1 Introduction

Let G be a topological group and P be a principal G-bundle over a base space X. The gauge group of P is the topological group of G-equivariant automorphisms of P which cover the identity map on X. If X is a finite \(\textit{CW}\)-complex, Crabb and Sutherland [2] showed that, despite there possibly being infinitely many inequivalent principal G-bundles over X, there are only finitely many homotopy types for the corresponding gauge groups. There has been an intensive effort recently to classify the homotopy types of gauge groups, particularly in cases of interest to physics and geometry.

In this paper, we consider the homotopy types of gauge groups of principal \(\mathrm{SO}(4)\)-bundles over \(S^{4}\). Our results are stated in more generality. Let Z be a subgroup of \((S^3)^n\) generated by an element \((-1,\ldots ,-1)\). Define

$$\begin{aligned} K_n=(S^3)^n/Z. \end{aligned}$$

Then, in particular, \(K_1=\mathrm{SO}(3)\) and \(K_2=\mathrm{SO}(4)\). Let \(\epsilon _i\) be the composite of the i-th inclusion \(S^3\!\rightarrow (S^3)^n\) and the projection \((S^3)^n\!\rightarrow K_n\) for \(1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}n\). Then \(\pi _3(K_n)\cong \mathbb {Z}^n\) is generated by \(\epsilon _1,\ldots ,\epsilon _n\). Let \(\mathscr {G}_{k_1,\ldots ,k_n}\) be the gauge group of a principal \(K_n\)-bundle over \(S^4\) corresponding to \(k_1\epsilon _1+\cdots +k_n\epsilon _n\in \pi _3(K_n)\). The aim of this paper is to classify the homotopy types of gauge groups \(\mathscr {G}_{k_1,\ldots ,k_n}\) as \(k_1,\ldots ,k_n\) range over all integers.

Let \(\{\{a_1,\ldots ,a_n\}\}\) denote a multiset consisting of elements \(a_1,\ldots ,a_n\). For integers a and b, let (a, b) be their greatest common divisor.

Theorem 1.1

The following hold:

  1. (a)

    if \(\{\{(k_1,12),\ldots ,(k_n,12)\}\}=\{\{(l_1,12),\ldots ,(l_n,12)\}\}\) then there is a homotopy equivalence \(\mathscr {G}_{k_1,\ldots ,k_n}\!\simeq \mathscr {G}_{l_1,\ldots ,l_n}\);

  2. (b)

    if there is a homotopy equivalence \(\mathscr {G}_{k_1,\ldots ,k_n}\!\simeq \mathscr {G}_{l_1,\ldots ,l_n}\) then \(\{\{(k_1,4),\ldots ,(k_n,4)\}\}=\{\{(l_1,4),\ldots ,(l_n,4)\}\}\) and \(\{\{(k_1,3),\ldots ,(k_n,3)\}\}=\{\{(l_1,3),\ldots ,(l_n,3)\}\}\).

The weaker statement in Theorem 1.1(b) stems from the fact that the homotopy equivalence induces an isomorphism of homotopy groups, but as we will see in Lemma 2.7, the isomorphism of second homotopy groups may take the form \(\mathbb {Z}/12{\oplus }\mathbb {Z}/1\cong \mathbb {Z}/3{\oplus }\mathbb {Z}/4\) when \(n=2\) (where \(\mathbb {Z}/1\) is the trivial group), which does not imply an equality between \(\{\{(1,12),(12,12)\}\}\) and \(\{\{(4,12),(3,12)\}\}\).

If one is willing to localize then a classification holds. For a nilpotent space X and a prime p, let \(X_{(p)}\) be the localization of X at p. For an integer m, let \(\nu _{p}(m)\) be the p-component of m.

Theorem 1.2

Let p be a prime. There is a p-local homotopy equivalence

$$\begin{aligned} (\mathscr {G}_{k_1,\ldots ,k_n})_{(p)}\simeq (\mathscr {G}_{l_1,\ldots ,l_n})_{(p)} \end{aligned}$$

if and only if \(\{\{\nu _{p}((k_1,12)),\ldots ,\nu _{p}((k_n,12))\}\}= \{\{\nu _{p}((l_1,12)),\ldots ,\nu _{p}((l_n,12))\}\}\).

