}\to\bigvee_{\lambda\in
f^{-1}(k)\cap\mathcal{S}}\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{\lambda}^{+}\,.$

The proof is finished by showing that, after inverting $m!$, this cofibre
sequence splits for every $k\leq mn$. This is done in the same fashion as in
the proof of Corollary 4.2. ∎

###  4.3. Proof of Theorem 4.1

We begin by constructing a suitable approximation of the stable summands
$U(m)_{+}\wedge_{U(\lambda)}\bigwedge_{a\in
I}C_{|a|}(\mathfrak{u}_{\lambda_{a}})^{+}$.

For the unitary group $U(k)$ we choose a maximal torus $T$ and denote by
$\mathfrak{t}\subseteq\mathfrak{u}_{k}$ its Lie algebra. Then $\mathfrak{t}$
is naturally a $\Sigma_{k}$-representation and we can form the vector bundle
$U(k)/T\times_{\Sigma_{k}}\mathfrak{t}^{n}$ associated with the representation
$\mathfrak{t}^{n}=\mathfrak{t}\oplus\cdots\oplus\mathfrak{t}$. As we show in
Lemma A.3, the adjoint representation can be used to define a map from the
Thom space

$\phi^{+}\colon\thinspace(U(k)/T)_{+}\wedge_{\Sigma_{k}}(\mathfrak{t}^{+})^{\wedge
n}\to C_{n}(\mathfrak{u}_{k})^{+}$

which is a homology isomorphism with coefficients in $\mathbb{Z}[1/p\mid p\leq
k]\subseteq\mathbb{Q}$.

Now fix $\lambda\in\mathcal{P}$. To keep track of the rank we write
$T(\lambda_{a})$ for the maximal torus of $U(\lambda_{a})$ and
$\mathfrak{t}_{\lambda_{a}}$ for its Lie algebra. For every $a\in I$ we define
a based $U(\lambda_{a})$-space $\mathfrak{X}_{a}$ by

$\mathfrak{X}_{a}=\begin{cases}S^{0}&\textnormal{if }|a|=0,\\\
\mathfrak{u}_{\lambda_{a}}^{+}&\textnormal{if }|a|=1,\\\
(U(\lambda_{a})/T(\lambda_{a}))_{+}\wedge_{\Sigma_{\lambda_{a}}}(\mathfrak{t}_{\lambda_{a}}^{+})^{\wedge|a|}&\textnormal{if
}|a|>1.\end{cases}$

The $U(\lambda_{a})$-action is the adjoint action when $|a|=1$ and the obvious
action on $U(\lambda_{a})/T(\lambda_{a})$ when $|a|>1$. The construction above
supplies us with based $U(\lambda_{a})$-equivariant maps

$\phi_{a}^{+}\colon\thinspace\mathfrak{X}_{a}\to
C_{|a|}(\mathfrak{u}_{\lambda_{a}})^{+}$

for all $a\in I$ when $|a|>1$. When $|a|=0,1$ we let $\phi_{a}^{+}$ be the
identity map.

Let $Q(\lambda)$ be the set of primes $p$ such that
$p\leq\max\\{\lambda_{a}\mid a\in I,\,|a|>1\\}$.

######  Lemma 4.5.

After inverting all primes $p\in Q(\lambda)$, the map

$id\wedge\bigwedge_{a\in I}\phi_{a}^{+}\colon\thinspace
U(m)_{+}\wedge_{U(\lambda)}\bigwedge_{a\in I}\mathfrak{X}_{a}\to
U(m)_{+}\wedge_{U(\lambda)}\bigwedge_{a\in
I}C_{|a|}(\mathfrak{u}_{\lambda_{a}})^{+}$

is a stable homotopy equivalence.

###### Proof.

Let $R$ denote the subring $\mathbb{Z}[1/p\mid p\in
Q(\lambda)]\subseteq\mathbb{Q}$. The space
$U(m)_{+}\wedge_{U(\lambda)}\bigwedge_{a\in
I}C_{|a|}(\mathfrak{u}_{\lambda_{a}})^{+}$ admits a CW-structure by Corollary
3.12. By Whitehead’s theorem it therefore suffices to show that
$id\wedge\bigwedge_{a\in I}\phi_{a}^{+}$ is a homology isomorphism with
coefficients in $R$.

By Lemma A.3 every $\phi_{a}^{+}$, $a\in I$ induces an isomorphism on homology
with $R$-coefficients. Then $\bigwedge_{a\in I}\phi_{a}^{+}$ is a homology
isomorphism by the Künneth theorem, and

$id\times\bigwedge_{a\in I}\phi_{a}^{+}\colon\thinspace
U(m)\times_{U(\lambda)}\bigwedge_{a\in I}\mathfrak{X}_{a}\to
U(m)\times_{U(\lambda)}\bigwedge_{a\in
I}C_{|a|}(\mathfrak{u}_{\lambda_{a}})^{+}$

is a homology isomorphism by comparison of Serre spectral sequences. Both of
these bundles have canonical sections at infinity. Collapsing these and using
a five lemma argument we see that $id\wedge\bigwedge_{a\in I}\phi_{a}^{+}$ is
a homology isomorphism with $R$-coefficients as well. ∎

The next step will be the passage from the Lie algebra to the Lie group level.

For the maximal torus $T=T(k)$ of $U(k)$ let $F^{j}(T)\subseteq T$ be the
$j$-th stage of the fat wedge filtration. Then

$T\backslash F^{k-1}(T)=\\{(t_{1},\dots,t_{k})\in T\mid t_{i}\neq
1\textnormal{ for all }1\leq i\leq k\\}\,.$

The Cayley transform of Definition 2.1 restricts to a $\Sigma_{k}$-equivariant
homeomorphism $\mathfrak{t}\cong T\backslash F^{k-1}(T)$, and this extends to
a $\Sigma_{k}$-equivariant homeomorphism

$\mathfrak{t}^{+}\cong T/F^{k-1}(T)\,.$

This identification yields a $\Sigma_{k}$-equivariant quotient map
$c_{T}\colon\thinspace T_{+}\to\mathfrak{t}^{+}$. Its $n$-th smash power is
the $\Sigma_{k}$-equivariant map

$c_{T}^{\wedge n}\colon\thinspace T^{n}_{+}\cong(T_{+})^{\wedge
n}\to(\mathfrak{t}^{+})^{\wedge n}$

that collapses the subspace $F^{nk-1}(T^{n})\subseteq T^{n}$.

The fat wedge filtration of $T^{n}\cong(S^{1})^{nk}$ splits
$\Sigma_{nk}$-equivariantly by [8, Proposition 5.2]. In particular, there is a
$\Sigma_{k}$-equivariant stable section up to homotopy of $c_{T}^{\wedge n}$.
More precisely, let $V=\mathbb{R}^{nk}$ be the standard permutation
representation of $\Sigma_{nk}$, and write $V$ also for its restriction to a
$\Sigma_{k}$-representation along the diagonal inclusion
$\Sigma_{k}\leq\Sigma_{nk}$. Then there is a $\Sigma_{k}$-equivariant map

(6) $s_{T^{n}}\colon\thinspace V^{+}\wedge(\mathfrak{t}^{+})^{\wedge n}\to
V^{+}\wedge T^{n}_{+}$

and a $\Sigma_{k}$-equivariant homotopy $(id\wedge c_{T}^{\wedge
n})s_{T^{n}}\simeq id$.

For every $a\in I$ we define a based $U(\lambda_{a})$-space $X_{a}$ by

$X_{a}=\begin{cases}S^{0}&\textnormal{if }|a|=0,\\\
U(\lambda_{a})_{+}&\textnormal{if }|a|=1,\\\
(U(\lambda_{a})/T(\lambda_{a}))_{+}\wedge_{\Sigma_{\lambda_{a}}}(T(\lambda_{a})_{+})^{\wedge|a|}&\textnormal{if
}|a|>1.\end{cases}$

The $U(\lambda_{a})$-action in the case $|a|=1$ is by conjugation. There are
based $U(\lambda_{a})$-equivariant maps, for all $a\in I$

$\pi_{a}\colon\thinspace X_{a}\to\mathfrak{X}_{a}$

defined as follows: When $|a|>1$ we let $\pi_{a}=id\wedge c_{T}^{\wedge|a|}$,
and when $|a|=0,1$ we let $\pi_{a}$ be the identity map, respectively the
projection $U(\lambda_{a})_{+}\to\mathfrak{u}_{\lambda_{a}}^{+}$ onto the top
filtration quotient of $U(\lambda_{a})$ described in (3).

######  Lemma 4.6.

The map

$id\wedge\bigwedge_{a\in I}\pi_{a}\colon\thinspace
U(m)_{+}\wedge_{U(\lambda)}\bigwedge_{a\in I}X_{a}\to
U(m)_{+}\wedge_{U(\lambda)}\bigwedge_{a\in I}\mathfrak{X}_{a}$

has a stable section up to homotopy.

For the proof we require the following lemma.

######  Lemma 4.7.

Let $G$ be a compact Lie group and $H\subseteq G$ a closed subgroup. Let $X,Y$
be based $H$-spaces and $f\colon\thinspace X\to Y$ a based $H$-equivariant
map. If $f$ has an $H$-equivariant stable section up to homotopy, then
$id\wedge f\colon\thinspace G_{+}\wedge_{H}X\to G_{+}\wedge_{H}Y$ has a
$G$-equivariant stable section up to homotopy.

###### Proof.

By assumption there is a finite dimensional $H$-representation $V$, an
$H$-equivariant map

$s\colon\thinspace V^{+}\wedge Y\to V^{+}\wedge X$

and an $H$-equivariant homotopy $(id\wedge f)s\simeq id$.

Consider the $G$-vector bundle

$E=G\times_{H}V\to G/H\,.$

If $p\colon\thinspace G\to G/H$ is the projection, then the pullback

$p^{\ast}E=\\{(g_{1},[g_{2},v])\in G\times E\mid g_{1}H=g_{2}H\\}$

is a $(G\times H)$-vector bundle with action defined by

$(g,h)\cdot(g_{1},[g_{2},v])=(gg_{1}h^{-1},[gg_{1},v])\,.$

There is a $(G\times H)$-equivariant bundle isomorphism

$p^{\ast}E\cong G\times V$

defined by $(g_{1},[g_{2},v])\mapsto(g_{1},g_{1}^{-1}g_{2}v)$ with inverse
$(g,v)\mapsto(g,[g,v])$. The action on $G\times V$ is given by
$(g,h)\cdot(g_{1},v)=(gg_{1}h^{-1},hv)$.

Since $G$ is compact, we can find a $G$-vector bundle $F\to G/H$ such that
$F\oplus E$ is a trivial $G$-vector bundle, i.e.,

$F\oplus E\cong G/H\times W$

for some $G$-representation $W$. There is then an isomorphism of $(G\times
H)$-vector bundles

$G\times W\cong p^{\ast}(F\oplus E)\cong p^{\ast}F\times V\,,$

where $V$ is the trivial $G$-bundle over a point with $H$ acting trivially on
it. The $(G\times H)$-action on $G\times W$ is given by
$(g,h)\cdot(g_{1},w)=(gg_{1}h^{-1},gw)$.

Taking Thom spaces on both sides gives a $(G\times H)$-equivariant
homeomorphism

$G_{+}\wedge W^{+}\cong Z\wedge V^{+}\,,$

where $Z$ is the Thom space of $p^{\ast}F$.

Now we consider the composite

$W^{+}\wedge G_{+}\wedge Y\cong Z\wedge V^{+}\wedge Y\xrightarrow{id\wedge
s}Z\wedge V^{+}\wedge X\cong W^{+}\wedge G_{+}\wedge X\,.$

By construction this is $(G\times H)$-equivariant and the composite with
$id\wedge f$ is $(G\times H)$-equivarianty homotopic to the identity. By
passing to the $H$-quotient we obtain the desired $G$-equivariant stable
section up to homotopy. ∎

###### Proof of Lemma 4.6.

By Lemma 4.7 it suffices to show that for every $a\in I$ the map
$\pi_{a}\colon\thinspace X_{a}\to\mathfrak{X}_{a}$ has a
$U(\lambda_{a})$-equivariant stable section
$\sigma_{a}\colon\thinspace\mathfrak{X}_{a}\to X_{a}$ up to homotopy.

For $|a|=0$ this is trivial, and for $|a|=1$ a stable section is provided by a
Pontryagin-Thom collapse (see Section 2). Now assume $|a|>1$. We can view
$\pi_{a}$ as the map

$id\wedge c_{T}^{\wedge|a|}\colon\thinspace
U(\lambda_{a})_{+}\wedge_{N(T(\lambda_{a}))}(T(\lambda_{a})_{+})^{\wedge|a|}\to
U(\lambda_{a})_{+}\wedge_{N(T(\lambda_{a}))}(\mathfrak{t}_{\lambda_{a}}^{+})^{\wedge|a|}$

where $N(T(\lambda_{a}))$ is the normaliser of $T(\lambda_{a})$ in
$U(\lambda_{a})$. Once again, by Lemma 4.7, it suffices to give a
$N(T(\lambda_{a}))$-equivariant stable section up to homotopy of
$c_{T}^{\wedge|a|}$. But such a section is given by (6). ∎

Finally, we will construct a map

$\varphi_{\lambda}\colon\thinspace U(m)_{+}\wedge_{U(\lambda)}\bigwedge_{a\in
I}X_{a}\to F^{\lambda}\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{+}\,.$

First we define, for every $a\in I$, a based map

$\varphi_{a}^{+}\colon\thinspace
X_{a}\to\operatorname{Hom}(\mathbb{Z}^{|a|},U(\lambda_{a}))_{+}\,.$

For $|a|>1$ we let $\varphi_{a}^{+}$ be the action map defined in (4),
extended to a based map over the disjoint basepoint. For $|a|=0,1$ we choose
$\varphi_{a}^{+}$ to be the identity map.

Let $\beta_{\lambda}$ be the composite

$\bigwedge_{a\in I}X_{a}\xrightarrow{\bigwedge_{a\in
I}\textnormal{sh}_{a}^{+}\varphi_{a}^{+}}\bigwedge_{a\in
I}\operatorname{Hom}(\mathbb{Z}^{n},U(\lambda_{a}))_{+}\xrightarrow{\oplus}\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{+}$

where $\textnormal{sh}_{a}$ is the map defined in (5) and the last map takes
the block sum of matrices. By construction, the image of $\beta_{\lambda}$ is
in the subspace $F^{\lambda}\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{+}$. We
then define $\varphi_{\lambda}$ as the composite

$U(m)_{+}\wedge_{U(\lambda)}\bigwedge_{a\in
I}X_{a}\xrightarrow{id\wedge\beta_{\lambda}}U(m)_{+}\wedge_{U(\lambda)}F^{\lambda}\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{+}\to
F^{\lambda}\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{+}\,,$

in which the second map is conjugation. This is well-defined, because
$F^{\lambda}\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{+}$ is invariant under
the conjugation action by $U(m)$.

To finish the proof of Theorem 4.1 we consider the diagram

$\textstyle{U(m)_{+}\wedge_{U(\lambda)}\bigwedge_{a\in
I}X_{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id\wedge\bigwedge_{a\in
I}\pi_{a}}$$\scriptstyle{\varphi_{\lambda}}$$\textstyle{U(m)_{+}\wedge_{U(\lambda)}\bigwedge_{a\in
I}\mathfrak{X}_{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{id\wedge\bigwedge_{a\in
I}\phi_{a}^{+}}$$\textstyle{F^{\lambda}\operatorname{Hom}(\mathbb{Z}^{n},U(m))_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta_{\lambda}}$$\textstyle{U(m)_{+}\wedge_{U(\lambda)}\bigwedge_{a\in
I}C_{|a|}(\mathfrak{u}_{\lambda_{a}})^{+}\,.}$

A simple diagram chase shows that the diagram commutes. Let $\alpha$ be a
stable homotopy inverse of the right hand vertical map, and let
$\tilde{\sigma}_{\lambda}$ be a stable section, up to homotopy, of the top
horizontal map. Then the composite
$\varphi_{\lambda}\tilde{\sigma}_{\lambda}\alpha$ is a stable section, up to
homotopy, of $\zeta_{\lambda}$.

###  4.4. Symplectic and orthogonal groups

Stable decompositions for spaces of commuting elements in the symplectic and
orthogonal groups can be obtained with a proof that is almost identical with
the one in the unitary case.

The Weyl group of $Sp(m)$ and of $SO(2m+1)$ is the signed symmetric group,
i.e., the wreath product $\mathbb{Z}/2\wr\Sigma_{m}$. It acts on a maximal
torus $T\cong(S^{1})^{m}$ by permutation and inversion of the factors. The
Weyl group of $SO(2m)$ is the subgroup of $\mathbb{Z}/2\wr\Sigma_{m}$
consisting of those signed permutations with an even number of minus signs.

Now let $W$ denote any of these Weyl groups. As before, we denote by
$\mathcal{P}$ the poset of partitions of $m$ indexed by $I=\\{0,1\\}^{n}$. The
$\mathcal{P}$-indexed filtration of $T^{n}$ constructed in Section 3.3 and
represented by the map $\Psi_{T}\colon\thinspace T^{n}\to\mathcal{P}$ is
invariant under the $W$-action and thus it induces a map
$\bar{\Psi}_{T}\colon\thinspace T^{n}/W\to\mathcal{P}$.

Let us focus first on the symplectic groups. Every abelian subgroup of $Sp(m)$
is contained in a maximal torus. This implies that
$\operatorname{Hom}(\mathbb{Z}^{n},Sp(m))$ is path-connected, and we can
identify the space $\operatorname{Hom}(\mathbb{Z}^{n},Sp(m))/Sp(m)$ with
$T^{n}/(\mathbb{Z}/2\wr\Sigma_{m})$. Just like in the unitary case (see
Construction 3.6) we obtain an induced filtration

$\Psi_{Sp(m)}\colon\thinspace\operatorname{Hom}(\mathbb{Z}^{n},Sp(m))\to\mathcal{P}\,.$

The next step is the identification of the strata with bundles of commuting
varieties. This works again by use of the Cayley transform. The Cayley
transform $X\mapsto(X-\mathds{1})(X+\mathds{1})^{-1}$ is defined on the Lie
algebra of $Sp(m)$,

$\mathfrak{sp}_{m}=\\{X\in\textnormal{Mat}_{m}(\mathbb{H})\mid
X^{\dagger}=-X\\}\,.$

As in the complex case, it is an equivariant diffeomorphism of
$\mathfrak{sp}_{m}$ with the open subspace of $Sp(m)$ consisting of those
matrices $g$ such that $g-\mathds{1}$ is invertible. This leads to the obvious
analogue of Corollary 3.10 in which each unitary group is replaced by the
corresponding symplectic group.

To show that the filtration splits stably, we proceed as in the unitary case.
However, this requires a stable splitting of the map $c_{T}^{\wedge
n}\colon\thinspace T^{n}_{+}\to(\mathfrak{t}^{+})^{\wedge n}$ which is
equivariant for the action of the wreath product $\mathbb{Z}/2\wr\Sigma_{m}$.
The stable splitting proved in [8, Proposition 5.2] does have this property,
see [8, Remark 5.3]. We arrive at the following theorem:

######  Theorem 4.8.

After inverting $2^{m}m!$ there is a stable splitting for all
$n\in\mathbb{N}$,

$\operatorname{Hom}(\mathbb{Z}^{n},Sp(m))_{+}\simeq\bigvee_{\lambda\in\mathcal{P}}\left(Sp(m)_{+}\wedge_{\prod_{a\in
I}Sp(\lambda_{a})}\bigwedge_{a\in
I}C_{|a|}(\mathfrak{sp}_{\lambda_{a}})^{+}\right)\,.$

######  Example 4.9.

Consider $Sp(1)=SU(2)$. In this case $\mathcal{P}$ is in bijection with
$I=\\{0,1\\}^{n}$, by sending $\lambda=(\lambda_{a})_{a\in I}$ to the unique
$a$ such that $\lambda_{a}=1$. The resulting decomposition away from $2$ takes
the form

$\operatorname{Hom}(\mathbb{Z}^{n},SU(2))_{+}\simeq\bigvee_{0\leq l\leq
n}\left(\bigvee^{\binom{n}{l}}C_{l}(\mathfrak{su}_{2})^{+}\right)\,.$

In fact, the Cayley transform
$SU(2)\backslash\\{\mathds{1}\\}\cong\mathfrak{su}_{2}$ yields an “accidental”
homeomorphism

$C_{l}(\mathfrak{su}_{2})^{+}\cong\operatorname{Hom}(\mathbb{Z}^{l},SU(2))/S_{l}(SU(2))$

which shows that the splitting holds even without inverting $2$, see [1,
Theorem 6.5].

Now we consider the orthogonal groups. Let $k\in\\{2m,2m+1\\}$. When $k\geq 3$
some extra care must be taken, because if in addition $n\geq 2$, then
$\operatorname{Hom}(\mathbb{Z}^{n},SO(k))$ is no longer path-connected. In
particular, the action map (defined as in (4))

$\varphi\colon\thinspace
SO(k)/T\times_{W}T^{n}\to\operatorname{Hom}(\mathbb{Z}^{n},SO(k))$

maps only onto the path-component of the trivial homomorphism. Let
$\operatorname{Hom}(\mathbb{Z}^{n},SO(k))_{\mathds{1}}$ denote this path-
component. Then
$\operatorname{Hom}(\mathbb{Z}^{n},SO(k))_{\mathds{1}}/SO(k)\cong T^{n}/W$ and
$\bar{\Psi}_{T}$ induces a filtration

$\Psi_{SO(k)}\colon\thinspace\operatorname{Hom}(\mathbb{Z}^{n},SO(k))_{\mathds{1}}\to\mathcal{P}\,.$

Observe that
$\operatorname{Hom}(\mathbb{Z}^{n},O(k))_{\mathds{1}}=\operatorname{Hom}(\mathbb{Z}^{n},SO(k))_{\mathds{1}}$,
which is why we need not consider the full orthogonal group.

A characterisation of the path-components of
$\operatorname{Hom}(\mathbb{Z}^{n},SO(k))$ was obtained in [17]. In terms of
representation theory, a representation $\mathbb{Z}^{n}\to SO(k)$ belongs to
the path-component $\operatorname{Hom}(\mathbb{Z}^{n},SO(k))_{\mathds{1}}$ if
and only if it is isomorphic to a direct sum of $m$ representations
$\mathbb{Z}^{n}\to SO(2)$, and an additional one-dimensional trivial
representation if $k=2m+1$.

Using the real Cayley transform
$\psi_{2m}\colon\thinspace\mathfrak{so}_{2m}\to SO(2m)$, defined by the usual
formula $X\mapsto(X-\mathds{1})(X+\mathds{1})^{-1}$, we arrive at the
following analogue of Corollary 3.10: There is an equivariant homeomorphism

$\Psi_{SO(k)}^{-1}(\lambda)^{+}\cong(O(k)/O(2\lambda_{0}+\varepsilon))_{+}\wedge_{\prod_{0\neq
a\in I}O(2\lambda_{a})}\bigwedge_{a\in
I}C_{|a|}(\mathfrak{so}_{2\lambda_{a}})^{+}\,,$

where $\varepsilon=0$ if $k$ is even and $\varepsilon=1$ if $k$ is odd.

Proceeding as before we obtain:

######  Theorem 4.10.

Let $k\in\\{2m,2m+1\\}$ and let $\varepsilon=0$ if $k$ is even and
$\varepsilon=1$ if $k$ is odd. After inverting $2^{m-1+\varepsilon}m!$ there
is a stable splitting for all $n\in\mathbb{N}$,

$\operatorname{Hom}(\mathbb{Z}^{n},SO(k))^{+}_{\mathds{1}}\simeq\bigvee_{\lambda\in\mathcal{P}}\left((O(k)/O(2\lambda_{0}+\varepsilon))_{+}\wedge_{\prod_{0\neq
a\in I}O(2\lambda_{a})}\bigwedge_{a\in
I}C_{|a|}(\mathfrak{so}_{2\lambda_{a}})^{+}\right)\,.$

######  Example 4.11.

Consider the case $k=3$. Then $m=1$ and thus, as in Example 4.9, the poset of
partitions is in bijection with $I=\\{0,1\\}^{n}$. The theorem implies a
stable decomposition

$\operatorname{Hom}(\mathbb{Z}^{n},SO(3))_{\mathds{1}}^{+}\simeq
S^{0}\vee\bigvee_{1\leq l\leq
n}\left(\bigvee^{\binom{n}{l}}(O(3)/O(1))_{+}\wedge_{O(2)}C_{l}(\mathfrak{so}_{2})^{+}\right)\,.$

The summand $S^{0}$ comes from the partition $\lambda\in\mathcal{P}$ with
$\lambda_{0}=1$. A priori the splitting holds away from $2$. However, the only
commuting varieties that appear are those for $\mathfrak{so}_{2}$. Because the
Weyl group is trivial for $\mathfrak{so}_{2}$, inspection of the proof then
shows that the splitting holds without inverting $2$ (namely, the step of
Lemma 4.5 can be achieved without inverting primes). To identify the summands
we note that $C_{l}(\mathfrak{so}_{2})=\mathfrak{so}_{2}^{l}$ and that the
subgroup $SO(2)\subset O(2)$ acts trivially on $\mathfrak{so}_{2}^{l}$. The
residual action of $O(2)/SO(2)\cong\mathbb{Z}/2$ on
$\mathfrak{so}_{2}\cong\mathbb{R}$ is via sign, so that

$O(3)/O(1)\times_{O(2)}\mathfrak{so}_{2}^{l}\cong L\oplus\cdots\oplus
L\;\textnormal{($l$ times)}\,,$

where $L$ is the tautological line bundle over $\mathbb{R}P^{2}$. The Thom
space of $lL$ is known to be homeomorphic with the stunted projective space
$\mathbb{R}P^{l+2}/\mathbb{R}P^{l-1}$ (see e.g. [4, Lemma 3.2]). Therefore, we
obtain a stable splitting

$\operatorname{Hom}(\mathbb{Z}^{n},SO(3))_{\mathds{1}}\simeq\bigvee_{1\leq
l\leq
n}\left(\bigvee^{\binom{n}{l}}\mathbb{R}P^{l+2}/\mathbb{R}P^{l-1}\right)\,.$

(Here we have omitted the disjoint basepoint.) This splitting agrees with the
one obtained in [2, Section 7] and with the homology computations given in
[18].

######  Remark 4.12.

Notice that when we choose $n=1$ in the previous example, the stable splitting
sees no interesting information as $SO(3)\cong\mathbb{R}P^{3}$. In contrast,
Miller’s splitting of $SO(n)$ produces the well-known stable equivalence
$SO(3)\simeq\mathbb{R}P^{2}\vee S^{3}$.

##  5\. Steenrod powers and a negative result

In this section we show that the stable splitting of Theorem 4.1 does not hold
integrally:

######  Theorem 5.1.

Let $p$ be a prime and let $n\geq 2$ be an integer. The quotient map

$\zeta\colon\thinspace\operatorname{Hom}(\mathbb{Z}^{n},U(p))_{+}\to
C_{n}(\mathfrak{u}_{p})^{+}$

does not have a stable section up to homotopy at the prime $p$.

If $X$ is a space and $n\in\mathbb{N}$, we denote by

$\textnormal{SP}^{n}(X):=X^{n}/\Sigma_{n}$

the $n$-th symmetric power of $X$. If $X$ is based, there is an evident
inclusion map $\textnormal{SP}^{n-1}(X)\to\textnormal{SP}^{n}(X)$ whose
cofibre is the reduced symmetric power

$\overline{\textnormal{SP}}^{n}(X):=\textnormal{SP}^{n}(X)/\textnormal{SP}^{n-1}(X)\cong
X^{\wedge n}/\Sigma_{n}\,.$

We will prove Theorem 5.1 by comparing
$\operatorname{Hom}(\mathbb{Z}^{n},U(p))$ and $C_{n}(\mathfrak{u}_{p})^{+}$ to
the $p$-th (reduced) symmetric powers of $S^{n}$. Steenrod operations give
obstructions for a stable splitting of the natural map
$\textnormal{SP}^{p}(S^{n})\to\overline{\textnormal{SP}}^{p}(S^{n})$. We will
lift those obstructions to $\operatorname{Hom}(\mathbb{Z}^{n},U(p))$.

Fix $s\geq 1$. We will carry out most of the proof of Theorem 5.1 for the
unitary group of rank $p^{s}$; only in Lemma 5.5 will we have to choose $s=1$.

