On the moduli spaces of commuting elements in the projective unitary groups

TL;DR

This paper describes the moduli spaces of commuting elements in projective unitary groups and applies these results to classify flat principal $PU(m)$-bundles over certain CW-complexes, advancing understanding of their topological structure.

Contribution

It provides explicit descriptions of the moduli spaces of commuting elements in $PU(m)$ for finitely generated abelian groups and applies these to classify flat bundles over CW-complexes with abelian fundamental groups.

Findings

Explicit descriptions of moduli spaces for finitely generated abelian groups.

Injectivity of the natural map for CW-complexes with abelian fundamental groups.

Complete enumeration of flat principal $PU(m)$-bundles over such spaces.

Abstract

We provide descriptions for the moduli spaces $\text{Rep}(\Gamma, PU(m))$, where $\Gamma$ is any finitely generated abelian group and $PU(m)$ is the group of $m\times m$ projective unitary matrices. As an application we show that for any connected CW-complex $X$ with $\pi_1(X)\cong \mathbf{Z}^n$, the natural map $\pi_0(\text{Rep}(\pi_1(X), PU(m)))\to [X, BPU(m)]$ is injective, hence providing a complete enumeration of the isomorphism classes of flat principal $PU(m)$-bundles over $X$.

Equations

$$H^2(BPU(m),\mathbb{Z}/m\mathbb{Z})\cong \mathbb{Z}/m\mathbb{Z}$$

$$1\to \mathbb{Z}/m\mathbb{Z}\to SU(m)\to PU(m)\to 1.$$

$$Rep(\mathbb{Z}^{n},PU(m))\cong\coprod\limits_{\begin{subarray}{c}D\in T(n,\mathbb{Z}/m\mathbb{Z})\\ \sigma(D)\big{|}m\end{subarray}}\widehat{SP^{m\over\sigma(D)}}(\mathbb{T}^{n}/R(D))$$

$${\Psi }_{{\mathbb{T}}^n}^{PU(m)}:{\pi }_0(Rep({\mathbb{Z}}^n,PU(m)))\to [{\mathbb{T}}^n,BPU(m)].$$

$${\Psi }_X^{PU(m)}:{\pi }_0(Rep({\pi }_1(X),PU(m)))\to [X,BPU(m)]$$

$${\Psi }_{{\mathbb{T}}^n}^{PU(m)}:{\pi }_0(Rep({\mathbb{Z}}^n,PU(m)))\to [{\mathbb{T}}^n,BPU(m)]$$

$$\rho :B_n(U(m))\to T(n,\mathbb{Z}/m\mathbb{Z})$$

$$B_{n}(U(m))=\coprod\limits_{\begin{subarray}{c}D\in T(n,\mathbb{Z}/m\mathbb{Z})\\ \sigma(D)\big{|}m\end{subarray}}B_{n}((U(m))_{D}.$$

$$d_{ij}=\{\begin{array}{ll}{d}_k& \text{mbox}if(i,j)=(k+t,k),1\le k\le t;\\ -d_k& \text{mbox}if(i,j)=(k,k+t),1\le k\le t;\\ 0& \text{mbox}otherwise.\end{array}$$

$$c_j=\frac{1}{2\pi \sqrt{-1}}\log {\lambda }_j\in \mathbb{R}/⟨\frac{1}{r_j(D)}⟩\cong {\mathbb{S}}^1/R_j(D).$$

$$B_n(U(m){)}_D/U(m)\cong {\mathbb{T}}^n/R(D)$$

$$B_n(U(m){)}_D\to B_n(U(m){)}_D/U(m)\cong {\mathbb{T}}^n/R(D)\to \prod _{j=1}^n{\mathbb{S}}^1/R_j(D)$$

$${\mathbb{T}}^n/R(D^{\prime })\stackrel{{\bar{\varphi }}^{-1}}{\cong }{\mathbb{T}}^n/R(D)\stackrel{\bar{f}}{\to }{B}_n(U(m){)}_D/U(m)\stackrel{\bar{g}}{\cong }{B}_n(U(m){)}_{D^{\prime }}/U(m)$$

$$B_n(U(m){)}_D/U(m)\cong (B_n(\sigma (D){)}_D/U(\sigma (D)){)}^l/{\Sigma }_l=S{P}^l(B_n(\sigma (D){)}_D/U(\sigma (D))).$$

$$B_{n}(U(m))/U(m)\cong\coprod\limits_{\begin{subarray}{c}D\in T(n,\mathbb{Z}/m\mathbb{Z})\\ \sigma(D)\big{|}m\end{subarray}}SP^{m\over\sigma(D)}(\mathbb{T}^{n}/R(D))$$