The \(K_{1}=\mathrm{SO}(3)\) case in Theorem 1.1 is already known [7]. In this case, as there is only one index involved, Theorem 1.1 implies the stronger statement that \(\mathscr {G}_{k}\simeq \mathscr {G}_{l}\) if and only if \((k,12)=(l,12)\). The key new case is for \(K_{2}=\mathrm{SO}(4)\).

The \(\mathrm{SO}(4)\) and \(\mathrm{Spin}(4)\) cases are the last to consider among the principal G-bundles over \(S^{4}\) when G is a connected, compact Lie group of type 2. A classification in the \(\mathrm{SU}(3)\) case was completed in [6] and the \(\mathrm{PU}(3)\) case in [5], a classification of the p-local homotopy types in the \(\mathrm{Sp}(2)\) case was completed in [10] and \(\mathrm{PSp}(2)\) in [5], the U(2)-case was classified in [3], and the p-local homotopy types in the \(G_{2}\) case were classified up to one factor of 2 in [8].

The overall strategy used to prove Theorem 1.1 is similar to that in the other type 2 cases, but distinctive features arise. The principal \(\mathrm{SO}(4)\)-bundles over \(S^{4}\) are in one-to-one correspondence with \([S^{4}\!,B\mathrm{SO}(4)]\cong \mathbb {Z}{\oplus }\mathbb {Z}\) and so require a multi-index, as opposed to the earlier cases which only required a single index. This leads to the use of multisets and the possibility that a homotopy equivalence \(\mathscr {G}_{k_1,k_2}\!\simeq \mathscr {G}_{l_1,l_2}\) may arise via a permutation of indices. It also leads to the issue mentioned above of a group decomposition preventing an identification of multisets.

2 Proof of Theorem 1.1

Recall from [4] (cf. [1]) that there is a homotopy equivalence

$$\begin{aligned} B\mathscr {G}_{k_1,\ldots ,k_n}\!\simeq \mathrm {map}(S^4\!, BK_n;k_1\epsilon _1+\cdots +k_n\epsilon _n), \end{aligned}$$

where the right-hand side is the path-connected component of the space of maps from \(S^4\) to \(BK_n\) containing \(k_1\epsilon _1+\cdots +k_n\epsilon _n\). Consider the homotopy fiber sequence

$$\begin{aligned} K_n\xrightarrow {\partial _{k_1,\ldots ,k_n}}\Omega ^3_0K_n\rightarrow \mathrm {map}(S^4\!,BK_n;k_1\epsilon _1+\cdots +k_n\epsilon _n)\rightarrow BK_n, \end{aligned}$$

where the last map is the evaluation at the basepoint. Then \(\mathscr {G}_{k_1,\ldots ,k_n}\) is homotopy equivalent to the homotopy fiber of \(\partial _{k_1,\ldots ,k_n}\). Let \(\pi :K_n\rightarrow K_1^n\) be the canonical projection. Since \(\Omega ^3\pi :\Omega ^3_0K_n\rightarrow (\Omega ^3_0K_1)^n\) is a homotopy equivalence, one obtains the following.

Lemma 2.1

The gauge group \(\mathscr {G}_{k_1,\ldots ,k_n}\) is homotopy equivalent to the homotopy fiber of   \(\Omega ^3\pi {\circ } \partial _{k_1,\ldots ,k_n}\).

We need to identify the map \(\Omega ^{3}\pi {\circ } \partial _{k_1,\ldots ,k_n}\). Whitehead [11] showed that the adjoint \(S^4{\wedge }K_n\rightarrow BK_n\) of \(\partial _{k_1,\ldots ,k_n}\) is the Whitehead product of the adjoint of \(k_1\epsilon _1+\cdots +k_n\epsilon _n\) and the canonical inclusion \(\Sigma K_n\rightarrow BK_n\). Thus by the adjointness of Whitehead products and Samelson products, we obtain the following.

Lemma 2.2

The adjoint \(S^3{\wedge }K_n\rightarrow K_n\) of the map \(\partial _{k_1,\ldots ,k_n}\) is the Samelson product \(\langle k_1\epsilon _1+\cdots +k_n\epsilon _n,1_{K_n}\rangle \).

Now we calculate \(\pi {\circ } \langle k_1\epsilon _1+\cdots +k_n\epsilon _n,1_{K_n}\rangle \). For \(1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}n\), let \(p_{i}:K_{1}^{n}\rightarrow K_{1}\) be the projection onto the \(i^\mathrm{th}\)-factor. Define \(\pi _{i}\) by the composite

$$\begin{aligned} \pi _{i}:K_{n}\xrightarrow {\,\,\pi \,\,} K_{1}^{n}\xrightarrow {\,\,p_{i}\,\,} K_{1}. \end{aligned}$$

Define \(\overline{\epsilon }_{i}\) and \(\lambda _{i}\) by the composites

$$\begin{aligned} \overline{\epsilon }_{i}:S^{3}\xrightarrow {\,\,\epsilon _{i}\,\,} K_{n}\xrightarrow {\,\,\pi _{i}\,\,} K_{1},\quad \lambda _{i}:S^3{\wedge }K_n\xrightarrow {\,\,\langle \overline{\epsilon }_{i},\pi _{i}\rangle \,\,} K_{1}. \end{aligned}$$

Observe that \(\pi _{i}{\circ } (k_{1}\epsilon _{1}+\cdots +k_{n}\epsilon _{n})\simeq k_{i}\overline{\epsilon }_{i}\). Therefore, as \(\pi _i\) is a homomorphism,

$$\begin{aligned} \pi _{i}{\circ } \langle k_{1}\epsilon _{1}+\cdots +k_{n}\epsilon _{n},1_{K_{n}}\rangle \simeq \langle \pi _i{\circ } (k_1\epsilon _1+\cdots +k_n\epsilon _n),\pi _i\rangle \simeq k_i\langle \overline{\epsilon }_i,\pi _i\rangle =k_{i}\lambda _{i}. \end{aligned}$$

Thus \(\pi {\circ } \langle k_1\epsilon _1+\cdots +k_n\epsilon _n,1_{K_n}\rangle \simeq (k_{1}\lambda _1,\ldots ,k_{n}\lambda _n)\). Let \(k:K_1\rightarrow K_1\) be the \(k^\mathrm{th}\)-power map. Then \((k_{1}\lambda _1,\ldots ,k_{n}\lambda _n)\simeq (k_{1}{\times } \cdots {\times } k_{n}) {\circ } (\lambda _1,\ldots ,\lambda _n)\). Hence the composite \(\pi {\circ } \langle k_1\epsilon _1+\cdots +k_n\epsilon _n,1_{K_n}\rangle \) has the following linearity property.

Proposition 2.3

There is a homotopy

$$\begin{aligned} \pi {\circ } \langle k_1\epsilon _1+\cdots +k_n\epsilon _n,1_{K_n}\rangle \simeq (k_1{\times }\cdots {\times } k_n){\circ } (\lambda _1,\ldots ,\lambda _n). \end{aligned}$$

Next, we determine the order of \(\lambda _i\).

Proposition 2.4

The order of \(\lambda _{i}=\langle \overline{\epsilon }_i,\pi _i\rangle \) is 12 for each \(1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}n\).

Proof

The diagonal map \(S^3\rightarrow (S^3)^n\) induces an inclusion \(j:K_1\rightarrow K_n\) which is a section of \(\pi _i\) for each \(1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}n\). On the one hand, since \(\pi _{i}{\circ } j\) is the identity map on \(K_{1}\), the composite

$$\begin{aligned} S^3{\wedge }K_{1}\xrightarrow {\,\,1_{S^{3}}{\wedge }j\,\,} S^{3}{\wedge }K_{n}\xrightarrow {\,\,\langle \overline{\epsilon }_{i},\pi _{i}\rangle \,\,} K_{1} \end{aligned}$$

is \(\langle \overline{\epsilon }_{i},1_{K_{1}}\rangle \). By [7] the order of \(\langle \overline{\epsilon }_{i},1_{K_{1}}\rangle \) is 12. Thus the order of \(\langle \overline{\epsilon }_{i},\pi _{i}\rangle \) is at least 12. On the other hand, observe that \(\langle \overline{\epsilon }_{i},\pi _{i}\rangle \) factors as the composite

$$\begin{aligned} S^{3}{\wedge }K_{n}\xrightarrow {\,\,1_{S^{3}}{\wedge }\pi _{i}\,\,} S^{3}{\wedge }K_{1}\xrightarrow {\,\,\langle \overline{\epsilon }_{i},1_{K_{1}}\rangle \,\,} K_{1}. \end{aligned}$$

Thus the order of \(\langle \overline{\epsilon }_{i},\pi _{i}\rangle \) is at most 12. Hence the order of \(\langle \overline{\epsilon }_{i},\pi _{i}\rangle \) is precisely 12. \(\square \)

Lemma 2.5

Let \(\alpha _i:X\rightarrow Y_i\) be a map of order \(p^{m_i}\) into a p-local H-space \(Y_i\) for \(1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}n\). If \((k_i,p^{m_i})=(l_i,p^{m_i})\) for all i, then there is a self-homotopy equivalence \(h_{p}\) of \(Y_1{\times } \cdots {\times } Y_n\) such that

$$\begin{aligned} h_p{\circ } (k_1{\times } \cdots {\times } k_n){\circ } (\alpha _1,\ldots ,\alpha _n)\simeq (l_1{\times } \cdots {\times } l_n){\circ } (\alpha _1,\ldots ,\alpha _n). \end{aligned}$$

Proof

If \((k_{i},p^{m_{i}})=(l_{i},p^{m_{i}})=p^{m_{i}}\) then the composites \(k_{i}{\circ } \alpha _{i}\) and \(l_{i}{\circ } \alpha _{i}\) are both null homotopic since \(\alpha _{i}\) has order \(p^{m_{i}}\). In that case let \(h_{p,i}\) be the identity map on \(Y_{i}\). If \((k_{i},p^{m_{i}})=(l_{i},p^{m_{i}})< p^{m_{i}}\), let \(a_i=\frac{k_i}{(k_i,p^{m_i})}\) and \(b_i=\frac{l_i}{(l_i,p^{m_i})}\). Then \(a_i\) and \(b_i\) are units in \(\mathbb {Z}_{(p)}\), and in particular, the power maps \(a_i,b_i:Y_i\rightarrow Y_i\) are homotopy equivalences. As \((k_i,p^{m_i})=(l_i,p^{m_i})\) we obtain \(\frac{b_i}{a_i}k_i=l_i\). Therefore if \(h_{p,i}=b_{i}{\circ } a_{i}^{-1}\) then \(h_{p,i}{\circ } k_i\simeq l_i\) as self-maps of \(Y_i\). Thus \(h_p=h_{p,1}{\times } \cdots {\times } h_{p,n}\) is the desired self-homotopy equivalence. \(\square \)

Proposition 2.6

If \(\{\{(k_1,12),\ldots ,(k_n,12)\}\}=\{\{(l_1,12),\ldots ,(l_n,12)\}\}\), then \(\mathscr {G}_{k_1,\ldots ,k_n}\) and \(\mathscr {G}_{l_1,\ldots ,l_n}\) are homotopy equivalent.

Proof

By assumption, there is a permutation \(\sigma \) such that \(((k_{\sigma (1)},12),\ldots ,(k_{\sigma (n)},12)) =((l_1,12),\ldots ,(l_n,12))\). We denote the permutation of \((S^3)^n\) induced from \(\sigma \) by the same symbol. This automorphism induces an automorphism of \(K_n\) which we denote by \(\overline{\sigma }\). The automorphism \(\overline{\sigma }\) induces a homotopy commutative diagram of Samelson products

Taking adjoints, by Lemma 2.2 we obtain \(\Omega ^{3}\overline{\sigma }{\circ } \partial _{k_{1},\ldots ,k_{n}}\!\simeq \partial _{k_{\sigma (1)},\ldots ,k_{\sigma (n)}}{\circ } \overline{\sigma }\). Composing with the map \(\Omega ^{3} K\xrightarrow {\Omega ^{3}\pi }\Omega ^{3} K_{1}^{n}\) and using the fact that \(\Omega ^{3}_{0} K_{1}^{n}\simeq (\Omega ^{3}_{0} S^{3})^{n}\), we obtain a homotopy commutative diagram

By Lemma 2.1, the homotopy fibre of \(\Omega ^{3}\pi {\circ } \partial _{k_{1},\ldots ,k_{n}}\) is \(\mathscr {G}_{k_{1},\ldots ,k_{n}}\). So as \(\sigma \) and \(\overline{\sigma }\) are homotopy equivalences, this diagram induces a homotopy equivalence \(\mathscr {G}_{k_{1},\ldots ,k_{n}}\!\simeq \mathscr {G}_{k_{\sigma (1)},\ldots ,k_{\sigma (n)}}\). Thus, for ease of notation in what follows, we may assume without loss of generality that the permutation \(\sigma \) is the identity.