Let $T\subseteq U(p^{s})$ be a maximal torus. Observe that every commuting
$n$-tuple in $U(p^{s})$ is conjugate to an $n$-tuple in $T$. It follows that
the quotient of $\operatorname{Hom}(\mathbb{Z}^{n},U(p^{s}))$ by the adjoint
action of $U(p^{s})$ is homeomorphic to the quotient of $T^{n}$ by the
diagonal action of the Weyl group $\Sigma_{p^{s}}$. Since
$T\cong(S^{1})^{p^{s}}$ and $\Sigma_{p^{s}}$ permutes the factors, we obtain a
homeomorphism

$\operatorname{Hom}(\mathbb{Z}^{n},U(p^{s}))/U(p^{s})\cong\textnormal{SP}^{p^{s}}((S^{1})^{n})\,.$

Collapsing further the $(n-1)$-skeleton of $(S^{1})^{n}$ we obtain a composite
map

$\pi_{n}\colon\thinspace\operatorname{Hom}(\mathbb{Z}^{n},U(p^{s}))\to\textnormal{SP}^{p^{s}}(S^{n})\,.$

The discussion for the commuting variety $C_{n}(\mathfrak{u}_{p^{s}})^{+}$ is
similar. Let $\mathfrak{t}\subseteq\mathfrak{u}_{p^{s}}$ be the Lie algebra of
$T$. As every commuting $n$-tuple in $\mathfrak{u}_{p^{s}}$ is contained in a
Cartan subalgebra of $\mathfrak{u}_{p^{s}}$ (see Proposition A.1 in the
appendix) we can identify $C_{n}(\mathfrak{u}_{p})^{+}/U(p)$ with the quotient
of $(\mathfrak{t}^{n})^{+}$ by the diagonal action of $\Sigma_{p^{s}}$. As
$(\mathfrak{t}^{n})^{+}\cong((S^{1})^{\wedge n})^{\wedge p^{s}}$ with
$\Sigma_{p^{s}}$ permuting the smash factors, there is a homeomorphism

$C_{n}(\mathfrak{u}_{p})^{+}/U(p)\cong\overline{\textnormal{SP}}^{p^{s}}(S^{n})\,.$

In particular, we have a quotient map

$\bar{\pi}_{n}\colon\thinspace
C_{n}(\mathfrak{u}_{p^{s}})^{+}\to\overline{\textnormal{SP}}^{p^{s}}(S^{n})\,.$

It is easily seen that the maps $\pi_{n}$, $\bar{\pi}_{n}$ and the canonical
projection
$\bar{\zeta}\colon\thinspace\textnormal{SP}^{p^{s}}(S^{n})\to\overline{\textnormal{SP}}^{p^{s}}(S^{n})$
fit into a commutative diagram

(7)
$\textstyle{\operatorname{Hom}(\mathbb{Z}^{n},U(p^{s}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{n}}$$\scriptstyle{\zeta}$$\textstyle{\textnormal{SP}^{p^{s}}(S^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\zeta}}$$\textstyle{C_{n}(\mathfrak{u}_{p^{s}})^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\pi}_{n}}$$\textstyle{\overline{\textnormal{SP}}^{p^{s}}(S^{n}).}$

Let $\mathscr{P}^{i}$ denote the Steenrod reduced $p$-th power; when $p=2$ we
interpret $\mathscr{P}^{i}$ as $\textnormal{Sq}^{2i}$. In what follows we will
omit the coefficient ring for cohomology; if not noted otherwise, it will be
$\mathbb{F}_{p}$.

The mod-$p$ cohomology of $\textnormal{SP}^{p^{s}}(S^{n})$ was computed by
Nakaoka [14]. Let $u_{n}\in
H^{n}(\textnormal{SP}^{p^{s}}(S^{n}))\cong\mathbb{F}_{p}$ be a generator. By
[14, Corollary 2] there exists a unique class

$v_{n}\in H^{n+2(p^{s}-1)}(\overline{\textnormal{SP}}^{p^{s}}(S^{n}))$

such that
$\bar{\zeta}^{\ast}(v_{n})=(\mathscr{P}^{p^{s-1}}\cdots\mathscr{P}^{1})(u_{n})$.
Setting $a_{n,s}:=\pi_{n}^{\ast}(u_{n})$ and
$b_{n,s}:=\bar{\pi}_{n}^{\ast}(v_{n})$ we have that

$\zeta^{\ast}(b_{n,s})=(\mathscr{P}^{p^{s-1}}\cdots\mathscr{P}^{1})(a_{n,s})\,.$

The class $b_{n,s}$ will be the obstruction used to prove Theorem 5.1:

######  Lemma 5.2.

Suppose that $b_{n,s}\neq 0$. Then the quotient map

$\zeta\colon\thinspace\operatorname{Hom}(\mathbb{Z}^{n},U(p^{s}))_{+}\to
C_{n}(\mathfrak{u}_{p^{s}})^{+}$

does not have a stable section up to homotopy at the prime $p$.

###### Proof.

Suppose that, at the prime $p$, there is a _stable_ map

$\sigma\colon\thinspace
C_{n}(\mathfrak{u}_{p^{s}})^{+}\to\operatorname{Hom}(\mathbb{Z}^{n},U(p^{s}))_{+}$

such that $\zeta\sigma\simeq id$. Stability of the Steenrod powers gives

$0\neq
b_{n,s}=(\zeta\sigma)^{\ast}(b_{n,s})=(\mathscr{P}^{p^{s-1}}\cdots\mathscr{P}^{1})(\sigma^{\ast}(a_{n,s}))\,.$

By Proposition A.9 in the appendix, $C_{n}(\mathfrak{u}_{p^{s}})^{+}$ is at
least $n$-connected. Since $a_{n,s}$ has degree $n$, we have that
$\sigma^{\ast}(a_{n,s})=0$ which is a contradiction. ∎

Next we reduce to the case $n=2$.

######  Lemma 5.3.

If $b_{2,s}$ is non-zero, then $b_{n,s}$ is non-zero for all $n\geq 2$.

###### Proof.

Recall that there are maps
$S^{1}\times\textnormal{SP}^{p^{s}}(S^{n})\to\textnormal{SP}^{p^{s}}(S^{n+1})$
defined by

$(t,(x_{1},\dots,x_{p^{s}}))\to(t\wedge x_{1},\dots,t\wedge x_{p^{s}})$

which descend to maps

$\tau_{n}\colon\thinspace\Sigma\textnormal{SP}^{p^{s}}(S^{n})\to\textnormal{SP}^{p^{s}}(S^{n+1})\,.$

It is easy to see that for each $n$, $\tau_{n}$ induces an isomorphism on
$H^{n+1}$. Thus, we may assume $u_{n}$ be induced from a single element
$u\in\varprojlim H^{n}(\textnormal{SP}^{p^{s}}(S^{n}))\cong\mathbb{F}_{p}$, so
that $\tau_{n}^{\ast}(u_{n+1})$ corresponds to $u_{n}$ under the suspension
isomorphism. The maps $\tau_{n}$ descend further to maps

$\bar{\tau}_{n}\colon\thinspace\Sigma\overline{\textnormal{SP}}^{p^{s}}(S^{n})\to\overline{\textnormal{SP}}^{p^{s}}(S^{n+1})\,.$

It follows that also $v_{n}$ is stable, i.e., that
$\bar{\tau}_{n}^{\ast}(v_{n+1})$ corresponds to $v_{n}$ under suspension. Now
let $\mathfrak{z}\subseteq\mathfrak{u}_{p^{s}}$ denote the centre. Upon
identifying $\mathfrak{z}^{+}$ with $S^{1}$, the map $\mathfrak{z}\times
C_{n}(\mathfrak{u}_{p^{s}})\to C_{n+1}(\mathfrak{u}_{p^{s}})$ obtained by
restricting the inclusion
$\mathfrak{z}\times\mathfrak{u}_{p^{s}}^{n}\subseteq\mathfrak{u}_{p^{s}}^{n+1}$
induces a map

$\sigma_{n}\colon\thinspace\Sigma C_{n}(\mathfrak{u}_{p^{s}})^{+}\to
C_{n+1}(\mathfrak{u}_{p^{s}})^{+}\,.$

This map fits into a commutative diagram

$\textstyle{\Sigma
C_{n}(\mathfrak{u}_{p^{s}})^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Sigma\bar{\pi}_{n}}$$\scriptstyle{\sigma_{n}}$$\textstyle{\Sigma\overline{\textnormal{SP}}^{p^{s}}(S^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\tau}_{n}}$$\textstyle{C_{n+1}(\mathfrak{u}_{p^{s}})^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\pi}_{n+1}}$$\textstyle{\overline{\textnormal{SP}}^{p^{s}}(S^{n+1})\,.}$

The diagram shows that if $\bar{\pi}_{n}^{\ast}(v_{n})\neq 0$, then
$\bar{\pi}_{n+1}^{\ast}(v_{n+1})\neq 0$. This gives the desired reduction to
the case $n=2$. ∎

######  Remark 5.4.

The $\\{\bar{\tau}_{n}\\}$ are the structure maps of the spectrum
$\textnormal{SP}^{p^{s}}(\mathbb{S})/\textnormal{SP}^{p^{s}-1}(\mathbb{S})$
where $\mathbb{S}$ is the sphere spectrum. The $\\{v_{n}\\}$ come from a class
in spectrum cohomology

$v\in\varprojlim
H^{n+2(p^{s}-1)}(\overline{\textnormal{SP}}^{p^{s}}(S^{n}))\cong
H^{2(p^{s}-1)}(\textnormal{SP}^{p^{s}}(\mathbb{S})/\textnormal{SP}^{p^{s}-1}(\mathbb{S}))\,.$

The proof showed that the spaces $\\{C_{n}(\mathfrak{u}_{p^{s}})^{+}\\}$
define a spectrum, too, with structure maps $\\{\sigma_{n}\\}$. The maps
$\\{\bar{\pi}_{n}\\}$ give a map of spectra

$\bar{\pi}\colon\thinspace\\{C_{n}(\mathfrak{u}_{p^{s}})^{+}\\}\to\textnormal{SP}^{p^{s}}(\mathbb{S})/\textnormal{SP}^{p^{s}-1}(\mathbb{S})$

which, when $s=1$, satisfies $\bar{\pi}^{\ast}(v)\neq 0$ as the next lemma
will show.