$$B_n(U(m))\to Hom({\mathbb{Z}}^n,PU(m))$$

$${\mathbb{T}}^n\backslash B_n(U(m))/U(m)\cong Rep({\mathbb{Z}}^n,PU(m)).$$

$$Rep(\mathbb{Z}^{n},PU(m))\cong\coprod\limits_{\begin{subarray}{c}D\in T(n,\mathbb{Z}/m\mathbb{Z})\\ \sigma(D)\big{|}m\end{subarray}}\widehat{SP^{m\over\sigma(D)}}(\mathbb{T}^{n}/R(D))$$

$$Rep({\mathbb{Z}}^2,PU(m))\cong \widehat{S{P}^m}({\mathbb{T}}^2)⨆\coprod _{\begin{array}{c}D\in \mathbb{Z}/m\mathbb{Z}\\ 1<|D|<m\end{array}}\widehat{S{P}^{\frac{m}{|D|}}}({\mathbb{T}}^2/R(D))⨆\{x_1,\dots ,x_{\Phi (m)}\}$$

$$N(n,p^2)=1+\frac{(p^{n-1}-1)(p^n-1)(p^{2n+1}-p^n-p^{n-1}+p^4+p^2-1)}{(p^2-1)(p^4-1)}$$

$$u(p)=\frac{(p^{n-1}-1)(p^n-1)}{p^2-1}$$

$$Rep({\mathbb{Z}}^n,PU(p^2))\cong \widehat{S{P}^{p^2}}({\mathbb{T}}^n)⨆\coprod _{\begin{array}{c}\frac{(p^{n-1}-1)(p^n-1)}{p^2-1}\end{array}}\widehat{S{P}^p}({\mathbb{T}}^n/R(D))⨆\{x_1,\dots ,x_{N(n,p^2)-u(p)-1}\}$$

$$\Gamma =\mathbb{Z}/k_1\oplus \dots \oplus \mathbb{Z}/k_s\oplus {\mathbb{Z}}^n\cong \mathrm{Tor}(\Gamma )\oplus {\mathbb{Z}}^n.$$

$$B(\Gamma ,U(m))=\{(A_1,\dots ,A_{s+n})\in B_{s+n}(U(m)):A_i^{k_i}=I_m\text{ for }1\le i\le s\}.$$

$$\{({\theta }_1,\dots ,{\theta }_{s+n}):{\theta }_i^{k_i}=1\text{ for }1\le i\le s\}\cong \mathrm{Tor}(\Gamma )\times {\mathbb{T}}^n\subset {\mathbb{T}}^{s+n}.$$

$$B(\Gamma ,U(m))=\coprod B(\Gamma ,U(m){)}_D$$

$$Hom(\Gamma ,PU(m){)}_D\subset Hom(\Gamma ,PU(m))\text{ and }Rep(\Gamma ,PU(m){)}_D\subset Rep(\Gamma ,PU(m))$$

$$B(\Gamma,U(m))/U(m)\cong\coprod\limits_{\begin{subarray}{c}D\in T(s+n,\mathbb{Z}/m\mathbb{Z})\\ \sigma(D)\big{|}m~{}{\rm and}~{}r_{i}(D)\big{|}k_{i}\end{subarray}}SP^{m\over\sigma(D)}((\operatorname{Tor}(\Gamma)\times\mathbb{T}^{n})/R(D))$$

$$Rep(\Gamma,PU(m))\cong\coprod\limits_{\begin{subarray}{c}D\in T(s+n,\mathbb{Z}/m\mathbb{Z})\\ \sigma(D)\big{|}m~{}{\rm and}~{}r_{i}(D)\big{|}k_{i}\end{subarray}}\widehat{SP^{m\over\sigma(D)}}((\operatorname{Tor}(\Gamma)\times\mathbb{T}^{n})/R(D))$$

$$B(\Gamma ,U(m))\to Hom(\Gamma ,PU(m)),$$

Full text

1. Introduction

The space of ordered commuting $n$–tuples in a compact, connected Lie group $G$ is by definition the subspace $\text{Hom}(\mathbb{Z}^{n},G)\subset G^{n}$ (see [4] for background and basic properties). Its orbit space under conjugation, denoted $Rep(\mathbb{Z}^{n},G)$, can be identified with the moduli space of isomorphism classes of flat connections on principal $G$–bundles over the $n$–dimensional torus $\mathbb{T}^{n}$. In the case when all of the maximal abelian subgroups of $G$ are path connected, it can be shown that this moduli space has a single connected component, corresponding to the identity element $(1,\dots,1)\in G^{n}$ (see [4], Proposition 2.3). For example, $Rep(\mathbb{Z}^{n},U(m))\cong SP^{m}(\mathbb{T}^{n})$, the $m$-fold symmetric product of the $n$–torus. However by a result due to Borel (see [6], page 216), for any prime number $p$ the fundamental group of $G$ has $p$–torsion if and only if there exists a rank two elementary abelian $p$–subgroup of $G$ which is not a subgroup of any torus. In this case $Rep(\mathbb{Z}^{n},G)$ fails to be path–connected for all $n\geq 2$ and determining the number and exact structure of the components can be fairly complicated. Borel also shows that $H^{*}(G,\mathbb{Z})$ has $p$–torsion if and only if there exists a subgroup of the form $(\mathbb{Z}/p\mathbb{Z})^{3}\subset G$ which is not contained in any torus. This can be used to show for example that $Rep(\mathbb{Z}^{3},Spin(7))$ is not path connected, even though $Spin(7)$ is simply connected.

