The homotopy types of $U(n)$-gauge groups over lens spaces

We analyse the homotopy types of gauge groups for principal $U(n)$-bundles over lens spaces.

Our main result is the following. As will be shown, there are isomorphisms [L(p, q), BU (n)] ∼ = [P 2 (p), BU (n)] ∼ = Z/pZ. For k ∈ Z/pZ, let G k (L(p, q)) and G k (P 2 (p)) be the gauge groups of the principal U (n)-bundles over L(p, q) and P 2 (p) respectively with first Chern class k. For integers a, b, let (a, b) be their greatest common divisor.

Isomorphism classes of bundles and components of mapping spaces
As a CW -complex L(p, q) ≃ P 2 (p) ∪ e 3 , so there is a homotopy cofibration where f attaches the top cell to L(p, q) and i is the inclusion of the 2-skeleton. Let π : P 2 (p) −→ S 2 be the pinch map to the top cell. Let j be the composite of inclusions j : S 1 −→ P 2 (p) i −→ L(p, q).
Then there is a homotopy cofibration diagram (1) that defines the space C and the maps g and h.
Lemma 2.1. The map π • f is null homotopic and there is a homotopy equivalence C ≃ S 2 ∨ S 3 .
Proof. The degree of π • f is detected by the Bockstein in the homology of C, but this Bockstein is zero since the corresponding Bockstein for L(p, q) is zero. Therefore π • f ≃ * , implying that Let π be the composite where the right map collapses S 3 to a point. Lemma 2.1 immediately implies the following.
If n = 1 then BU (1) is the Eilenberg-Mac Lane space K(Z, 2) and for any CW -complex X the set [X, BU (1)] has a group structure. If n > 1 then the standard inclusion BU (n) −→ BU (∞) has In particular, as BU (∞) is an infinite loop space, [X, BU (n)] has a group structure. In our case, each space in the homotopy cofibration for any n ≥ 1 we obtain exact sequence of groups Recall that [S 2 Lemma 2.3. Let n ≥ 1. The following hold: (a) there is a group isomorphism [P 2 (p), BU (n)] ∼ = Z/pZ; (b) the map π * is reduction mod-p; (c) there is a group isomorphism [L(p, q) (d) the map π * is reduction mod-p.
In general, if X is a pointed CW -complex then the isomorphism classes of principal U (n)-bundles over X are classified by the homotopy classes in [X, BU (n)]. If P is such a bundle, classified by a map α, let G α (X) be its gauge group. This group has a classifying space BG α (X) and by [G, AB] there is a homotopy equivlalence BG α (X) ≃ Map α (X, BU (n)), where Map α (X, BU (n)) is the component of the space of continuous maps from X to BU (n) that contains α. The subgroup G * α (X) of G-equivariant automorphisms of P that pointwise fix the fibre at the basepoint is the pointed gauge group. There is a corresponding component of the pointed mapping space, Map * α (X, BU (n)), and a homotopy equivalence BG * α (X) ≃ Map * α (X, BU (n)). Evaluation of maps at the basepoint gives a homotopy fibration sequence The homotopy fibre of the connecting map ∂ α is G α (X).
In our case, we have [S 2 , BU (n)] ∼ = Z and, by Lemma 2.3, [P 2 (p), BU (n)] ∼ = [L(p, q), BU (n)] ∼ = Z/pZ. Note that, for dimensional reasons, the principal U (n)-bundles over S 2 , P 2 (p) and L(p, q) are classified by the value of the first Chern class. Fork ∈ Z, let Gk(S 2 ) be the gauge group of the isomorphism class of principal U (n)-bundles over S 2 whose first Chern class isk. For k ∈ Z/pZ, let G k (P 2 (p)) and G k (L(p, q)) be the gauge groups of the isomorphism classes of principal U (n)bundles over P 2 (p) and L(p, q) respectively whose first Chern class is k. Lemma 2.3 implies that if k ≡ k mod p then there is a commutative diagram of fibration sequences The homotopy fibres of ∂k(S), ∂ L k and ∂ P k are Gk(S 2 ), G k (L(p, q)) and G k (P 2 (p)) respectively. The goal is to find information about the gauge groups G k (L(p, q)) via the middle homotopy fibration in (4). However, it is not so easy to study this fibration directly, one issue being that it is unclear whether the components Map * k (L(p, q), BU (n)) are all homotopy equivalent. A similar issue appeared in work of the first author [MS] in dealing with gauge groups for principal G-bundles over S 3 -bundles over S 4 , where G is a simply-connected, simple compact Lie group. The approach in that case involved localization, which needs to be avoided here since P 2 (p) need not be nilpotent.
Instead, we obtain information indirectly: by [S], information about Gk(S 2 ) via the top fibration in (4) is known and in Section 3 we will determine information about G k (P 2 (p)) via the bottom fibration in (4). In Section 4 a splitting result is proved that lets us use the information about G k (P 2 (p)) to deduce information about G k (L(p, q)).
3. The homotopy types of G k (P 2 (p)) By Corollary 2.2, the pinch map P 2 (p) (4) we obtain a homotopy commutative diagram of fibration sequences First, we show that all the components Map * k (P 2 (p), BU (n)) are homotopy equivalent, and in a way that is compatible with a similar result from [S] about the components Map * k (S 2 , BU (n)).
Proof. This was essentially proved in [S] but not stated in this form. An argument is given for the sake of completeness. Let ǫ : S 2 −→ BU (n) be a fixed map with first Chern class −k. Define by sending a map f : S 2 −→ BU (n) with first Chern classk to the composite where σ is the comultiplication on S 2 and ∇ is the folding map. Similarly, define by sending g to ∇ • (g ∨ (−ǫ)) • σ. Then θ and φ are continuous and the homotopy associativity of σ implies that φ • θ and θ • φ are homotopic to the identity maps.
The space P 2 (p) is not a co-H-space. However, as π is a homotopy cofibration connecting map there is a coaction ψ : P 2 (p) −→ P 2 (p) ∨ S 2 which, when pinched to P 2 (p) is the identity map, and when pinched to S 2 is π. Further, this coaction has a homotopy associativity property: by sending a map f ′ : P 2 (p) −→ BU (n) with first Chern class k to the composite and define φ ′ : Map * 0 (P 2 (p), BU (n)) −→ Map * k (P 2 (p), BU (n)) by sending g to ∇ • (g ∨ (−ǫ)) • ψ. Then, as before, θ ′ is a homotopy equivalence.
Finally, the coaction ψ satisfies a homotopy commutative diagram This implies that θ and θ ′ , and φ and φ ′ , are compatible, implying the homotopy commutative diagram asserted by the lemma.
Using ∂ S k to also denote the composite U (n) , and similarly for ∂ P k , by Lemma 3.1 the left square in (5) may be replaced with a homotopy commutative square By (5), the homotopy fibres of ∂ S k and ∂ P k are Gk(S 2 ) and G k (P 2 (p)) respectively. We next identify certain self-homotopy equivalences of Map * k (P 2 (p), BU (n)). Since P 2 (p) is not a co-H-space it is not immediately clear that it has a degree d map for any integer d. However, we may define an analogue as follows. The degree p map on S 1 commutes with the degree d map for any d, so we obtain a cofibration diagram (7) for some map d. The cofibration diagram implies that, upon taking integral homology, (d) * is multiplication by d on H 1 (P 2 (p); Z) ∼ = Z/pZ. Suspending, since ΣP n (p) ≃ P n+1 (p), we see that Consequently, we obtain the following.
Lemma 3.2. If d is a unit in Z/pZ then for any m ≥ 1 the map P m+2 (p) Proof. Since d is a unit in Z/pZ, the map d induces an isomorphism in H m+1 (P m+2 (p); Z) and hence induces an isomorphism on H * (P m+2 (p); Z). Since m ≥ 1, P m+2 (p) is simply-connected, so Whitehead's Theorem implies that d is a homotopy equivalence.

Lemma 3.3. If d is a unit in Z/pZ then d * is a homotopy equivalence.
Proof. The first step is to show that (d) * induces an isomorphism on homotopy groups. Note that Map * 0 (P 2 (p), BU (n)) is path-connected since it is one component of Map * (P 2 (p), BU (n)) and for all m ≥ 1 we have π m (Map * 0 (P 2 (p), BU (n)) ∼ = π m (Map * (P 2 (p), BU (n)). By the pointed Exponential Law, π m (Map * (P 2 (p), BU (n)) ∼ = Map * (P m+2 (p), BU (n)). The effect of d * on π m is therefore determined by applying Map * ( , BU (n)) to the map P m+2 (p) Σ m d −→ P m+2 (p). Since d is a unit in Z/pZ, by Lemma 3.2, Σ m d is a homotopy equivalence. Thus d * induces an isomorphism on π m .
As this is true for all m ≥ 1, d * is a weak homotopy equivalence.
Observe that P 2 (p) is a CW -complex and BU (n) may be given the structure of a CW -complex.
Recall that, by elementary number theory, if (u, n) = 1 then u is a unit mod n, and if (k, n) = (ℓ, n) then k ≡ uℓ mod n for some integer u satisfying (u, n) = 1.
Lemma 3.4. Suppose that (k, n) = (ℓ, n), implying that k ≡ uℓ mod n for some integer satisfying (u, n) = 1. Then there is a homotopy commutative diagram Proof. In [Th], refining work in [S], it was shown that U (n) ∂ S 1 −→ Ω 0 U (n) has order n. Therefore, ∂ S 1 generates a cyclic subgroup of order n in the group [U (n), Ω 0 U (n)]. By [L], Lemma 3.5. Suppose that (k, n) = (ℓ, n), implying that k ≡ uℓ mod n for some integer satisfying (u, n) = 1. Suppose that (u, p) = 1 as well. Then there is a homotopy commutative diagram where u * is a homotopy equivalence.
Proof. Consider the diagram The left square homotopy commutes by Lemma 3.4 and the right square homotopy commutes by applying Map * ( , BU (n)) to the right square in (7). Observe that the composite along the top row is ∂ P ℓ and the composite along the bottom row is ∂ P k . Thus u * • ∂ P ℓ ≃ ∂ P k . Finally, the hypothesis (u, p) = 1 implies that u is a unit mod-p, so by Lemma 3.3, u * is a homotopy equivalence.
Note that the condition (u, n) = 1 and (u, p) = 1 is equivalent to the condition (u, np) = 1.
Proof. The homotopy fibres of ∂ P k and ∂ P ℓ are G k (P 2 (p)) and G ℓ (P 2 (p)) respectively. Taking homotopy fibres in the homotopy commutative diagram in the statement of Lemma 3.5 gives an induced map of fibres ϕ : G ℓ (P 2 (p)) −→ G k (P 2 (p)). The fact that u * is a homotopy equivalence implies that ϕ is a homotopy equivalence.
There is a partial converse to Proposition 3.6 in a limited number of cases.
Sutherland [S] showed that the image of (∂ S k ) * is generated by (n − 1)!(n,k) γ. Let δ = π * (γ). Then the image of (∂ P k ) * is generated by (n − 1)!(n, k) δ. Since δ generates a cylic group of order (n!, p), the only way that (n − 1)!(n, k)δ is not going to be trivial is if (n!, p) has factors of n which are not factors of (n − 1)! or (n, k). The only way n has factors that are not factors of (n − 1)! is if n is a prime p. So from now on assume that n = p. This leaves two cases: (n, k) = (p, k) is 1 or p.
Observe that Proposition 3.6 and Lemma 3.7 both hold when n = p = p, noting in Proposition 3.6 that (u, np) = (u, p 2 ) = 1 implies that (u, p) = 1. Therefore there is a complete classification of gauge groups in this case.
Proposition 3.8. For a prime p, consider the gauge groups of principal U (p)-bundles over P 2 (p).