By Proposition 2.3, \(\pi {\circ } \langle k_1\epsilon _1+\cdots +k_n\epsilon _n,1_{K_n}\rangle \simeq (k_1{\times } \cdots {\times } k_n){\circ } (\lambda _1,\ldots ,\lambda _n)\). Since \(\pi _i(\Omega ^3_0S^3)\) is finite for each i, there is a homotopy equivalence

$$\begin{aligned} \Omega ^3_0S^3\simeq \prod _{p\in P}\Omega ^3_0S^3_{(p)} \end{aligned}$$

where P is the set of all primes and \(S^{3}_{(p)}\) is the localization of \(S^{3}\) at p. Let \(\widehat{\lambda }_i:K_{n}\rightarrow \Omega ^{3}_{0} S^{3}\) be the adjoint of \(\lambda _i\) and let

$$\begin{aligned} \big (\,\widehat{\lambda }_1,\ldots ,\widehat{\lambda }_n\big )_{p}:K_{n}\rightarrow \big (\Omega ^{3}_{0} S^{3}_{(p)}\big )^{n} \end{aligned}$$

be the composite of \((\widehat{\lambda }_1,\ldots ,\widehat{\lambda }_n)\) and the map localizing \((\Omega ^{3}_{0} S^{3})^{n}\) to \((\Omega ^{3}_{0} S^{3}_{(p)})^{n}\). For convenience, the \(m^\mathrm{th}\)-power map on \(\Omega ^{3}_{0} S^{3}\) will also be denoted by m. By Lemma 2.5, for each prime p there is a self-homotopy equivalence \(h_p\) of \((\Omega ^3_0S^3_p)^n\) satisfying a homotopy commutative diagram

(2.1)

Note that since each \(k_{i}{\circ } \widehat{\lambda }_i\) is a divisor of 12, the map \(h_{p}\) is a homotopy equivalence if \(p\hbox {\,\,\char 062\,\,}5\). Let h be the composite

$$\begin{aligned} h:\;\; (\Omega ^{3}_{0} S^{3})^{n}\xrightarrow {\,\,\simeq \, \ } \prod _{p\in P}\bigl (\Omega ^{3}_{0} S^{3}_{(p)}\bigr )^{n}\xrightarrow {\,\,\prod _{p\in P} h_{p}\,\,} \prod _{p\in P}\bigl (\Omega ^{3}_{0} S^{3}_{(p)}\bigr )^{n}\xrightarrow {\,\,\simeq \, \ } (\Omega ^{3}_{0} S^{3})^{n}. \end{aligned}$$

Since each \(h_{p}\) is a homotopy equivalence, so is h. From diagram (2.1) at each p we obtain a homotopy commutative diagram

(2.2)

On the one hand, by definition, \(\widehat{\lambda }_{i}\) is the adjoint of \(\lambda _{i}\), so the adjoint of \((k_{1}{\times } \cdots {\times } k_{n}) {\circ } (\widehat{\lambda }_1,\ldots ,\widehat{\lambda }_n)\) is \((k_{1}{\times } \cdots {\times } k_{n}){\circ } (\lambda _{1},\ldots ,\lambda _{n}\)), which by Proposition 2.3 is homotopic to \(\pi {\circ } \langle k_{1}\epsilon _{1}+\cdots + k_{n}\epsilon _{n},1_{K_{n}}\rangle \). On the other hand, by Lemma 2.2, \(\pi {\circ } \partial _{k_{1},\ldots ,k_{n}}\) is also the adjoint of \(\pi {\circ } \langle k_{1}\epsilon _{1}+\cdots + k_{n}\epsilon _{n},1_{K_{n}}\rangle \). This implies that \((k_{1}{\times } \cdots {\times } k_{n}){\circ } (\widehat{\lambda }_1,\ldots ,\widehat{\lambda }_n)\simeq \pi {\circ } \partial _{k_{1},\ldots ,k_{n}}\). Hence (2.2) may be rewritten as a homotopy commutative diagram

