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

We examine the relation between the gauge groups of $\mathrm{SU}(n)$- and $\mathrm{PU}(n)$-bundles over $S^{2i}$, with $2\leq i\leq n$, particularly when $n$ is a prime. As special cases, for $\mathrm{PU}(5)$-bundles over $S^4$, we show that there is a rational or $p$-local equivalence $\mathcal{G}_k\simeq_{(p)}\mathcal{G}_l$ for any prime $p$ if, and only if, $(120,k)=(120,l)$, while for $\mathrm{PU}(3)$-bundles over $S^6$ there is an integral equivalence $\mathcal{G}_k\simeq\mathcal{G}_l$ if, and only if, $(120,k)=(120,l)$.


I
Let G be a topological group and X a space. e gauge group 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 [8,16]. Provided P is understood, we will use the notation G(G) if we need to emphasize the structure group G.
e following problem is of interest: having fixed a topological group G and a space X , classify the possible homotopy types of the gauge groups G(P) of principal G-bundles over X .
Crabb and Sutherland showed in [3] that if G is a compact, connected, Lie group and X is a connected, finite complex, then the number of distinct homotopy types of G(P), as P ranges over all principal G-bundles over X , is finite.
is 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. e first result of this kind was obtained by Kono [12] in 1991. Using the fact that principal SU(2)-bundles over S 4 are classified by k ∈ π 3 (SU(2)) and denoting by G k the gauge group of the principal bundle P k → S 4 corresponding to the integer k, Kono showed that there is a homotopy equivalence G k ≃ 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 SU(2)gauge groups over S 4 .
In this le er, we generalise certain results relating the classification of PU(n)gauge groups to that of SU(n)-gauge groups from [10] for the case n = 2, and from [7] for the case n = 3.
Samelson products play a crucial role in the homotopy classification of gauge groups. In Section 3, we show that the Samelson products on SU(p) and PU(p), with p ≥ 3 a prime, are related as follows. eorem 1.1. e orders of the Samelson products ϵ i , 1 : In Section 4, we prove the follwing result, which gives a sufficient condition for certain homotopy invariants of SU(n)-and PU(n)-gauge groups to coincide.
As special cases of our results, we obtain the following complete classifications. (5)), then (120, k) = (120, l); (b) if (120, k) = (120, l), then G k (PU(5)) ≃ G l (PU(5)) when localised rationally or at any prime.  It is well known that there is a homotopy equivalence PU(n) ≃ PSU(n) for each n, and henceforth we shall not distinguish between the two.
For each 2 ≤ i ≤ n, the set of isomorphism classes of principal PU(n)-bundles over S 2i is in bijection with the set .
Let ϵ i : S 2i−1 → PU(n) denote a generator of π 2i−1 (PU(n)). en, each isomorphism class is represented by the bundle P k → S 2i induced by pulling back the universal PU(n)-bundle along the classifying map kϵ i : S 2i → BPU(n), where ϵ i denotes the adjoint of ϵ i and generates π 2i (BPU(n)).
Let G k denote the gauge group of P k → S 2i . By [1,4], there is a homotopy equivalence BG k ≃ Map k (S 2i , BPU(n)), the la er space being the connected component of Map(S 2i , BPU(n)) containing the classifying map kϵ i . e evaluation fibration extends to a homotopy fibration sequence By [17], there is a homotopy equivalence leading to the following homotopy fibration sequence which exhibits the gauge group G k as the homotopy fibre of the map ∂ k . By the pointed exponential law, there is a bijection where 1 denotes the identity map on PU(n). By the bilinearity of the Samelson product, we find and hence, taking adjoints once more, ∂ k ≃ k∂ 1 .
us, every one of the gauge groups is the homotopy fibre of the single map ∂ 1 post-composed with the appropriate power map on Ω 2i−1 0 PU(n). Hence, if ∂ 1 can be determined to have finite order in [PU(n), Ω 2i−1 0 PU(n)], then the following lemma applies. [18]). Let X be a connected CW-complex and let Y be an Hspace with a homotopy inverse. Suppose that f ∈ [X , Y ] has finite order n. en, for any k, l ∈ such that (n, k) = (n, l), the homotopy fibres of k f and l f are homotopy equivalent when localised rationally or at any prime.
Additionally, in the special case of principal PU(n)-bundles over S 2n , as the homotopy groups of Ω 2n−1 0 PU(n) are all finite, the following stronger lemma applies. Lemma 2.3 (Hamanaka, Kono [5]). Let X be a connected CW-complex and let Y be an H-space such that π j (Y ) is finite for all j. Let f ∈ [X , Y ] be such that n f ≃ * for some finite n and let k, l ∈ satisfy (n, k) = (n, l). en, there exists a homotopy equivalence h : Y → Y satisfying hk f ≃ l f .
Note that the order of ∂ 1 coincides with the order of ϵ i , 1 .
In this section we wish to compare the orders of the Samelson products δ i , 1 and ϵ i , 1 on SU(n) and PU(n), respectively. First, observe that there is a commutative diagram and recall the following well known property of the map q.  Proof. Let p be a prime which does not divide n. en q is a p-local homotopy equivalence by Lemma 3.1, and hence the commutativity of (⋆) yields , so the p-primary components of the orders of δ i , 1 and ϵ i , 1 coincide.
Hence, when n is prime, the orders of δ i , 1 and ϵ i , 1 coincide up to at most their n-primary component. Proof. Recall that q : SU(n) → PU(n) fits into a homotopy fibration sequence Since /n is discrete, applying the functor Corollary 3.4. Let p be a prime. If p k divides the order of δ i , 1 for some k ≥ 1, then the order of ϵ i , 1 is at least p k .
Proof. If p k divides the order of δ i , 1 , then p k also divides the order of the composite q • δ i , 1 (p) by Lemma 3.3. It then follows, by the commutativity of (⋆), that the order of ϵ i , 1 (p) is at least p k .
Hence, the order of δ i , 1 certainly divides that of ϵ i , 1 . For the remainder of this section, we shall restrict to considering PU(n) when n is an odd prime p.
Since the universal cover of PU(p) is SU(p) and H * (SU(p); ) is torsion-free, by [11] we have the following decomposition of PU(p).
Lemma 3.5. ere is a p-local homotopy equivalence where L is the lens space S 2p−1 /( /p ).
Let α : L (p) → PU(p) (p) be the inclusion. en we can write the equivalence of Lemma 3.5 as where µ is the group multiplication in PU(p) (p) . We note that this composite is equal to the product , where pr j denotes the projection onto the jth factor. Lemma 3.6. With the above notation, the localised Samelson product is trivial if, and only if, each of ϵ i (p) , α and ϵ i , ϵ j (p) , for 2 ≤ j ≤ p − 1, are trivial.
Proof. By Lemmas 3.3 and 3.4 in [7], ϵ i , 1 (p) is trivial if, and only if, both ϵ i (p) , α and ϵ i , j ϵ j (p) are trivial. Applying the same lemmas to the second factor a further p − 3 times gives the statement.
Furthermore, we find: Proof. By [13], there is a p-local equivalence where P 2k+1 (p) is the mod-p Moore space given by the cofibre of the degree p map on the sphere S 2k . Note that, by extending the cofibre sequence to the right, we see that S 2i−2 ∧ P 2k+1 (p) ≃ P 2k+2i−1 (p). Hence, we have consists of elements of order at most p by the same argument as in Lemma 3.7.
On the other hand, by [15, eorem 7.1], the groups [P 2k+2i−1 (p), PU(p)] are annihilated by multiplication by p (since, for any m ≥ 3, the identity on P m (p) has order p), whence the statement. Lemmas 3.6 to 3.8 combine to give: Corollary 3.9. e order of the Samelson product We now have all the ingredients necessary to prove eorem 1.1.
Proof of eorem 1.1. Consider the following commutative diagram where ι : SU(p) → U(p) is the inclusion and η i := ι * (δ i ). By Bo [2], the map η i , η p−i−1 is non-trivial and p divides its order. Hence, the order of δ i , 1 (p) is at least p. e result now follows from Lemma 3.2 and Corollaries 3.4 and 3.9.

H PU(n)
e content of eorem 1.2 is a straightforward observation about how certain homotopy invariants of SU(n)-gauge groups relate to the corresponding invariants of PU(n)-gauge groups.
Applying now the functor [Σ 2 X , −] to the homotopy fibration sequence described in Section 2, as well as to its SU(n) analogue, yields the following commutative diagram where the two le most vertical maps are isomorphisms. e statement now follows from the five lemma.

C
We have shown, particularly with eorems 1.1 and 1.2, how the close relationship between the groups SU(p) and PU(p) is reflected in the homotopy properties of the corresponding gauge groups. Indeed, it is worth noting that, should further classifications of gauge groups of SU(p)-bundles over even-dimensional spheres be obtained, the aforementioned theorems would provide the corresponding results for PU(p)-gauge groups as immediate corollaries, provided the original results were arrived at via the standard methods.
We also note that in [7], the PU(3)-gauge group G k is shown to be homotopy equivalent to G k × S 1 , where G k is a space whose homotopy groups are all finite.
is allows the authors to apply Lemma 2.3 and obtain a classification result for G k that holds integrally. We expect the same result to apply more generally to gauge groups of PU(n)-bundles over S 2n−2 . However, there are currently no other cases, beside that of [7], in which such a result would be applicable.