In this note we consider the case when $G=PU(m)$, the group of $m\times m$ projective unitary matrices, which has fundamental group isomorphic to $\mathbb{Z}/m\mathbb{Z}$ and

[TABLE]

has a canonical generator $\nu$ of order $m$, corresponding to the central extension

[TABLE]

Given a homomorphism $h:\mathbb{Z}^{n}\to PU(m)$, one can associate to it the cohomology class $\alpha=h^{*}(\nu)\in H^{2}(\mathbb{Z}^{n},\mathbb{Z}/m\mathbb{Z})$. For our purposes it’s convenient to identify $H^{2}(\mathbb{Z}^{n},\mathbb{Z}/m\mathbb{Z})$ with the set $T(n,\mathbb{Z}/m\mathbb{Z})$ of $n\times n$ skew–symmetric matrices over $\mathbb{Z}/m\mathbb{Z}$: given a basis $z_{1},\dots,z_{n}$ for $\mathbb{Z}^{n}$, then $D=(d_{ij})\in T(n,\mathbb{Z}/m\mathbb{Z})$ corresponds to $\sum_{i<j}d_{ij}z_{i}^{*}z_{j}^{*}$.

Now given a skew–symmetric $n\times n$ matrix $D$ over $\mathbb{Z}/m\mathbb{Z}$ representing $\alpha$, we define $\sigma(\alpha)=\sigma(D)=\sqrt{|R(D)|}$, where $R(D)\subset(\mathbb{Z}/m\mathbb{Z})^{n}$ is the row space of $D$ (see [1], Definition 2). Alternatively, for $\alpha\in H^{2}(\mathbb{Z}^{n},\mathbb{Z}/m\mathbb{Z})$, by [8], Proposition 4.1 we can find a basis $g_{1},\dots,g_{n}$ of $\mathbb{Z}^{n}$ such that $\alpha=c_{1}g_{1}^{*}g_{2}^{*}+\dots+c_{r}g_{2r-1}^{*}g_{2r}^{*}$ where $2r\leq n$, $c_{1},\dots,c_{r}\in\mathbb{Z}/m\mathbb{Z}$ and |c_{r}|~{}\Big{|}~{}|c_{r-1}|~{}\Big{|}~{}\dots~{}\Big{|}~{}|c_{1}|. Using this basis it follows that $\sigma(\alpha)=\prod_{i=1}^{r}|c_{i}|$.

For a topological group $G$, let $\widehat{SP^{n}}(G)$ denote the reduced $n$-fold symmetric product of $G$, defined as the quotient $SP^{n}(G)/G$, where $G$ acts by translation on each unordered coordinate. Then our main result can be stated as follows:

Theorem A**.**

For all $m,n\geq 1$ there are homeomorphisms

[TABLE]

Note that $\mathbb{T}^{n}/R(D)\cong\mathbb{T}^{n}$, so our formula expresses the moduli space as a disjoint union of reduced symmetric products of the $n$–torus. The number of path-connected components of $Rep(\mathbb{Z}^{n},PU(m))$ is equal to $N(n,m)=|\{D\in T(n,\mathbb{Z}/m\mathbb{Z})~{}|~{}\sigma(D)\text{ divides }m\}|$, a rather intricate number that has been computed in Adem-Cheng (see [1], Corollary 3.9). We also show how to apply our methods to provide a description of $Rep(\Gamma,PU(m))$ for any finitely generated abelian group $\Gamma$ (see Theorem 2.11). Note that by the results in [5], for these groups there is a homotopy equivalence $Rep(\Gamma,PGL(m,\mathbb{C}))\simeq Rep(\Gamma,PU(m))$.

Recall that for $G$ a topological group and $X$ a CW–complex, the homotopy classes of maps from $X$ to $BG$, denoted $[X,BG]$, classify isomorphism classes of principal $G$ bundles over $X$. Representations play a key in this through the theory of flat bundles; in our setting the key connection is via the induced map on components

[TABLE]

Taking composition with the classifying map $c_{X}:X\to B\pi_{1}(X)$, we obtain the following general result

Theorem B**.**

Let $X$ denote a connected CW–complex with $\pi_{1}(X)=\mathbb{Z}^{n}$; then the map

[TABLE]

is injective for all $n,m\geq 1$ and so there are $N(n,m)$ distinct isomorphism classes of flat $PU(m)$–bundles on $X$.

