Homotopy types of gauge groups of $\mathrm{PU}(p)$-bundles over spheres

Let $G$ be a topological group and $X$ a space. The gauge group $\mathcal{G}(P)$ of a principal $G$-bundle $P$ over $X$ is defined as the group of $G$-equivariant bundle automorphisms of $P$ which cover the identity on $X$. A detailed introduction to gauge groups can be found in husemoller ; piccinini1998conjugacy .

The following problem is of interest: having fixed a topological group $G$ and a space $X$, classify the possible homotopy types of the gauge groups $\mathcal{G}(P)$ of principal $G$-bundles $P$ over $X$.

Crabb and Sutherland showed (crabb-sutherland, , Theorem 1.1) that if $G$ is a compact, connected, Lie group and $X$ is a connected, finite CW-complex, then the number of distinct homotopy types of $\mathcal{G}(P)$, as $P$ ranges over all principal $G$-bundles over $X$, is finite. This is often in contrast with the fact that the number of isomorphisms classes of principal $G$-bundles over $X$ may be infinite. However, their methods did not lead to an enumeration of the classes of gauge groups. Classification results require a different kind of analysis.

The first classification result was obtained by Kono kono91 in 1991. Using the fact that principal $\mathrm{SU}(2)$-bundles over $S^{4}$ are classified by $k\in\mathbb{Z}\cong\pi_{3}(\mathrm{SU}(2))$ and denoting by $\mathcal{G}_{k}$ the gauge group of the principal $\mathrm{SU}(2)$-bundle $P_{k}\to S^{4}$ corresponding to the integer $k$, Kono showed that there is a homotopy equivalence $\mathcal{G}_{k}\simeq\mathcal{G}_{l}$ if, and only if, $(12,k)=(12,l)$, where $(m,n)$ denotes the greatest common divisor of $m$ and $n$. It thus follows that there are precisely six homotopy types of $\mathrm{SU}(2)$-gauge groups over $S^{4}$.

In this paper, we examine how the close relationship between the groups $\mathrm{SU}(n)$ and $\mathrm{PU}(n)$ is reflected in the homotopy properties of the gauge groups of the corresponding bundles, particularly when $n$ is a prime. We do this by generalising certain results relating the classification of $\mathrm{PU}(n)$-gauge groups to that of $\mathrm{SU}(n)$-gauge groups from the paper so3 for the case $n=2$ (observe that $\mathrm{PU}(2)\cong\operatorname{SO}(3)$), and from hasui16 for the case $n=3$.

Our first main result compares certain Samelson products on $\mathrm{SU}(p)$ and $\mathrm{PU}(p)$, with $p\geq 3$ a prime. As will be illustrated in Section 2, the finiteness of the orders of Samelson products (in the appropriate groups of homotopy classes of maps) plays a crucial role in the homotopy classification of gauge groups. In Section 3, we will show the following.

Theorem 1.1 is the key ingredient for “if” direction of our classification results. The converse direction, on the other hand, will require that suitable homotopy invariants of the gauge groups be identified. In Section 2, we introduce the notation $\mathcal{G}_{i,k}(\mathrm{PU}(n))$, with $k\in\mathbb{Z}$, for gauge groups of $\mathrm{PU}(n)$-bundles over $S^{2i}$ in analogy with the notation used by Kono kono91 and others. In Section 4, we give a sufficient condition for certain homotopy invariants of $\mathrm{SU}(n)$- and $\mathrm{PU}(n)$-gauge groups to coincide.

Our methods then allow us to deduce classification results for $\mathrm{PU}(p)$-gauge groups from the corresponding classification results for $\mathrm{SU}(p)$-gauge groups. As examples of applications of our results, we obtain the following complete classifications.

We should note that in hasui16 , the $\mathrm{PU}(3)$-gauge group $\mathcal{G}_{2,k}$ is shown to be homotopy equivalent to $\widehat{\mathcal{G}}_{2,k}\times S^{1}$, where $\widehat{\mathcal{G}}_{2,k}$ is a space whose homotopy groups are all finite. This allows the authors to apply Lemma 2 and obtain a classification result for $\mathcal{G}_{k}$ that holds integrally. We expect the same result to apply more generally to gauge groups of $\mathrm{PU}(n)$-bundles over $S^{2n-2}$. However, there are currently no other cases, beside that of hasui16 , in which such a result would be applicable.