The following lemma shows that $b_{2,1}\neq 0$, and thus $b_{n,1}\neq 0$ for
all $n\geq 2$ according to Lemma 5.3. In view of Lemma 5.2 this finishes the
proof of Theorem 5.1.

######  Lemma 5.5.

The map $\bar{\pi}_{2}\colon\thinspace
C_{2}(\mathfrak{u}_{p})^{+}\to\overline{\textnormal{SP}}^{p}(S^{2})\cong
S^{2p}$ is an isomorphism in $H^{2p}$.

###### Proof.

The homeomorphism $\overline{\textnormal{SP}}^{p}(S^{2})\cong S^{2p}$ follows
from Proposition A.7. It also follows directly from the well-known fact that
$\textnormal{SP}^{p}(\mathbb{C}P^{1})\cong\mathbb{C}P^{p}$ (see e.g. [19,
Section 22]).

Let $i\colon\thinspace(\mathfrak{t}^{2})^{+}\to C_{2}(\mathfrak{u}_{p})^{+}$
be the map induced by the inclusion $\mathfrak{t}\subseteq\mathfrak{u}_{p}$.
To prove the lemma it will be enough to show that

$i^{\ast}\colon\thinspace H^{2p}(C_{2}(\mathfrak{u}_{p})^{+})\to
H^{2p}((\mathfrak{t}^{2})^{+})$

is trivial with mod-$p$ coefficients. Indeed, consider the composite of $i$
and $\bar{\pi}_{2}$ on cohomology with $\mathbb{Z}_{(p)}$\- and
$\mathbb{F}_{p}$-coefficients. By Theorems A.4 and A.11 we have that
$H^{2p}(C_{2}(\mathfrak{u}_{p})^{+};\mathbb{Z}_{(p)})\cong\mathbb{Z}_{(p)}$
and $H^{2p}(C_{2}(\mathfrak{u}_{p})^{+};\mathbb{F}_{p})\cong\mathbb{F}_{p}$,
so that we have a diagram of the following form:

$\textstyle{\mathbb{Z}_{(p)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\pi}_{2}^{\ast}}$$\scriptstyle{\cong}$$\textstyle{\mathbb{Z}_{(p)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\ast}}$$\textstyle{\mathbb{Z}_{(p)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{F}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\pi}_{2}^{\ast}}$$\textstyle{\mathbb{F}_{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\ast}}$$\scriptstyle{0}$$\textstyle{\mathbb{F}_{p}}$

Here the vertical maps are reduction modulo $p$. If we assume that the lower
right map is zero, then the degree of
$i^{\ast}\colon\thinspace\mathbb{Z}_{(p)}\to\mathbb{Z}_{(p)}$ must be
divisible by $p$. Since the composite of the top horizontal maps is
multiplication by $p!$ (see Remark A.8), the top left map must be an
isomorphism. It follows from the diagram that the lower left map is an
isomorphism, too.

It remains to show that $i^{\ast}$ is trivial with
$\mathbb{F}_{p}$-coefficients. Let $S\subseteq SU(p)$ be a maximal torus and
$\mathfrak{s}$ its Lie algebra, so that
$\mathfrak{t}\cong\mathfrak{z}\oplus\mathfrak{s}$. By Proposition A.9 the map
$i$ is the double suspension of the inclusion $(\mathfrak{s}^{2})^{+}\to
C_{2}(\mathfrak{su}_{p})^{+}$. By Lemma A.5, the latter is the (unreduced)
suspension of the inclusion

$j\colon\thinspace S(\mathfrak{s}^{2})\to C_{2}^{1}(\mathfrak{su}_{p})\,,$

where $S(\mathfrak{s}^{2})\subseteq\mathfrak{s}^{2}$ is the unit sphere and
$C_{2}^{1}(\mathfrak{su}_{p})$ is the intersection of
$C_{2}(\mathfrak{su}_{p})$ with the unit sphere in $\mathfrak{su}_{p}^{2}$.
Thus, we may equivalently show that $j^{\ast}$ is trivial with
$\mathbb{F}_{p}$-coefficients.

The inclusion $j$ can be factored as

$S(\mathfrak{s}^{2})\to
SU(p)/S\times_{\Sigma_{p}}S(\mathfrak{s}^{2})\xrightarrow{\phi^{\prime}}C_{2}^{1}(\mathfrak{su}_{p})\,,$

where $\phi^{\prime}$ is the action map defined in (9) and the first map is
the inclusion of the fibre of the sphere bundle

(8) $SU(p)/S\times_{\Sigma_{p}}S(\mathfrak{s}^{2})\to(SU(p)/S)/\Sigma_{p}\,.$

We claim that the fibre inclusion is zero in mod-$p$ cohomology. Let
$(E_{r}^{\ast,\ast},d_{r})$ be the Serre spectral sequence of (8). Then
$E^{0,2p-3}_{2}\cong\mathbb{F}_{p}$ and $d_{2}\colon\thinspace
E^{0,2p-3}_{2}\to E^{2p-2,0}_{2}$ is determined by the mod-$p$ Euler class of
(8). In Lemma A.14 we show that the Euler class is non-zero, so $d_{2}$ is
injective and thus $E_{\infty}^{0,2p-3}=0$. This proves the claim. ∎

######  Remark 5.6.

Let us explain why the same line of arguments followed in this section fails
to work for $SU(p)$. In the case of $\operatorname{Hom}(\mathbb{Z}^{2},U(p))$
diagram (7) takes the form

$\textstyle{\operatorname{Hom}(\mathbb{Z}^{2},U(p))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{C}P^{p}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{2}(\mathfrak{u}_{p})^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{2p}}$

where the right vertical map is the projection collapsing $\mathbb{C}P^{p-1}$.
The obstruction for splitting the left vertical map comes from the $p$-th
power operation

$\mathscr{P}^{1}\colon\thinspace H^{2}(\mathbb{C}P^{p})\to
H^{2p}(\mathbb{C}P^{p})\,.$

In contrast, for $SU(p)$ the moduli space of representations is

$\operatorname{Hom}(\mathbb{Z}^{2},SU(p))/SU(p)\cong\mathbb{C}P^{p-1}$

(see e.g. [2, Proposition 6.5]), and $\mathbb{C}P^{p-1}$ does not have any
non-trivial mod-$p$ Steenrod operations. Indeed, when $p=2$ we know that
$C_{2}(\mathfrak{su}_{p})^{+}$ is a stable summand of
$\operatorname{Hom}(\mathbb{Z}^{2},SU(p))_{+}$.

##  Appendix A Topology of commuting varieties

Suppose that $G$ is a compact connected Lie group of rank $r\geq 1$ and
dimension $d$ and let $\mathfrak{g}$ be its Lie algebra. In this appendix we
make some basic observations about the one-point compactification of the
commuting variety

$C_{n}(\mathfrak{g})=\\{(X_{1},\dots,X_{n})\in\mathfrak{g}^{n}\mid[X_{i},X_{j}]=0\textnormal{
for all }1\leq i,j\leq n\\}$

which we view as a subspace of the Euclidean space
$\mathfrak{g}^{n}\cong\mathbb{R}^{nd}$. As before, we let $G$ act on
$C_{n}(\mathfrak{g})^{+}$ diagonally via the adjoint representation, leaving
invariant the point $+$ at infinity.

###  A.1. General results

We start by observing that the $G$-action on $C_{n}(\mathfrak{g})^{+}$ has
isotropy groups of maximal rank, i.e., the isotropy group at any point of
$C_{n}(\mathfrak{g})^{+}$ contains a maximal torus of $G$.

######  Proposition A.1.

Any $(X_{1},\dots,X_{n})\in C_{n}(\mathfrak{g})$ is contained in a Cartan
subalgebra of $\mathfrak{g}$.

###### Proof.

Suppose that $(X_{1},\dots,X_{n})\in C_{n}(\mathfrak{g})$ and let
$g_{i}:=\exp(X_{i})$ for every $1\leq i\leq n$. Note that
$(g_{1},\dots,g_{n})\in\operatorname{Hom}(\mathbb{Z}^{n},G)_{\mathds{1}}$, the
path–component of the space of commuting $n$-tuples
$\operatorname{Hom}(\mathbb{Z}^{n},G)$ that contains the trivial
representation $\mathds{1}=(1,\dots,1)$. Indeed, the Baker–Campbell–Hausdorff
formula shows that for every $1\leq i,j\leq n$

$g_{i}g_{j}=\exp(X_{i}+X_{j})=\exp(X_{j}+X_{i})=g_{j}g_{i}.$

Now the map
$\gamma\colon\thinspace[0,1]\to\operatorname{Hom}(\mathbb{Z}^{n},G)$ given by
$t\mapsto(\exp(tX_{1}),\dots,\exp(tX_{n}))$ provides a path from $\mathds{1}$
to $(g_{1},\dots,g_{n})$. In particular, for every $t\in[0,1]$ the $n$-tuple
$(\exp(tX_{1}),\dots,\exp(tX_{n}))$ is contained in a maximal torus of $G$, by
[3, Lemma 4.2]. Choose $t>0$ small enough such that, for all $1\leq i\leq n$,
$tX_{i}$ lies in a neighbourhood of $0\in\mathfrak{g}$ on which the
exponential map is injective. Let $T$ be a maximal torus containing
$(\exp(tX_{1}),\dots,\exp(tX_{n}))$. Let $\mathfrak{t}\subseteq\mathfrak{g}$
be the Lie algebra of $T$. It follows that $tX_{i}\in\mathfrak{t}$ and thus
$X_{i}\in\mathfrak{t}$ for all $1\leq i\leq n$. We conclude that any
$\underline{X}:=(X_{1},\dots,X_{n})\in C_{n}(\mathfrak{g})$ is contained in
some Cartan subalgebra of $\mathfrak{g}$. ∎

######  Corollary A.2.

The $G$-action on $C_{n}(\mathfrak{g})^{+}$ has isotropy groups of maximal
rank.

###### Proof.

The point $+$ at infinity is fixed by $G$. By Proposition A.1, any
$\underline{X}=(X_{1},\dots,X_{n})\in C_{n}(\mathfrak{g})$ lies in a Cartan
subalgebra $\mathfrak{t}$, which we view as the Lie algebra of a maximal torus
$T\subseteq G$. Then the isotropy subgroup

$G_{\underline{X}}=\left\\{g\in
G~{}|~{}\textnormal{Ad}_{g}(X_{i})=X_{i}\,,\text{ for all
}i=1,\dots,n\right\\}$

where $\textnormal{Ad}_{g}\colon\thinspace\mathfrak{g}\to\mathfrak{g}$ is the
adjoint action of $g\in G$, contains the maximal torus $T$. ∎

Next we consider the homology of $C_{n}(\mathfrak{g})^{+}$. Let $T\subseteq G$
be a maximal torus, $\mathfrak{t}\subseteq\mathfrak{g}$ its Lie algebra, and
$W$ the Weyl group relative to $T$. Consider the vector bundle

$\mathfrak{t}^{n}\to G/T\times_{W}\mathfrak{t}^{n}\to(G/T)/W$

associated with the $W$-representation
$\mathfrak{t}^{n}=\mathfrak{t}\oplus\cdots\oplus\mathfrak{t}$. The map

$\displaystyle\phi\colon\thinspace G/T\times_{W}\mathfrak{t}^{n}$
$\displaystyle\to C_{n}(\mathfrak{g})$ $\displaystyle[gT,X_{1},\dots,X_{n}]$
$\displaystyle\mapsto(\textnormal{Ad}_{g}(X_{1}),\dots,\textnormal{Ad}_{g}(X_{n}))\,,$

where $\textnormal{Ad}_{g}\colon\thinspace\mathfrak{g}\to\mathfrak{g}$ is the
adjoint action of $g\in G$, is proper. Therefore it extends to a map from the
Thom space

$\phi^{+}\colon\thinspace(G/T)_{+}\wedge_{W}(\mathfrak{t}^{n})^{+}\to
C_{n}(\mathfrak{g})^{+}\,.$

Let $\tilde{\phi}\colon\thinspace G/T\times_{W}(\mathfrak{t}^{n})^{+}\to
C_{n}(\mathfrak{g})^{+}$ be the composition of $\phi^{+}$ and the projection

$G/T\times_{W}(\mathfrak{t}^{n})^{+}\to(G/T)_{+}\wedge_{W}(\mathfrak{t}^{n})^{+}$

that collapses the section at infinity.

######  Lemma A.3.

The map $\tilde{\phi}\colon\thinspace G/T\times_{W}(\mathfrak{t}^{n})^{+}\to
C_{n}(\mathfrak{g})^{+}$ induces an isomorphism on homology with coefficients
in $\mathbb{Z}[1/|W|]$, and so does the map $\phi^{+}$.

###### Proof.

By Corollary A.2 the isotropy group of any point of $C_{n}(\mathfrak{g})^{+}$
under the adjoint action has maximal rank. Moreover,
$(C_{n}(\mathfrak{g})^{+})^{T}=(\mathfrak{t}^{n})^{+}$. Thus, it follows from
[3, Theorem 3.3] that $\tilde{\phi}$ induces an isomorphism with coefficients
in $\mathbb{Z}[1/|W|]$. Furthermore, the homology of $(G/T)/W$ with
$\mathbb{Z}[1/|W|]$-coefficients is trivial, so the map collapsing the section
at infinity is a homology isomorphism as well. ∎

######  Theorem A.4.

Let $n\geq 1$. Then $C_{n}(\mathfrak{g})^{+}$ is a
$\mathbb{Z}[1/|W|]$-homology sphere of dimension $nr$ if $n$ is even, and of
dimension $d+(n-1)r$ if $n$ is odd.

###### Proof.

Let $k=\mathbb{Z}[1/|W|]$. According to Lemma A.3 there is an isomorphism

$H^{\ast}(C_{n}(\mathfrak{g})^{+};k)\cong
H^{\ast}(G/T\times(\mathfrak{t}^{n})^{+};k)^{W}\,.$

If $n$ is even, then $W$ acts orientation-preservingly on
$(\mathfrak{t}^{n})^{+}$ and thus trivially on
$H^{\ast}((\mathfrak{t}^{n})^{+};k)$. As an ungraded $W$-module $H^{*}(G/T;k)$
is isomorphic to the group ring $kW$, and thus the trivial representation
$H^{\ast}(G/T;k)^{W}$ is one-dimensional and must occur in degree $0$. This
shows that

$H^{i}(C_{n}(\mathfrak{g})^{+};k)\cong\begin{cases}k&\textnormal{if }i=0,nr\\\
0&\textnormal{otherwise,}\end{cases}$

and hence that $C_{n}(\mathfrak{g})^{+}$ is a $k$-homology sphere of dimension
$nr$.

On the other hand, if $n$ is odd, then $W$ acts orientation-reversingly on
$(\mathfrak{t}^{n})^{+}$, and so $H^{\ast}((\mathfrak{t}^{n})^{+};k)$ is the
sum of the trivial $W$-module $k$ in degree zero and the sign representation
$k^{\sigma}$ in degree $nr$. Since $|W|$ is invertible in $k$, there is an
isomorphism

$\Bigl{(}H^{\ast}(G/T;k)\otimes k^{\sigma}\Bigr{)}^{W}\cong
H^{\ast}((G/T)/W;k^{\sigma})$

and, by Poincaré duality, this is $k$ in degree $d-r$ and zero in all other
degrees. Taking into account the fact that $k^{\sigma}$ is in degree $nr$, it
follows that

$\Bigl{(}H^{\ast}(G/T;k)\otimes
H^{\ast}((\mathfrak{t}^{n})^{+};k)\Bigr{)}^{W}$

is $k$ in degrees $0$ and $d+(n-1)r$, and zero otherwise. This finishes the
argument when $n$ is odd. ∎

Since $G$ is compact, we can fix an Ad-invariant inner product on
$\mathfrak{g}^{n}$. Let $S(\mathfrak{g}^{n})\subseteq\mathfrak{g}^{n}$ denote
the unit sphere with respect to this inner product. Let

$C^{1}_{n}(\mathfrak{g}):=C_{n}(\mathfrak{g})\cap S(\mathfrak{g}^{n})\,.$

We use the superscript $\lozenge$ to denote the unreduced suspension.

######  Lemma A.5.

There is a $G$-equivariant homeomorphism

$C_{n}(\mathfrak{g})^{+}\cong C_{n}^{1}(\mathfrak{g})^{\lozenge}$

where $G$ acts trivially on the suspension coordinate.

###### Proof.

Fix a homeomorphism
$\gamma\colon\thinspace[0,1]\stackrel{{\scriptstyle\cong}}{{\to}}\mathbb{R}_{\geq
0}^{+}$ such that $\gamma(0)=0$ and $\gamma(1)=+$. Using this define a
$G$-equivariant homeomorphism

$S(\mathfrak{g}^{n})^{\lozenge}\stackrel{{\scriptstyle\cong}}{{\to}}(\mathfrak{g}^{n})^{+}\,,\quad(t,X)\mapsto\begin{cases}\gamma(t)X&\textnormal{if
}t<1\\\ +&\textnormal{if }t=1.\end{cases}$

Since rescaling preserves commutativity, this restricts to a $G$-equivariant
homeomorphism $C_{n}^{1}(\mathfrak{g})^{\lozenge}\cong
C_{n}(\mathfrak{g})^{+}$. ∎

Let us view $S(\mathfrak{t}^{n})\subseteq(\mathfrak{t}^{n})^{+}$ as the
equatorial sphere. The map $\tilde{\phi}\colon\thinspace
G/T\times_{W}(\mathfrak{t}^{n})^{+}\to C_{n}(\mathfrak{g})^{+}$ then restricts
to a map

(9) $\phi^{\prime}\colon\thinspace G/T\times_{W}S(\mathfrak{t}^{n})\to
C_{n}^{1}(\mathfrak{g})\,.$

######  Proposition A.6.

For every $n\geq 1$ the space $C_{n}(\mathfrak{g})^{+}$ is simply-connected,
unless $n=1$ and $\dim(\mathfrak{g})=1$ in which case
$C_{n}(\mathfrak{g})^{+}\cong S^{1}$.

###### Proof.

First assume $n=1$. Then $C_{n}(\mathfrak{g})^{+}\cong\mathfrak{g}^{+}\cong
S^{\dim(\mathfrak{g})}$ which is simply-connected, unless
$\dim(\mathfrak{g})=1$. Now assume $n\geq 2$. By Proposition A.1 the map
$\phi^{\prime}$ is surjective and hence, since $S(\mathfrak{t}^{n})$ is path-
connected, $C^{1}_{n}(\mathfrak{g})$ is path-connected as well. Since
$C_{n}(\mathfrak{g})^{+}$ is the suspension of $C^{1}_{n}(\mathfrak{g})$, it
is thus simply-connected. ∎

Next we describe, for the particular case of commuting pairs, the quotient of
$C_{2}(\mathfrak{g})^{+}$ by the adjoint action of $G$.

######  Proposition A.7.

Let $G$ be a compact connected Lie group of rank $r$. Then there is a
homeomorphism $C_{2}(\mathfrak{g})^{+}/G\cong S^{2r}$.