Regarding surjectivity, we obtain that for all $m\geq 2$,

[TABLE]

is surjective if and only if $n\leq 3$ (Proposition 3.2). It follows that there exists a principal $PU(m)$-bundle on the $n$–torus $\mathbb{T}^{n}$ which does not admit a flat structure if and only if $n\geq 4$. Here we apply the results in [9], which provide a classification of principal $PU(m)$–bundles over low–dimensional complexes.

The results in this paper can be viewed as an application and refinement of the analysis carried out in our previous work [1], where we described the space $B_{n}(U(m))$ of almost commuting $n$–tuples in $U(m)$.

2. Projective Representations and Almost Commuting Elements

In this section we will apply the methods from [1] to give a description of $Rep(\mathbb{Z}^{n},PU(m))$. The almost commuting elements will play a crucial role.

Definition 2.1**.**

We define $B_{n}(G)$, the almost commuting elements in a Lie group $G$, as the set of all ordered $n$-tuples $(A_{1},A_{2},\ldots,A_{n})\in G^{n}$ such that the commutators $[A_{i},A_{j}]\in Z(G)$, the centre of $G$, for all $1\leq i,j\leq n$.

Let $F_{n}$ denote the free group on $n$ generators $a_{1},\ldots a_{n}$. We will identify a map from $F_{n}$ to $G$ with the $n$–tuple of images of these generators in $G$. Suppose that $f:F_{n}\to U(m)$ is in $B_{n}(U(m))$. For any $u,v\in F_{n}$, $[f(u),f(v)]=\gamma I_{m}$ for some $m$-th root of unity $\gamma$, as the determinant of a commutator of invertible matrices is equal to one. The exponential function $z\mapsto e^{2\pi\sqrt{-1}z}$ establishes a group isomorphism between $\mathbb{R}/\mathbb{Z}$ and $\mathbb{S}^{1}\subset\mathbb{C}$ with inverse $w\mapsto\frac{1}{2\pi\sqrt{-1}}\log w$. The multiplicative groups of $m$-th roots of unity and all roots of unity correspond to the subgroup $\mathbb{Z}[\frac{1}{m}]/\mathbb{Z}\cong\mathbb{Z}/m\mathbb{Z}$ and $\mathbb{Q}/\mathbb{Z}$ of $\mathbb{R}/\mathbb{Z}$ respectively under this isomorphism. We will be using this identification from now on. Hence there is a map $F_{n}\times F_{n}\to\mathbb{Z}/m\mathbb{Z}\subset\mathbb{Q}/\mathbb{Z}$ defined by $(u,v)\mapsto\frac{1}{2\pi\sqrt{-1}}\log\gamma$. Since $f([F_{n},F_{n}])\subset Z(U(m))$, the map factors through the abelianization of $F_{n}\times F_{n}$ and thus gives rise to a $\mathbb{Z}/m\mathbb{Z}$-valued skew-symmetric bilinear form $\omega_{f}:\mathbb{Z}^{n}\times\mathbb{Z}^{n}\to\mathbb{Z}/m\mathbb{Z}$.

Define a map

[TABLE]

by $\rho(f)=(d_{ij})$, where $[f(a_{i}),f(a_{j})]=e^{2d_{ij}\pi\sqrt{-1}}I_{m}$. For $D\in T(n,\mathbb{Z}/m\mathbb{Z})$, let $B_{n}(U(m))_{D}=\rho^{-1}(D)$. For $f\in B_{n}(U(m))_{D}$, the ordered $n$-tuple $(f(a_{1}),\ldots,f(a_{n}))$ is said to be $D$-commuting. Note that $\rho(f)$ is the skew-symmetric matrix associated to the bilinear form $\omega_{f}$.

Definition 2.2**.**

For any $n\times n$ matrix $D\in M_{n\times n}(\mathbb{Z}/m\mathbb{Z})$, the row space $R(D)$ is the sub-module of $(\mathbb{Z}/m\mathbb{Z})^{n}$ generated by the rows of $D$ over $\mathbb{Z}$. Let $R_{i}(D)\subset\mathbb{Z}/m\mathbb{Z}$ be the image of $R(D)$ under the projection onto the $i$-th factor. Let $r_{i}(D)=|R_{i}(D)|$ for $i=1,\dots,n$ and $\sigma(D)=\sqrt{|R(D)|}$.

We recall the structure of the almost commuting $n$-tuples in $U(m)$, established in [1], Corollary 3.6.

Proposition 2.3**.**

For $D\in T(n,\mathbb{Z}/m\mathbb{Z})$, the space $B_{n}(U(m))_{D}$ is non-empty and path connected if $\sigma(D)$ divides $m$, and is empty otherwise. The space $B_{n}(U(m))$ can be expressed as a disjoint union of path connected components

[TABLE]

We begin our analysis by focusing on the basic case when $m=\sigma(D)$. Let $0\leq t\leq n/2$ and $d_{1},d_{2},\ldots,d_{t}\neq 0\in\mathbb{Z}/m\mathbb{Z}$. Define $D_{n}(d_{1},d_{2},\ldots d_{t})=(d_{ij})\in T(n,\mathbb{Z}/m\mathbb{Z})$ be the skew-symmetric matrix with

[TABLE]

Lemma 2.4**.**

Suppose $D\in T(n,\mathbb{Z}/m\mathbb{Z})$, $m=\sigma(D)$. Let $(A_{1},\ldots A_{n})\in B_{n}(U(m))_{D}$. If $B\in U(m)$ commutes with $A_{i}$ for all $i=1,\ldots,n$, then $B$ is a scalar matrix.

Proof.

By [8, Proposition 4.1], there exists $Q=(q_{ij})\in GL(n,\mathbb{Z})$ such that $Q^{T}DQ=D^{\prime}=D_{n}(d_{1},\ldots,d_{t})$. Define $D^{\prime\prime}=D_{n+1}(d_{1},\ldots,d_{t})$ and $A_{j}^{\prime}=A_{1}^{q_{1j}}A_{2}^{q_{2j}}\ldots A_{n}^{q_{nj}}$ for $j=1,\ldots,n$. Then $\sigma(D^{\prime\prime})=m$ and $(A_{1}^{\prime},\ldots A_{n}^{\prime},B)\in B_{n+1}(U(m))_{D^{\prime\prime}}$. By [1, theorem 3.3], there exists an orthonormal basis of $\mathbb{C}^{m}$ consisting of eigenvectors of $B$ corresponding to a common eigenvalue. Hence, $B$ is a scalar matrix. ∎

Lemma 2.5**.**

Suppose $D\in T(n,\mathbb{Z}/m\mathbb{Z})$ and $m=\sigma(D)$. Let $(A_{1},\ldots A_{n})\in B_{n}(U(m))_{D}$ and $\omega=e^{2\pi\sqrt{-1}/r_{j}(D)}$. If $\lambda$ is an eigenvalue of $A_{j}$, then $\omega^{q}\lambda$, where $q=0,\ldots,r_{j}(D)-1$, are all the distinct eigenvalues of $A_{j}$.

Proof.

Let $D=(d_{ij})$. There exists $a_{1},\ldots,a_{n}\in\mathbb{Z}$ such that $a_{1}d_{1j}+\ldots+a_{n}d_{nj}=\left[-\frac{1}{r_{j}(D)}\right]$. Let $B=A_{1}^{a_{1}}\ldots A_{n}^{a_{n}}$. Then $BA_{j}=\omega^{-1}A_{j}B$. If $v$ is an eigenvector of $A_{j}$ corresponding to eigenvalue $\lambda$, then $A_{j}Bv=\omega BA_{j}v=\omega B(\lambda v)=\omega\lambda Bv$ and so $\omega\lambda$ is also an eigenvalue of $A_{j}$. Inductively $\omega^{q}\lambda$ is also an eigenvalue of $A_{j}$ for any $q$. On the other hand, $A_{j}^{r_{j}(D)}$ commutes with $A_{1},\ldots,A_{n}$. By lemma 2.4, $A_{j}^{r_{j}(D)}$ is a scalar matrix. Since $\lambda$ is an eigenvalue of $A_{j}$, $A_{j}^{r_{j}(D)}=\lambda^{r_{j}(D)}I_{m}$ and so any eigenvalue of $A_{j}$ is of the form $\omega^{q}\lambda$. ∎

In the situation of lemma 2.5, take an eigenvalue $\lambda_{j}$ of $A_{j}$. Define

[TABLE]

This element $c_{j}$ is independent of the choice of $\lambda_{j}$.

Theorem 2.6**.**

Let $m=\sigma(D)$. There exists a $\mathbb{T}^{n}$-equivariant homeomorphism

[TABLE]

such that its composition with the quotient map

[TABLE]

sends $(A_{1},\ldots A_{n})$ to $(c_{1},\ldots c_{n}).$

Proof.

If $D$ is of the special form $D=D_{n}(d_{1},\ldots,d_{t})$, the theorem follows easily from the proof of theorem 3.4 in [1]. In the general case, let $(C_{1},\ldots C_{n})\in B_{n}(U(m))_{D}$. By multiplying by scalars if necessary, we assume that 1 is an eigenvalue of each $C_{i}$. The map $f:\mathbb{T}^{n}\to B_{n}(U(m))_{D}$ with $f(\theta_{1},\ldots,\theta_{n})=(\theta_{1}C_{1},\ldots,\theta_{n}C_{n})$ is $\mathbb{T}^{n}$-equivariant. Also, there exists $Q=(q_{ij})\in GL(n,\mathbb{Z})$ such that $Q^{T}DQ=D^{\prime}=D_{n}(d_{1},\ldots,d_{t})$. Define $\phi:\mathbb{T}^{n}\to\mathbb{T}^{n}$ by $\phi(\theta_{1},\ldots,\theta_{n})=(\theta_{1}^{\prime},\ldots,\theta_{n}^{\prime})$ and $g:B_{n}(U(m))_{D}\to B_{n}(U(m))_{D^{\prime}}$ by $g(A_{1},\ldots,A_{n})=(A^{\prime}_{1},\ldots,A^{\prime}_{n})$, where $\theta_{j}^{\prime}=\theta_{1}^{q_{1j}}\cdot\dots\cdot\theta_{n}^{q_{nj}}$ and $A_{j}^{\prime}=A_{1}^{q_{1j}}A_{2}^{q_{2j}}\ldots A_{n}^{q_{nj}}$ for $j=1,\ldots,n$. Note that $g$ is a $\phi$-equivariant and $U(m)$-equivariant homeomorphism. Using the result for the special case $D^{\prime}$, it can be deduced that the composition

[TABLE]

is a homeomorphism and so is $\bar{f}$. It is clear that $\bar{f}^{-1}$ has the desired properties. ∎

Note that $R(D)$ is a finite subgroup of $\mathbb{T}^{n}$, acting by translation; thus $\mathbb{T}^{n}\to\mathbb{T}^{n}/R(D)$ is a covering space and $\mathbb{T}^{n}/R(D)$ is homeomorphic to $\mathbb{T}^{n}$.

For the general case $m=l\cdot\sigma(D)$, we recall that from [1], Corollary 3.10 that

[TABLE]

Hence, we obtain that

Theorem 2.7**.**

The moduli space of ordered almost commuting $n$–tuples in $U(m)$ can be expressed as a disjoint union of symmetric products of the $n$–torus $\mathbb{T}^{n}$:

[TABLE]

Now we apply the following result.

Lemma 2.8** ([2], Lemma 2.3).**

The projection map $U(m)\to PU(m)$ induces a $U(m)$–equivariant, $\mathbb{T}^{n}$–principal bundle

[TABLE]

which gives rise to a homeomorphism

[TABLE]

In particular it induces a bijection between $\pi_{0}(B_{n}(U(m))/U(m))$ and $\pi_{0}(Rep(\mathbb{Z}^{n},PU(m)))$.

The components $B_{n}(U(m))_{D}$ each give rise to a principal $\mathbb{T}^{n}$–bundle, and after dividing out by conjugation we see that the action of $\mathbb{T}^{n}$ is given by the simultaneous action on each unordered coordinate in the symmetric product through the homomorphism $\mathbb{T}^{n}\to\mathbb{T}^{n}/R(D)$; we denote these reduced symmetric products by $\widehat{SP^{m\over\sigma(D)}}(\mathbb{T}^{n}/R(D))$. Note that they are all in fact homeomorphic to the corresponding reduced symmetric product of $\mathbb{T}^{n}$. Combining the two results above we obtain

Theorem A**.**

[TABLE]

The labelling of components can be understood using cohomology. As before we can identify $T(n,\mathbb{Z}/m\mathbb{Z})$ with $H^{2}(\mathbb{Z}^{n},\mathbb{Z}/m\mathbb{Z})$ using a basis. For a projective representation $h:\mathbb{Z}^{n}\to PU(m)$, there is a cohomology class in $H^{2}(\mathbb{Z}^{n},\mathbb{Z}/m\mathbb{Z})$ associated to it, defined as the pullback $\alpha=h^{*}(\nu)$, where $\nu\in H^{2}(BPU(m),\mathbb{Z}/m\mathbb{Z})\cong\mathbb{Z}/m\mathbb{Z}$ is the canonical generator associated to $SU(m)$. The component corresponding to $h$ is precisely the one labelled by a skew symmetric matrix $D$ which represents $\alpha$. As this component is non-empty we must have \sigma(\alpha)\big{|}m.

Example 2.9**.**

In the case when $n=2$, $H^{2}(\mathbb{Z}^{2},\mathbb{Z}/m\mathbb{Z})\cong\mathbb{Z}/m\mathbb{Z}$ and $\sigma(D)=|D|$, the order of $D$, which always divides $m$ and so $N(2,m)=m$. Our decomposition can be written as

[TABLE]

where $\Phi$ is Euler’s function (see [3], Proposition 9).