A homotopy decomposition for
In this section we prove Theorem 1.1 (a). Consider the homotopy cofibration sequence Lemma 4.1. There is a homotopy equivalence Σ 2 L(p, q) ≃ P 4 (p) ∨ S 5 .
Proof. Consider the homotopy cofibration In general, any closed, orientable 3-manifold M is parallelizable so by [A] it has the property that its top cell splits off stably. In our case, as L(p, q) is such a manifold, the attaching map f is stably trivial. As f is in the stable range after two suspensions, we have Σ 2 f null homotopic. This implies that there is a homotopy equivalence Σ 2 L(p, q) ≃ P 4 (p) ∨ S 5 .
Lemma 4.1 implies that the map Σ 2 L(p, q) Σ 2 g −→ S 5 has a right homotopy inverse. We wish to choose this right homotopy inverse carefully; the next lemma does this given a condition on p. Recall from Lemma 2.1 that there is a homotopy cofibration S 1 j −→ L(p, q) h −→ S 2 ∨ S 3 and g is homotopic to h composed with the map collapsing S 2 to a point.
Lemma 4.2. Suppose that 2 | p. Then the map Σ 2 L(p, q) Σ 2 g −→ S 5 has a right homotopy inverse r : S 5 −→ Σ 2 L(p, q) with the property that the composite S 5 r −→ Σ 2 L(p, q) to the inclusion of the right wedge summand.
Proof. Let F be the homotopy fibre of P 3 (p) Σπ −→ S 3 and consider the homology Serre exact sequence for the homotopy fibration ΩS 3 −→ F −→ P 3 (p). This is an exact sequence We claim that Σλ ′′ is null homotopic, implying that ΣD ≃ S 3 ∨ S 5 . If so, let r be the composite r : S 5 ֒→ ΣD −→ ΣG −→ ΣL(p, q). Observe that as G −→ L(p, q) Σπ −→ S 3 is a fibration, the composite Σ 2 π•r is null homotopic. Further, the Blakers-Massey Theorem implies that the homotopy fibration N −→ P 3 (p) Σi −→ ΣL(p, q) is the same as the homotopy cofibration S 3 Σf −→ P 3 (p) Σi −→ ΣL(p, q) in dimensions ≤ 3, and λ is the inclusion of the bottom cell. A homology argument then shows that the composite S 5 r −→ Σ 2 L(p, q) Σ 2 g −→ S 5 is homotopic to the identity map. This proves the second assertion of the lemma.