Finally, it is worth noting that, should any further classifications of gauge groups of $\mathrm{SU}(p)$-bundles over even-dimensional spheres be obtained, our results would provide the corresponding classification results for $\mathrm{PU}(p)$-gauge groups as immediate corollaries, provided the original results were arrived at via the standard methods.

The projective unitary group $\mathrm{PU}(n)$ is defined as the quotient of $\mathrm{U}(n)$ by its centre

In marked contrast with the orthogonal case, there is a homotopy equivalence between $\mathrm{PU}(n)$ and $\mathrm{PSU}(n)$ for every $n$. For the purposes of this paper, one can therefore equivalently define $\mathrm{PU}(n)$ as the quotient of $\mathrm{SU}(n)$ by its centre

We let $q\colon\mathrm{SU}(n)\to\mathrm{PU}(n)$ denote the quotient map corresponding to the latter definition, which we shall use throughout. Note that $q$ is an $n$-fold covering map and $\mathrm{SU}(n)$, being simply-connected, is the universal cover of $\mathrm{PU}(n)$.

Having fixed $n\geq 3$ and $2\leq i\leq n$, let $\delta_{i}\colon S^{2i-1}\to\mathrm{SU}(n)$ denote the generator of $\pi_{2i-1}(\mathrm{SU}(n))\cong\mathbb{Z}$ corresponding to the generator $\epsilon_{i}$ of $\pi_{2i-1}(\mathrm{PU}(n))$. That is, such that $q_{*}(\delta_{i})=\epsilon_{i}$, where $q$ denotes the quotient map $q\colon\mathrm{SU}(n)\to\mathrm{PU}(n)$ introduced in Section 2.

In this section, we wish to compare the orders of the Samelson products $\langle\delta_{i},1\rangle$ and $\langle\epsilon_{i},1\rangle$ on $\mathrm{SU}(n)$ and $\mathrm{PU}(n)$, respectively. First, observe that there is a commutative diagram

and recall the following property of the quotient map $q$.

Hence, when $n$ is prime, the orders of $\langle\delta_{i},1\rangle$ and $\langle\epsilon_{i},1\rangle$ coincide up to at most their $n$-primary component.

For the remainder of this section, we shall restrict to considering $\mathrm{PU}(n)$ when $n$ is an odd prime $p$.

Since $\mathrm{SU}(p)$ is the universal cover of $\mathrm{PU}(p)$ and $H_{*}(\mathrm{SU}(p);\mathbb{Z})$ is torsion-free, by (kishimoto08, , Theorem 1.1) we have the following decomposition of $\mathrm{PU}(p)$.

Let $\alpha\colon L_{(p)}\to\mathrm{PU}(p)_{(p)}$ be the inclusion. Then we can write the equivalence of Lemma 7 as

where $\mu$ is the group multiplication in $\mathrm{PU}(p)_{(p)}$. We note that this composite is equal to the product

in the group $[L_{(p)}\times\prod_{j=2}^{p-1}S^{2j-1}_{(p)},\mathrm{PU}(p)_{(p)}]$, where $\mathrm{pr}_{j}$ denotes the projection onto the $j$th factor.

We therefore calculate the groups $[S^{2i-1}\land L,\mathrm{PU}(p)]_{(p)}$ and, for $2\leq j\leq p-1$, the homotopy groups $\pi_{2i+2j-2}(\mathrm{PU}(p))_{(p)}$ in order to get an upper bound on the order of $\langle\epsilon_{i},1\rangle_{(p)}$.

For the next part of our calculation, we will need a certain mod-$p$ decomposition of $\Sigma^{2}L$ which will, in turn, require some cohomological information. The mod-$p$ cohomology algebra of $\mathrm{PU}(n)$, with $p$ any prime and $n$ arbitrary, was determined by Baum and Browder in (baum-browder, , Corollary 4.2). In particular, we have:

For $m\geq 2$, denote by $P^{m}(p)$ the mod-$p$ Moore space defined as the homotopy cofibre of the degree $p$ map

on the sphere $S^{m-1}$. In other words, $P^{m}(p)=S^{m-1}\cup_{p}e^{m}$. Note that, by extending the cofibre sequence to the right, we see that $\Sigma P^{m}(p)\simeq P^{m+1}(p)$.

With $A$ as in Lemma 11, we have:

Lemmas 8, 9, and 13 together imply: