The cohomology of the classifying space of $PU(4)$

Let $G$ be a topological group. The classifying space $BG$ of $G$ is characterized as the base space of a universal $G$-bundle $G\to EG\to BG$. The cohomology of $BG$ is the main tool in the theory of classifying principal $G$-bundles.

Let $U(n)$ be the group of $n\times n$ unitary matrices. The projective unitary group $PU(n)$ is the quotient group $U(n)/S^{1}$ of $U(n)$ by its center subgroup $S^{1}=\{e^{i\theta}I_{n}:\theta\in[0,2\pi]\}$, where $I_{n}$ denotes the identity matrix. $PU(n)$ is homotopy equivalent to $PGL(n,\mathbb{C})$, the projective general linear group $GL(n,\mathbb{C})/\mathbb{C}^{*}$, which is the automorphism group of the algebra of $n\times n$ complex matrices. Therefore, for a pointed space $X$, the set of pointed homotopy classes of maps $[X,BPU(n)]$ also classifies bundles of $n\times n$ complex matrix algebras over $X$, known as the topological Azumaya algebras of degree $n$ over $X$. (See [9].)

The cohomology ring of $BPU(n)$ also plays a significant role in the study of the topological period-index problem, which was introduced by Antieau and Williams [1, 2] as an analogue of period-index problems in algebraic geometry (cf. [3, 7, 19]). Gu [10, 11], Crowley and Grant [6] also investigated this problem for certain topological spaces.

Since $PU(n)$ can also be viewed as the quotient group of the special unitary group $SU(n)$ by its center subgroup $\mathbb{Z}/n$ generated by $e^{2\pi i/n}I_{n}$, there is an induced fibration of classifying spaces:

Hence, for a commutative ring $R$, if $1/n\in R$, then

It follows that if $x\in H^{*}(BPU(n);\mathbb{Z})$ is a torsion element, then there exists $k\geq 0$ such that $n^{k}x=0$. (In the case of Chow rings, Vezzosi [23] proved the stronger result that all torsion classes in the Chow ring of $BPGL(n,\mathbb{C})$ are $n$-torsion.)

However, if $n$ is not invertible in the coefficient ring $R$, the ring structure of $H^{*}(BPU(n);R)$ is very complicated for general $n$. Here we list some of the known results on the cohomology of $BPU(n)$ for special values of $n$.

Since $PU(2)=SO(3)$, the ring $H^{*}(BPU(2);\mathbb{Z})$ is well understood as a special case of the result of Brown [4].

The ring $H^{*}(BPU(3);\mathbb{Z}/3)$ is computed in Kono-Mimura-Shimada [16]. The Steenrod algebra structure of $H^{*}(BPU(3);\mathbb{Z}/3)$ and the Brown-Peterson cohomology of $BPU(3)$ are determined in Kono-Yagita [17].

The ring $H^{*}(BPU(n);\mathbb{Z}/2)$ is known if $n\equiv 2$ mod $4$ (Kono-Mimura [15] and Toda [21]). In [21], Toda also computed the ring $H^{*}(BPU(4);\mathbb{Z}/2)$.

In [24], Vistoli provided a description of $H^{*}(BPU(p);\mathbb{Z})$, as well as the Chow ring of $BPGL(p,\mathbb{C})$, for any odd prime $p$. In particular, the ring structure of $H^{*}(BPU(3);\mathbb{Z})$ is given in terms of generators and relations there. We will explicitly explain Vistoli’s result in the next Section.

More interesting results on the ordinary cohomology ring $H^{*}(BPU(p);\mathbb{Z}/p)$ (resp. on the Brown-Peterson cohomology of $BPU(p)$) for an arbitrary prime $p$ can be found in Vavpetič-Viruel [22] (resp. in Masaki-Yagita [20]).

For an arbitrary integer $n>3$, the cohomology of $BPU(n)$ is only known in certain finite range of dimensions. For instance, Gu [12] determined the ring structure of $H^{*}(BPU(n);\mathbb{Z})$ in dimensions less than or equal to $10$. Other partial results on the $p$-local cohomology groups of $BPU(n)$ for an odd prime $p$, $p\mid n$, can be found in Gu-Zhang-Zhang-Zhong [14] and Zhang-Zhang-Zhong [25].

Summarizing these results, the only examples of $BPU(n)$, whose integral cohomology rings can be described via generators and relations, are the cases $n=2,3$. Also, these are the only cases for which the Steenrod algebra structure of $H^{*}(BPU(n);\mathbb{Z}/p)$, $p$ a prime such that $p\mid n$, are known. The main goal of this paper is to determine the integral cohomology ring of $BPU(4)$ and the Steenrod algebra structure of its mod $2$ cohomology.

For the graded polynomial ring $\mathbb{Z}[v_{1},\dots,v_{n}]$ with $\deg(v_{i})=2$, define a graded algebraic homomorphism $\Theta_{n}:\mathbb{Z}[v_{1},\dots,v_{n}]\to\mathbb{Z}[\eta]/(n\eta)$, $\deg(\eta)=2$, by $\Theta(v_{i})=i\eta$. In fact, this is the ring homomorphism $H^{*}(BT^{n};\mathbb{Z})\to H^{*}(B(\mathbb{Z}/n);\mathbb{Z})$ induced by the embedding $\mathbb{Z}/n\hookrightarrow T^{n}$, $\omega\mapsto(\omega,\omega^{2},\dots,\omega^{n-1},1)$, $\omega=e^{2\pi i/n}$, $T^{n}=(S^{1})^{n}$. Let $\Theta^{\prime}_{n}$ be the restriction of $\Theta_{n}$ to $K_{n}$, and let $K^{\prime}_{n}=\ker\Theta^{\prime}_{n}$. The result of Vistoli [24] can be described as follows.

For the element $\delta=\prod_{i\neq j}(v_{i}-v_{j})\in K_{p}$, one easily checks that

It is shown in [24, Proposition 3.1] that the image of $\Theta^{\prime}_{p}$ is generated by $\Theta^{\prime}_{p}(\delta)$, so for $p>2$, as $\mathbb{Z}$-module,

where $E_{\mathbb{Z}/p}[x_{3}]$ means the exterior algebra over $\mathbb{Z}/p$ with the generator $x_{3}$. Since the rank of $K_{p}$ is the same as the rank of $H^{*}(BSU(n);\mathbb{Z})\cong\mathbb{Z}[c_{2},\dots,c_{n}]$, the additive structure of $H^{*}(BPU(p);\mathbb{Z})$ are determined completely.

One reason for the difficulty of describing $H^{*}(BPU(n);\mathbb{Z})$ is that the ring $K_{n}$ is hard to compute for large $n$. For small values of $n$, here are some results. $K_{2}\cong\mathbb{Z}[\alpha_{2}]$, $\alpha_{2}=\sigma_{1}^{2}-4\sigma_{2}\in\Lambda_{2}^{2}$, is obvious. Vezzosi [23, Lemma 3.2] (see also [24, Theorem 14.2]) proved that

Here we give the description of $K_{4}$.

For $n\geq 5$, the complexity of $K_{n}$ grows very quickly as $n$ grows. To the author’s knowledge, there is no one computed $K_{n}$ for $n\geq 5$ by writing down the generators and relations explicitly so far.

As in the case of $K_{p}$ for $p$ a prime, $K_{4}$ can be viewed as the quotient ring of $H^{*}(BPU(4);\mathbb{Z})$ by torsion elements.

One of the results in Toda [21] gives $H^{*}(BPSO(4m+2);\mathbb{Z}/2)$, $m\geq 1$. In particular, this gives the mod $2$ cohomology of $BPU(4)$, since it is known that $PU(4)$ is homeomorphic to $PSO(6)$.

In the notation of Theorem 2.4, we determine the Steenrod algebra structure of $H^{*}(BPU(4);\mathbb{Z}/2)$ as follows.

In the next two sections, we review Gu’s work in [12], where he constructs a fibration with $BPU(n)$ as the total space, and provides an approach to calculate the Serre spectral sequence associated to this fibration.

Throughout this section, we write $R[x;k]$, $E_{R}[y;k]$ for the polynomial algebra and exterior algebra respectively, with coefficients in a commutative ring $R$, and with one generator of degree $k$. We also use the simplified notation $E[y;k]$ for $R=\mathbb{Z}$.

Let $C(3)$ be the (differential graded) $\mathbb{Z}$-algebra

For notational convenience, we set $y_{2,0}=x_{1}^{2}$ and $y_{2,k}=x_{2,k-1}^{2}$ for $k\geq 1$. The differential of $C(3)$ is given by

Gu [12] showed that the integral cohomology ring of $K(\mathbb{Z},3)$ is isomorphic to the cohomology of $C(3)$, which is dual to the DGA construction of Cartan in [5].

Let ${}_{p}A$ denote the $p$-primary component of a commutative ring $A$, $p$ a prime. For $R=\bigoplus_{i>0}R^{i}$ a graded commutative ring without unit, let $\hat{R}=R\oplus R^{0}$ be the (graded commutative) unital ring, where the degree zero summand $R^{0}=\mathbb{Z}$ has a generator acting as the unit of the ring.

For $p$ a prime, the mod $p$ cohomology ring of $K(\mathbb{Z},3)$ is also given by the cohomology of the differential graded $\mathbb{Z}/p$-algebra $(C(3)\otimes\mathbb{Z}/p,d)$, which has an easier description (see [12, (2.13)]). Furthermore, the Steenrod algebra structure of $H^{*}(K(\mathbb{Z},3);\mathbb{Z}/p)$ can be determined from the fact that $x_{1}$ (resp. $x_{p,k}$) is the the transgression image of $v$ (resp. $v^{p^{k+1}}$) $\in H^{*}(BS^{1};\mathbb{Z}/p)\cong\mathbb{Z}/p[v]$ in the mod $p$ Serre spectral sequence ${}^{K}E$, using Kudo’s transgression theorem [18]. In particular, for $p=2$, we have the following analogue of [13, Proposition 2.4].

The differentials of the integral spectral sequence ${}^{K}E$ are described in [12] as follows. Here we only list a part of the results in [12, Corollary 2.16], which will be sufficient for our purposes.

To describe the differential of ${}^{T}E^{*,*}_{*}$, we first rewrite $H^{*}(BT^{n};\mathbb{Z})=\mathbb{Z}[v_{1},\dots,v_{n}]$ as $\mathbb{Z}[v_{1}-v_{n},\dots,v_{n-1}-v_{n},v_{n}]$. Then ${}^{T}E^{*,*}_{*}$ is generated by elements of the form $\xi\cdot f(v_{n})$, where $\xi\in H^{*}(K(\mathbb{Z},3);\mathbb{Z})$ and $f(v_{n})=\sum_{i}a_{i}v_{n}^{i}$ is a polynomial in $v_{n}$ with coefficients $a_{i}\in\mathbb{Z}[v_{1}-v_{n},\dots,v_{n-1}-v_{n}]$. Let $v_{j}^{\prime}=v_{j}-v_{n}$ for $1\leq j\leq n-1$, and write $a_{i}=\sum_{t_{1},\dots,t_{n-1}}k_{i,t_{1},\dots,t_{n-1}}(v^{\prime}_{1})^{t_{1}}\cdots(v_{n-1}^{\prime})^{t_{n-1}}$, $k_{i,t_{1},\dots,t_{n-1}}\in\mathbb{Z}$.

Since the operator $\nabla$ preserves symmetric polynomials, for $f\in H^{*}(BU(n);\mathbb{Z})\cong\Lambda_{n}$ and $\xi\in H^{*}(K(\mathbb{Z},3);\mathbb{Z})$, we have the following corollary of Proposition 4.5 by the Leibniz rule.

In particular, the fact that $\nabla(\sigma_{k})=\sum_{i=1}^{n}\frac{\partial\sigma_{k}}{\partial v_{i}}=(n-k+1)\sigma_{k-1}$ shows that for $c_{k}\in H^{*}(BU(n);\mathbb{Z})\cong{{}^{U}}E_{3}^{0,*}$, $1\leq k\leq n$, we have ([12, Corollary 3.4])

The proof of Theorem 2.2 is essentially a $K_{4}$ analogue of the proof of Lemma 3.2 for $K_{3}$ in [23] (also Theorem 14.2 in [24]).

Firstly, we give the explicit expression of $\alpha_{i}$ in $\Lambda_{4}^{i}$. Let

A direct calculation shows that $\nabla(\alpha_{i})=0$ for all $\alpha_{i}$ above, and

Consider the ring homomorphism

It is easy to see that this map is injective. After tensoring with $\mathbb{Z}[\frac{1}{2}]$, this map becomes an isomorphism, and the inverse is obtained by sending $v_{i}$ to $v_{i}-\frac{\sigma_{1}}{4}$. The commutative diagram below shows that $K_{4}\otimes\mathbb{Z}[\frac{1}{2}]\hookrightarrow\mathbb{Z}[\frac{1}{2}][\sigma_{2},\sigma_{3},\sigma_{4}]$.