Example 2.10**.**

We now consider the case when $m=p^{2}$ and $n\geq 4$. From the analysis in [1], section 3, we see that $Rep(\mathbb{Z}^{n},PU(p^{2}))$ has

[TABLE]

components, of which

[TABLE]

correspond to $\sigma(D)=p$ and so we have

[TABLE]

Our analysis can be applied to describe the projective representations of any finitely generated abelian group. Let

[TABLE]

Define

[TABLE]

It is a subspace of $B_{s+n}(U(m))$ and is invariant under the action of the subgroup

[TABLE]

This group action commutes with the conjugation action of $U(m)$. Hence, $B(\Gamma,U(m))$ is a $U(m)$–equivariant, $(\operatorname{Tor}(\Gamma)\times\mathbb{T}^{n})$–principal bundle over $Hom(\Gamma,PU(m))$. Also, there is a decomposition of

[TABLE]

into a disjoint union of subspaces indexed by $D\in T(s+n,\mathbb{Z}/m\mathbb{Z})$. The subspaces

[TABLE]

can be similarly defined.

Note that for $A_{i}\in U(m)$, $A_{i}^{k_{i}}=I_{m}$ if and only if all the eigenvalues of $A_{i}$ are $k_{i}$-th roots of unity. By Theorem 2.6, we obtain the following results similar to Theorem 2.7 and Theorem A.

Theorem 2.11**.**

Let $\Gamma=\mathbb{Z}/k_{1}\oplus\ldots\oplus\mathbb{Z}/k_{s}\oplus\mathbb{Z}^{n}\cong\operatorname{Tor}(\Gamma)\oplus\mathbb{Z}^{n}.$ Then

[TABLE]

and

[TABLE]

Remark 2.12**.**

Suppose \sigma(D)\big{|}m and r_{i}(D)\big{|}k_{i}. Let $H$ be the cokernel of the composition $R(D)\hookrightarrow\operatorname{Tor}(\Gamma)\times\mathbb{T}^{n}\stackrel{{\scriptstyle proj}}{{\longrightarrow}}\operatorname{Tor}(\Gamma)$ of inclusion followed by projection. Then, if we focus on the images of the components associated to $D$ under the quotient

[TABLE]

we have

[TABLE]

The number of orbits on the right hand side can be computed using Burnside’s lemma:

[TABLE]

where

[TABLE]

Example 2.13** (Projective representations of finite abelian groups).**

Our results allow us to recover results about projective representations of finite abelian groups. Consider a finite abelian group $\Gamma=\mathbb{Z}/k_{1}\oplus\ldots\oplus\mathbb{Z}/k_{s}$, where $k_{i}$ divides $k_{i+1}$ for $i=1,\ldots,s-1$. By Theorem 2.11, the space of degree $m$ projective unitary representation modulo projective equivalence is given by

[TABLE]

If $\sigma(D)=m$, then $\widehat{SP^{m\over\sigma(D)}}(\Gamma/R(D))$ is a single point and corresponds to an irreducible projective representation.

Removing the restriction on the dimension, we see that projective equivalence classes of irreducible projective representations of $\Gamma$ are in one–to–one correspondence with the $D\in T(n,\mathbb{Q}/\mathbb{Z})$ such that $r_{i}(D)|k_{i}$ for any $i=1,\ldots,s$. Note that since $D$ is skew-symmetric, the condition $r_{i}(D)|k_{i}$ is equivalent to $|d_{ij}|=|-d_{ji}|$ divides $k_{i}$ for $i<j$.

In terms of cohomology, this indexing can be seen to arise from pulling back using the projection $p:\mathbb{Z}^{s}\to\Gamma$, which yields a factorization

[TABLE]

Note that $p^{*}:H^{2}(\Gamma,\mathbb{Q}/\mathbb{Z})\to H^{2}(\mathbb{Z}^{s},\mathbb{Q}/\mathbb{Z})$ is injective, therefore if we consider all possible dimensions $m$ we see that the total indexing is in one-to-one correspondence with elements in $H^{2}(\Gamma,\mathbb{Q}/\mathbb{Z})\cong H^{3}(\Gamma,\mathbb{Z})$. Moreover, for the projective equivalence class corresponding to $D$, its set of representatives, up to linear equivalence, is indexed by $\Gamma/R(D)$, and each such representation has degree $\sigma(D)$.

3. Projective Representations and Flat Bundles

We now reformulate the computation of path components using homotopy theory, following the approach in [4], Lemma 2.5. Recall that given a group homomorphism $h:\Gamma\to G$, it induces a continuous (pointed) map on classifying spaces $Bh:B\Gamma\to BG$. For $G$ a compact connected Lie group, the correspondence

[TABLE]

is continuous and so induces a map on path components. As conjugation by $G$ is homotopically trivial on $BG$, it gives rise to a map

[TABLE]

If $X$ is a path–connected CW–complex with $\pi_{1}(X)=\Gamma$, then composing with the classifying map $c:X\to B\pi_{1}(X)$ of the universal cover $\tilde{X}\to X$ we obtain a map

[TABLE]

which measures the flat principal $G$–bundles on $X$.