(2.3)

By Lemma 2.1, the homotopy fibres of \(\Omega ^{3}\pi {\circ } \partial _{k_{1},\ldots ,k_{n}}\) and \(\Omega ^{3}\pi {\circ } \partial _{k_{1},\ldots ,k_{n}}\) respectively are \(\mathscr {G}_{k_{1},\ldots ,k_{n}}\) and \(\mathscr {G}_{l_{1},\ldots ,l_{n}}\). From (2.3) there is an induced map of homotopy fibres \(\mathscr {G}_{k_{1},\ldots ,k_{n}}\!\rightarrow \mathscr {G}_{l_{1},\ldots ,l_{n}}\). As h is a homotopy equivalence, this induced map of homotopy fibres is also a homotopy equivalence, completing the proof. \(\square \)

Let \(\tilde{\epsilon }_i:S^3\rightarrow (S^3)^n\) be the i-th inclusion for \(1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}n\). Let \(\widetilde{\mathscr {G}}_{k_1,\ldots ,k_n}\) be the gauge group of a principal \((S^3)^n\)-bundle over \(S^4\) corresponding to \(k_1\tilde{\epsilon }_1+\cdots +k_n\tilde{\epsilon }_n\). Then there is an isomorphism

$$\begin{aligned} \widetilde{\mathscr {G}}_{k_1,\ldots ,k_n}\!\cong \widetilde{\mathscr {G}}_{k_1}{\times } \cdots {\times } \widetilde{\mathscr {G}}_{k_n}, \end{aligned}$$

where each \(\widetilde{\mathscr {G}}_{k_{i}}\) is the gauge group of the principal \(S^{3}\) bundle over \(S^{4}\) classified by \(k_{i}\in \mathbb {Z}\cong \pi _{4}(BS^{3})\).

On the other hand, there is a two-sheeted covering \(\widetilde{\mathscr {G}}_{k_1,\ldots ,k_n}\!\rightarrow \mathscr {G}_{k_1,\ldots ,k_n}\). Therefore for all \(m\hbox {\,\,\char 062\,\,}2\) there is an isomorphism

$$\begin{aligned} \pi _{m}(\mathscr {G}_{k_{1},\ldots ,k_{n}})\cong \bigoplus _{i=1}^{n}\,\pi _{m}(\widetilde{\mathscr {G}}_{k_{i}}). \end{aligned}$$
(2.4)

In particular, by [9], \(\pi _2(\widetilde{\mathscr {G}}_k)\cong \mathbb {Z}/\frac{12}{(k,12)}\), so we obtain the following.

Lemma 2.7

\(\pi _2(\mathscr {G}_{k_1,\ldots ,k_n})\cong \mathbb {Z}/\frac{12}{(k_1,12)}{\oplus }\cdots {\oplus }\mathbb {Z}/\frac{12}{(k_n,12)}.\)

Observe that the isomorphism in Lemma 2.7 does not imply that there is an equality of multisets \(\{\{(k_1,12),\ldots ,(k_n,12)\}\}=\{\{(l_1,12),\ldots ,(l_n,12)\}\}\). For example, if \(n=2\) then \(\mathbb {Z}/12{\oplus }\mathbb {Z}/1\cong \mathbb {Z}/3{\oplus }\mathbb {Z}/4\) but \(\{\{(1,12),(12,12)\}\}\ne \{\{(4,12),(3,12)\}\}\). However, if we work one prime at a time we do get an equality of multisets.

Proposition 2.8

If   \(\mathscr {G}_{k_{1},\ldots ,k_{n}}\!\simeq \mathscr {G}_{l_{1},\ldots ,l_{n}}\) then we obtain \(\{\{(k_1,4),\ldots ,(k_n,4)\}\}=\{\{(l_1,4),\ldots ,(l_n,4)\}\}\) and \(\{\{(k_1,3),\ldots ,(k_n,3)\}\}=\{\{(l_1,3),\ldots ,(l_n,3)\}\}\).

Proof

By Lemma 2.7, the homotopy equivalence \(\mathscr {G}_{k_{1},\ldots ,k_{n}}\!\simeq \mathscr {G}_{l_{1},\ldots ,l_{n}}\) implies that there is an isomorphism of groups

$$\begin{aligned} \mathbb {Z}\big /\frac{12}{(k_1,12)}{\oplus }\cdots {\oplus }\mathbb {Z}\big /\frac{12}{(k_n,12)}\cong \mathbb {Z}\big /\frac{12}{(l_1,12)}{\oplus }\cdots {\oplus }\mathbb {Z}\big /\frac{12}{(l_n,12)}. \end{aligned}$$
(2.5)

For a prime p, let \(\mathbb {Z}_{(p)}\) be the integers localized at p. Tensoring the isomorphism (2.5) with \(\mathbb {Z}_{(2)}\) gives a group isomorphism

$$\begin{aligned} \mathbb {Z}\big /\frac{4}{(k_1,4)}{\oplus }\cdots {\oplus }\mathbb {Z}\big /\frac{4}{(k_n,4)}\cong \mathbb {Z}\big /\frac{4}{(l_1,4)}{\oplus }\cdots {\oplus }\mathbb {Z}\big /\frac{4}{(l_n,4)}. \end{aligned}$$

The groups \(\mathbb {Z}/1\), \(\mathbb {Z}/2\) and \(\mathbb {Z}/4\) appearing on either side of this isomorphism are indecomposable, so each side must have the same number of generators of each order. Hence \(\{\{(k_1,4),\ldots ,(k_n,4)\}\}=\{\{(l_1,4),\ldots ,(l_n,4)\}\}\). The same argument applies if (2.5) is tensored with \(\mathbb {Z}_{(3)}\). \(\square \)

Finally, we prove Theorems 1.1 and 1.2.

Proof of Theorem 1.1

Combine Propositions 2.6 and 2.8. \(\square \)

Proof of Theorem 1.2

Suppose there is a p-local homotopy equivalence \((\mathscr {G}_{k_1,\ldots ,k_n})_{(p)} \simeq (\mathscr {G}_{l_1,\ldots ,l_n})_{(p)}\). Notice that \(\nu _{2}((k_{i},12))=(k_{i},4)\) and \(\nu _{3}((k_{i},12))=(k_{i},3)\), so Proposition 2.8 proves the \(p=2\) and \(p=3\) cases. If \(p\hbox {\,\,\char 062\,\,}5\) then \(\nu _{p}((k_{i},12))=1\) and \(\nu _{p}((l_{i},12))=1\) for all \(1\hbox {\,\,\char 054\,\,}i\hbox {\,\,\char 054\,\,}n\), so the asserted equality of multisets holds. The converse is proved using the same argument as for Proposition 2.6; in fact, it is easier since we need only consider the factor \(\Omega ^{3}_{0} S^{3}_{(p)}\) of \(\Omega ^{3}_{0} S^{3}=\prod _{p\in P}\Omega ^{3}_{0} S^{3}_{(p)}\). \(\square \)

Remark 2.9

Some generalization is possible. Let G be a simply-connected, simple compact Lie group with centre Z(G). Let \(L_{n}=G^{n}/Z\) where Z is the subgroup generated by the image of the diagonal map \(Z(G)\rightarrow Z(G)^{n}\). Replacing \(K_{n}\) with \(L_{n}\), the material in Sect. 2 through to Proposition 2.4 generalizes, where the order of \(\lambda _{i}=\langle \overline{\epsilon }_{i},\pi _{i}\rangle \) may no longer be 12 but it is a fixed number M for all \(1\hbox {\,\,\char 054\,\,}k\hbox {\,\,\char 054\,\,}n\). The proof of Proposition 2.6 leading to (2.1) holds, giving the statement that if \(\{\{(k_1,M),\ldots ,(k_n,M)\}\}=\{\{(l_1,M),\ldots ,(l_n,M)\}\}\) then \(\mathscr {G}_{k_1,\ldots ,k_n}\) and \(\mathscr {G}_{l_1,\ldots ,l_n}\) are p-locally homotopy equivalent for each prime p. However, as \(\Omega ^{3}_{0} G\) may not only have torsion homotopy groups, the argument for (2.2) leading to an integral homotopy equivalence of gauge groups will not hold. Further, the converse statement in Proposition 2.8 requires the homotopy type of \(\widetilde{\mathscr {G}}_{k}\) to be determined by a homotopy set that depends on (k, M), which is not known to hold in general.