###### Proof.

In Proposition A.1 we showed that every $(X_{1},X_{2})\in C_{2}(\mathfrak{g})$
is contained in a Cartan subalgebra $\mathfrak{t}\subseteq\mathfrak{g}$.
Therefore, the quotient $C_{2}(\mathfrak{g})^{+}/G$ may be identified with the
quotient of $(\mathfrak{t}^{2})^{+}$ by the diagonal action of the Weyl group.
We can identify the $W$-module $\mathfrak{t}^{2}$ with
$\mathfrak{t}\otimes{\mathbb{C}}$, the complexification of $\mathfrak{t}$. By
Chevalley’s theorem the ring of $W$-invariant polynomial functions on
$\mathfrak{t}\otimes\mathbb{C}$ is a polynomial algebra generated by $r$
homogeneous elements. These homogeneous elements provide a system of global
coordinates defining a homeomorphism of $(\mathfrak{t}\otimes\mathbb{C})/W$
with $\mathbb{C}^{r}$ (cf. [11, Prop. 9.3]), and hence a homeomorphism of
$(\mathfrak{t}\otimes\mathbb{C})^{+}/W$ with $(\mathbb{C}^{r})^{+}\cong
S^{2r}$. ∎

######  Remark A.8.

The composition of the inclusion $(\mathfrak{t}^{2})^{+}\to
C_{2}(\mathfrak{g})^{+}$ with the projection $C_{2}(\mathfrak{g})^{+}\to
C_{2}(\mathfrak{g})^{+}/G$ may be identified with the projection
$(\mathfrak{t}^{2})^{+}\to(\mathfrak{t}^{2})^{+}/W$. By computing local
degrees one sees that this map has degree $|W|$.

Finally, using the classification of compact Lie algebras, it follows that

$\mathfrak{g}\cong\mathfrak{z}\oplus\mathfrak{g}_{1}\oplus\cdots\oplus\mathfrak{g}_{s}\,,$

where $\mathfrak{z}$ denotes the center of $\mathfrak{g}$ and
$\mathfrak{g}_{1},\dots\mathfrak{g}_{s}$ are simple Lie algebras. From here we
get the following statement.

######  Proposition A.9.

For all $n\geq 0$,
$C_{n}(\mathfrak{g})^{+}\cong\Sigma^{n\dim(\mathfrak{z})}\left(C_{n}(\mathfrak{g}_{1})^{+}\wedge\cdots\wedge
C_{n}(\mathfrak{g}_{s})^{+}\right)$ where the $\mathfrak{g}_{i}$ are the
simple factors of $\mathfrak{g}$.

###### Proof.

By definition of the Lie bracket on a direct sum of Lie algebras we have that

$C_{n}(\mathfrak{g})\cong C_{n}(\mathfrak{z})\times
C_{n}(\mathfrak{g}_{1})\times\cdots\times C_{n}(\mathfrak{g}_{s})\,.$

Since $\mathfrak{z}$ is abelian, we further have
$C_{n}(\mathfrak{z})=\mathfrak{z}^{n}$, and the statement follows. ∎

######  Remark A.10.

We clarify the comment made in the introduction that $C_{n}(\mathfrak{g})$ is
a possibly infinite subspace arrangement. The space
$C_{n}(\mathfrak{g})\subseteq\mathfrak{g}^{n}$ can be written, tautologically,
as a union of lines in $\mathfrak{g}^{n}$. However, unless $\mathfrak{g}$ is
abelian, it cannot be written as a union of _finitely_ many linear subspaces
of $\mathfrak{g}^{n}$. For example,
$C_{2}(\mathfrak{su}_{2})\subseteq\mathbb{R}^{6}$ is the determinantal variety
of real $2\times 3$ matrices of rank less than two. Elementary linear algebra
shows that a linear subspace $V\subseteq\mathbb{R}^{6}$ which is contained in
$C_{2}(\mathfrak{su}_{2})$ can have dimension at most three. However, with the
singular point deleted, $C_{2}(\mathfrak{su}_{2})$ is a $4$-dimensional
manifold, namely the complement of the zero section in a rank two vector
bundle over $\mathbb{R}P^{2}$ (see Example 4.3). More generally, since every
commuting $n$-tuple in $\mathfrak{g}$ lies in a Cartan subalgebra of
$\mathfrak{g}$, an argument like the one in [16, Corollary 2.5] shows that
$C_{n}(\mathfrak{g})$ is an irreducible real algebraic variety, and thus it
cannot be written as a union of finitely many proper linear subspaces of
$\mathfrak{g}^{n}$.

###  A.2. Mod-$p$ cohomology of $C_{n}(\mathfrak{u}_{p})^{+}$ for $n$ even

Let $p$ be a prime. To lighten notation we will omit the coefficient ring and
all cohomology groups in this section are understood with
$\mathbb{F}_{p}$-coefficients. For a space $X$ with mod-$p$ cohomology of
finite type, define the mod-$p$ Poincaré series by

$\Pi_{X}^{p}(t)=\sum_{k\geq 0}\dim_{\mathbb{F}_{p}}H^{k}(X)\,t^{k}\,.$

The objective of this section is to prove the following theorem.

######  Theorem A.11.

For any prime $p$

$\Pi_{C_{n}(\mathfrak{u}_{p})^{+}}^{p}(t)=\begin{cases}1-t^{2p+1}+(t^{2p}+t^{2p+1})\prod_{k=1}^{p-1}(1+t^{2k-1})&\textnormal{if
}n=2\,,\\\\[4.30554pt]
1+t^{np}+(t^{n+1}+t^{np})(t^{2p-3}+t^{2p-2})\prod_{k=1}^{p-2}(1+t^{2k-1})&\textnormal{if
}n>2\textnormal{ and even.}\end{cases}$

Our computation is based on a recent computation due to Guerra and Jana [9] of
the mod-$p$ cohomology of the unordered complete flag manifold in
$\mathbb{C}^{p}$ which we restate here for convenience. Let $T\subseteq U(p)$
be a maximal torus and $N$ its normaliser.

######  Theorem A.12 ([9, Theorem 6.8]).

If $p$ is an odd prime, there is an isomorphism

$H^{\ast}(U(p)/N;\mathbb{F}_{p})\cong\mathbb{F}_{p}[\alpha_{S},\gamma_{S}\,|\,S\subset\\{1,\dots,p-2\\}]/I\,,$

where
$I=(\alpha_{S}^{2},\gamma_{S}^{2},\alpha_{S}\gamma_{S}\,|\,S\subset\\{1,\dots,p-2\\})$,
and the degrees of the generators are

$|\alpha_{S}|=2p-3+\sum_{i\in S}(2i-1)\,,\quad|\gamma_{S}|=2p-2+\sum_{i\in
S}(2i-1)\,.$

It follows that the mod-$p$ Poincaré series of $U(p)/N$ is (see [9, Corollary
6.9])

$\Pi^{p}_{U(p)/N}(t)=1+(t^{2p-3}+t^{2p-2})\prod_{k=1}^{p-2}(1+t^{2k-1})\,.$

This formula remains valid when $p=2$ in which case
$U(2)/N\cong\mathbb{R}P^{2}$.

To compute the cohomology of $C_{n}(\mathfrak{u}_{p})^{+}$ we first recall
from Lemma A.5 and Proposition A.9 that

$C_{n}(\mathfrak{u}_{p})^{+}\cong\Sigma^{n}C_{n}(\mathfrak{su}_{p})^{+}\cong\Sigma^{n}C_{n}^{1}(\mathfrak{su}_{p})^{\lozenge}\,,$

so it will suffice to compute the cohomology of
$C_{n}^{1}(\mathfrak{su}_{p})$. This will be achieved by an argument similar
to the one used in the proof of Lemma A.3.

Let $S\subseteq SU(p)$ be a maximal torus with Lie algebra
$\mathfrak{s}\subseteq\mathfrak{su}_{p}$, and write
$S(\mathfrak{s}^{n})\subseteq\mathfrak{s}^{n}$ for the unit sphere.

######  Lemma A.13.

For every $n\geq 1$ the map defined in (9)

$\phi^{\prime}\colon\thinspace
SU(p)/S\times_{\Sigma_{p}}S(\mathfrak{s}^{n})\to C_{n}^{1}(\mathfrak{su}_{p})$

induces an isomorphism on mod-$p$ cohomology.

###### Proof.

This is an application of the Vietoris-Begle mapping theorem in the form
stated, _e.g._ , in [3, Theorem 2.1]. Since $\phi^{\prime}$ is induced by the
restriction of an action map of a compact group, it is closed. As every
$(X_{1},\dots,X_{n})\in C_{n}^{1}(\mathfrak{su}_{p})$ is contained in a Cartan
subalgebra of $\mathfrak{su}_{p}$, the map $\phi^{\prime}$ is surjective. Then
we need only show that $\phi^{\prime}$ has $\mathbb{F}_{p}$-acyclic fibres.