In the case when $G=PU(m)$ we have a canonical homotopy class

[TABLE]

associated to the central extension

[TABLE]

For a CW–complex $X$, this gives rise to the composition

[TABLE]

We have

Theorem B**.**

Let $X$ denote a connected CW–complex with $\pi_{1}(X)=\mathbb{Z}^{n}$; then the map $\Psi^{PU(m)}_{X}:\pi_{0}(Rep(\pi_{1}(X),PU(m))\to[X,BPU(m)]$ is injective for all $n,m\geq 1$ and so there are $N(n,m)$ distinct isomorphism classes of flat principal $PU(m)$–bundles on $X$.

Proof.

First we consider the basic case $X=B\mathbb{Z}^{n}=\mathbb{T}^{n}$: the composition

[TABLE]

is the map on path components described in Section 2. We know that its image has precisely $N(n,m)$ elements, corresponding to the skew symmetric matrices $D$ with $\sigma(D)$ dividing $m$. In other words, this map distinguishes components, and is injective. If

[TABLE]

is induced by the classifying map, by naturality we have

[TABLE]

From the five-term exact sequence in cohomology for the fibration $\tilde{X}\to X\to B\pi_{1}X$ we infer that $c^{*}:H^{2}(B\pi_{1}(X),\mathbb{Z}/m\mathbb{Z})\to H^{2}(X,\mathbb{Z}/m\mathbb{Z})$ is in fact injective. Therefore the composition $\Omega\circ\Psi^{PU(m)}_{X}$ is injective and so is $\Psi^{PU(m)}_{X}$. ∎

Next we study the surjectivity of the map

[TABLE]

We begin with the following lemma

Lemma 3.1**.**

Let $m\geq 2$. If $n\leq 3$, $[\mathbb{T}^{n},BPU(m)]$ is finite, of cardinality equal to that of $T(n,\mathbb{Z}/m\mathbb{Z})$. If $n\geq 4$, the set $[\mathbb{T}^{n},BPU(m)]$ has infinitely many elements.

Proof.

This follows from Woodward’s classification of principal $PU(m)$–bundles for low dimensional complexes (see [9], page 514). For $\mathbb{T}^{n}$, $n\leq 3$ he shows that the map

[TABLE]

is a bijection. We outline a direct proof that $[\mathbb{T}^{n},BPU(m)]$ must be infinite for $n\geq 4$. For $n=4$, there is a map $\Phi:\mathbb{T}^{4}\to\mathbb{S}^{4}$ which induces an isomorphism on $H_{4}$. This arises from using the 4-dimensional cell in a CW-complex decomposition for $\mathbb{T}^{4}$ from its structure as a product of circles, each having a single 0-cell and a single 1-cell. The map $BSU(m)\to BPU(m)$ induces an isomorphism $\mathbb{Z}=\pi_{4}(BSU(m))\cong\pi_{4}(BPU(m))$ and so it is possible to choose a map $\rho:\mathbb{S}^{4}\to BPU(m)$ realizing this isomorphism. The Hurewicz map $\pi_{4}(BPU(m))\to H_{4}(BPU(m),\mathbb{Z})$ can be identified with the monomorphism $H_{4}(BSU(m),\mathbb{Z})\to H_{4}(BPU(m),\mathbb{Z})$, hence the composition $\rho\circ\Phi:\mathbb{T}^{4}\to BPU(m)$ is a map inducing an injection on the fundamental class in $\mathbb{T}^{4}$. It follows that $[\mathbb{T}^{4},BPU(m)]$ cannot be finite. For $n\geq 4$, we can use the split surjection $[\mathbb{T}^{n},BPU(m)]\to[\mathbb{T}^{4},BPU(m)]$ to verify the claim. ∎

Proposition 3.2**.**

Let $m\geq 2$, then $\Psi^{PU(m)}_{\mathbb{T}^{n}}:\pi_{0}(Rep(\mathbb{Z}^{n},PU(m))\to[\mathbb{T}^{n},BPU(m)]$ is surjective if and only if $n\leq 3$.

Proof.

From the definition in cohomology (and using an appropriate basis when $n=3$) it is easy to see that for $n=1,2,3$, we have that \{D\in T(n,\mathbb{Z}/m\mathbb{Z}):\sigma(D)~{}\big{|}~{}m\}=T(n,\mathbb{Z}/m\mathbb{Z}). Thus we conclude that $\Omega\circ\Psi^{PU(m)}_{\mathbb{T}^{n}}$ is a bijection and therefore by a cardinality argument so is $\Psi^{PU(m)}_{\mathbb{T}^{n}}$ for $n=1,2,3$. For $n\geq 4$ we have verified that $[\mathbb{T}^{n},BPU(m)]$ is infinite, whence the result follows. ∎

Corollary 3.3**.**

There exists a principal $PU(m)$-bundle on the $n$–torus $\mathbb{T}^{n}$ which does not admit a flat structure if and only if $n\geq 4$.

Acknowledgements

The first author was funded by NSERC. We are grateful to M. Bergeron, J.M. Gómez, Z. Reichstein and B. Williams for their helpful comments.