Let $\underline{X}=(X_{1},\dots,X_{n})\in C_{n}^{1}(\mathfrak{su}_{p})$.
Without loss of generality we may assume that $\underline{X}\in
S(\mathfrak{s}^{n})$ and that $X_{1}\neq 0$. It is easy to check (see [3,
Lemma 3.2]) that

$(\phi^{\prime})^{-1}(\underline{X})\cong
SU(p)_{\underline{X}}^{0}/N_{SU(p)_{\underline{X}}^{0}}(S)\,,$

where $SU(p)_{\underline{X}}^{0}$ is the connected component of the identity
of the isotropy group of $\underline{X}$ (in fact, these isotropy groups are
connected, but we shall not need this here). The Weyl group of
$SU(p)_{\underline{X}}^{0}$ is contained in the isotropy group of
$X_{1}\in\mathfrak{s}$ for the action of $\Sigma_{p}$ on $\mathfrak{s}$. As
$X_{1}\neq 0$, this isotropy group is a proper Young subgroup of $\Sigma_{p}$
and thus has order prime to $p$. It follows that
$(\phi^{\prime})^{-1}(\underline{X})$ is $\mathbb{F}_{p}$-acyclic, by the
usual argument that the cohomology of the quotient of a compact connected Lie
group by the normaliser of a maximal torus is $k$-acyclic for any field $k$
whose characteristic is prime to the order of the Weyl group. ∎

Now we consider the fibre bundle

(10) $S(\mathfrak{s}^{n})\to
SU(p)/S\times_{\Sigma_{p}}S(\mathfrak{s}^{n})\to(SU(p)/S)/\Sigma_{p}\,,$

whose fibre is a sphere of dimension $n(p-1)-1$ and whose base may be
identified with $U(p)/N$. When $n$ is even, the bundle is orientable. In this
case, the mod-$p$ cohomology of the total space, and hence that of
$C_{n}^{1}(\mathfrak{su}_{p})$, can be computed using the Gysin exact sequence

$\cdots\to H^{k}(U(p)/N)\xrightarrow{\cdot\chi_{n}}H^{k+n(p-1)}(U(p)/N)\to
H^{k+n(p-1)}(E)\to H^{k+1}(U(p)/N)\xrightarrow{\cdot\chi_{n}}\cdots\,.$

Here $E=SU(p)/S\times_{\Sigma_{p}}S(\mathfrak{s}^{n})$ is the total space and
$\chi_{n}\in H^{n(p-1)}(U(p)/N)$ the mod-$p$ Euler class of (10).

Next, we determine $\chi_{n}$. Let $C_{p}\subseteq\Sigma_{p}$ be a Sylow-$p$
subgroup and let $x\in H^{2}(C_{p})\cong\mathbb{F}_{p}$ be a generator. The
inclusion induces an isomorphism

$H^{2p-2}(\Sigma_{p})\cong H^{2p-2}(C_{p})$

by which we may view $x^{p-1}$ as a generator of $H^{2p-2}(\Sigma_{p})$. Since
$\Sigma_{p}$ is the fundamental group of $U(p)/N$, $x^{p-1}$ determines an
element in $H^{2p-2}(U(p)/N)$. Inspection of the proof of Theorem A.12 shows
that this element is, up to a unit, the generator $\gamma_{\emptyset}$ (see
[9, p. 4] where $\gamma_{\emptyset}$ is referred to as $\gamma_{1,1}$).

######  Lemma A.14.

For the mod-$p$ Euler class of (10) we have

$\chi_{n}=\begin{cases}u\gamma_{\emptyset}&\textnormal{if }n=2,\\\
0&\textnormal{if }n>2\end{cases}$

where $u\in\mathbb{F}_{p}^{\times}$.

###### Proof.

From the viewpoint of representation theory $\mathfrak{s}$ is the
$(p-1)$-dimensional real standard representation of $\Sigma_{p}$. Since $n$ is
even, we have that

$\mathfrak{s}^{n}\cong(\mathfrak{s}\otimes\mathbb{C})^{n/2}$

as $\Sigma_{p}$-representations. Now $\chi_{n}$ is the Euler class of the
vector bundle over $U(p)/N$ defined by the representation $\mathfrak{s}^{n}$,
thus

$\chi_{n}=\chi_{2}^{n/2}\,.$

If $n>2$, then this implies $\chi_{n}=0$, since the cup product structure on
the mod-$p$ cohomology of $U(p)/N$ is trivial by Theorem A.12.

To show that $\chi_{2}$ equals $\gamma_{\emptyset}$ up to a unit, it is enough
to show, by the discussion preceding the lemma, that the Euler class in
$H^{2p-2}(C_{p})$ associated with the restriction of
$\mathfrak{s}\otimes\mathbb{C}$ to a $C_{p}$-representation is $ux^{p-1}$ for
some $u\in\mathbb{F}_{p}^{\times}$. But it is well-known that this restriction
is the direct sum of $p-1$ non-trivial one-dimensional
$C_{p}$-representations, each of which has Euler class a non-zero multiple of
the generator $x\in H^{2}(C_{p})$. ∎

###### Proof of Theorem A.11.

We consider the Gysin sequence displayed above. If $n>2$, then $\chi_{n}=0$ by
Lemma A.14, and we deduce for all $k\geq 0$

$\tilde{H}^{k}(E)\cong\tilde{H}^{k}(U(p)/N)\oplus H^{k-n(p-1)+1}(U(p)/N)\,.$

Thus, by Lemma A.13 and the remarks preceding it, we obtain for the reduced
Poincaré polynomial
$\tilde{\Pi}_{C_{n}(\mathfrak{u}_{p})^{+}}^{p}(t):=\Pi_{C_{n}(\mathfrak{u}_{p})^{+}}^{p}(t)-1$

$\tilde{\Pi}_{C_{n}(\mathfrak{u}_{p})^{+}}^{p}(t)=t^{n+1}\,\tilde{\Pi}^{p}_{E}(t)=t^{n+1}\,\tilde{\Pi}^{p}_{U(p)/N}(t)+t^{np}\,\Pi_{U(p)/N}^{p}(t)\,.$

Plugging in the Poincaré series for $U(p)/N$ yields the desired result.

If $n=2$, then by Lemma A.14 the map
$H^{k}(U(p)/N)\xrightarrow{\cdot\chi_{2}}H^{k+2p-2}(U(p)/N)$ in the Gysin
sequence is injective for $k=0$, but trivial for all $k>0$, because the cup
product structure of $U(p)/N$ is trivial. Thus, we have

$\tilde{\Pi}_{C_{2}(\mathfrak{u}_{p})^{+}}^{p}(t)=t^{3}\,\tilde{\Pi}^{p}_{E}(t)=t^{3}\,(\tilde{\Pi}^{p}_{U(p)/N}(t)-t^{2p-2}+t^{2p-3}\,\tilde{\Pi}^{p}_{U(p)/N}(t))\,,$

and plugging in the Poincaré series for $U(p)/N$ finishes the proof. ∎

######  Remark A.15.

The mod-$p$ cohomology of $C_{n}(\mathfrak{u}_{p})^{+}$ for odd $n$ is not
computed as easily; in this case there is no Euler class and the differentials
in the Serre spectral sequence associated with (10) must be determined by
other means. These differentials are non-trivial as can be seen in the
simplest case $n=1$.

## References

  * [1] A. Adem and F. R. Cohen. ”Commuting elements and spaces of homomorphisms.” Math. Ann. 338 (2007), no. 3, 587–626.
  * [2] A. Adem, F. R. Cohen and J.M Gómez. ”Stable splittings, spaces of representations and almost commuting elements in Lie Groups.” Math. Proc. Cambridge Philos. Soc. 149 (2010), no. 3, 455–490.
  * [3] T. J. Baird. ”Cohomology of the space of commuting $n$-tuples in a compact Lie group.” Algebr. Geom. Topol. 7 (2007), 737–754.
  * [4] T. J. Baird, L. C. Jeffrey, and P. Selick. ”The space of commuting $n$-tuples in $SU(2)$.” Illinois Journal of Mathematics 55 (2011), no. 3, 805–813.
  * [5] G. Bredon. ”Introduction to compact transformation groups.” Pure Appl. Math., Vol. 46. Academic Press, New York-London, 1972. xiii+459 pp.
  * [6] M. C. Crabb. ”Spaces of commuting elements in $SU(2)$.” Proc. Edinb. Math. Soc. (2) 54 (2011), no. 1, 67–75.
  * [7] M. C. Crabb. ”On the stable splitting of $U(n)$ and $\Omega U(n)$.” Algebraic topology, Barcelona, 1986, 35–53. Lecture Notes in Math., 1298. Springer-Verlag, Berlin, 1987.
  * [8] M. C. Crabb, J. R. Hubbuck and J. A. W. McCall. ”Stable summands of $U(n)$.” Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 5, 963–973.
  * [9] L. Guerra and S. Jana. ”Cohomology of complete unordered flag manifolds.” arXiv:2309.00429.
  * [10] S. Illman. ”The equivariant triangulation theorem for actions of compact Lie groups.” Mathematische Annalen 262 (1983), 487–501.
  * [11] G. I. Lehrer and D. E. Taylor. ”Unitary reflection groups.”, Austral. Math. Soc. Lect. Ser., 20 Cambridge University Press, Cambridge, 2009. viii+294 pp.
  * [12] S. Lojasiewicz. ”Triangulation of semi-analytic sets.” Annali della Scuola Normale Superiore di Pisa 18 (1964), no. 4, 449–474.
  * [13] H. Miller. ”Stable splittings of Stiefel manifolds.” Topology 24 (1985), no. 4, 411–419.
  * [14] M. Nakaoka. ”Cohomology mod $p$ of symmetric products of spheres II.” J. Inst. Polytech. Osaka City Univ. Ser. A 10 (1959), 67–89.
  * [15] D. H. Park and D. Y. Suh. ”Semialgebraic $G$-CW-complex structure of semialgebraic $G$-spaces.” J. Korean Math. Soc. 35 (1998), no. 2, pp. 371–386.
  * [16] R. W. Richardson. ”Commuting varieties of semisimple Lie algebras and algebraic groups.” Compositio Mathematica 38 (1979), no. 3, 311–327.
  * [17] G. H. Rojo. ”On the space of commuting orthogonal matrices.” Journal of Group Theory 17 (2014), no. 2, 291–316.
  * [18] D. Sjerve, and E. Torres-Giese. ”Fundamental groups of commuting elements in Lie groups.” Bulletin of the London Mathematical Society 40 (2008), no. 1, 65–76.
  * [19] N. E. Steenrod. ”Cohomology operations, and obstructions to extending continuous functions.” Advances in Math. 8 (1972), 371–416.