Homotopy types of $SU(n)$-gauge groups over non-spin 4-manifolds

Let $M$ be an orientable, simply-connected, closed, non-spin 4-manifold and let $\mathcal{G}_k(M)$ be the gauge group of the principal $G$-bundle over $M$ with second Chern class $k\in\mathbb{Z}$. It is known that the homotopy type of $\mathcal{G}_k(M)$ is determined by the homotopy type of $\mathcal{G}_k(\mathbb{CP}^2)$. In this paper we investigate properties of $\mathcal{G}_k(\mathbb{CP}^2)$ when $G = SU(n)$ that partly classify the homotopy types of the gauge groups.


Introduction
Let G be a simple, simply-connected, compact Lie group and let M be an orientable, simply-connected, closed 4-manifold. Then the isomorophism class of a principal Gbundle P over M is classified by its second Chern class k ∈ Z. In particular, if k = 0, then P is a trivial G-bundle. The associated gauge group G k (M) is the topological group of G-equivariant automorphisms of P which fix M.
A simply-connected 4-manifold is spin if and only if its intersection form is even. In the case of simply-connected 4-manifolds, the spin condition is equivalent to all cup product squares being trivial in mod 2 cohomology. In this paper, we consider the homotopy types of gauge groups G k (M), where M is a non-spin 4-manifold such as CP 2 . When M is a spin 4-manifold, topologists have been studying the homotopy types of gauge groups over M extensively over the last twenty years. On the one hand, Theriault showed in [16] that there is a homotopy equivalence where d is the second Betti number of M. Therefore to study the homotopy type of G k (M) it suffices to study G k (S 4 ). On the other hand, many cases of homotopy types of G k (S 4 )'s are known. For examples, there are 6 distinct homotopy types of G k (S 4 )'s for G = SU (2) [11], and 8 distinct homotopy types for G = SU (3) [5]. When localized rationally or at any prime, there are 16 distinct homotopy types for G = SU (5) [19] and 8 distinct homotopy types for G = Sp (2) [17].
When M is a non-spin 4-manifold, the author in [14] showed that there is a homotopy equivalence so the homotopy type of G k (M) depends on the special case G k (CP 2 ). Compared to the extensive work on G k (S 4 ), only two cases of G k (CP 2 ) have been studied, which are the SU (2)and SU (3)-cases [12,18]. As a sequel to [14], this paper investigates the homotopy types of G k (CP 2 )'s in order to explore gauge groups over non-spin 4manifolds.
A common approach to classifying the homotopy types of gauge groups is as follows. Atiyah, Bott and Gottlieb [1,3] showed that the classifying space where ∂ k : G → Map   (Theriault,[17]) Let m be the order of ∂ 1 . If (m, k) = (m, l), then G k (S 4 ) is homotopy equivalent to G l (S 4 ) when localized rationally or at any prime.
For most cases of G, the exact value of the order of ∂ 1 is difficult to compute. When G = SU (n), the exact value or a partial result of the order of ∂ 1 was worked out for certain cases. For any number a = p r q where q is coprime to p, the p-component of a is p r and is denoted by ν p (a). Theorem 1.2 ([2,5,9,11,19,20]) Let G be SU (n) and let m be the order of ∂ 1 . Then • m = 12 for n = 2 • m = 24 for n = 3 • m = 120 for n = 5 • m = 60 or 120 for n = 4 • ν p (m) = ν p (n(n 2 − 1)) for n < (p − 1) 2 + 1.

Theorem 1.3 (Hamanaka and Kono
In this paper we consider gauge groups over CP 2 . Take M = CP 2 in (1) and denote the boundary map by ∂ k : is not a group so the order of ∂ k makes no sense. However, we can still define an "order" of ∂ k [18], which will be described in Sect. 2. We show that the "order" of ∂ 1 helps distinguish the homotopy type of G k (CP 2 ) as in Theorem 1.1.

Theorem 1.4 Let m be the "order
when localized rationally or at any prime.
We study the SU (n)-gauge groups over CP 2 and use unstable K -theory to give a lower bound on the "order" of ∂ 1 that is in the spirit of Theorem 1.2. Theorem 1.5 When G is SU (n), the "order" of ∂ 1 is at least 1 2 n(n 2 − 1) for n odd, and n(n 2 − 1) for n even.
Localized rationally or at an odd prime, we have G k (CP 2 ) G k (S 4 ) × 2 G [16]. The homotopy types of G k (CP 2 ) are then completely determined by that of G k (S 4 ), which have been investigated in many cases when the localizing prime is relatively large [6,7,9,10,20]. A large part of the remaining cases can be understood by studying the 2-localized order of ∂ 1 , on which Theorem 1.5 gives bounds for the SU (n) case. For example, combining Theorem 1.5 with Lemma 2.2 implies the order of ∂ 1 is either 120 or 60 for G = SU (5). Furthermore, when G = SU (4) since the order of ∂ 1 is either 120 or 60, the order of ∂ 1 is either 60 or 120.
Finally we prove a necessary condition for the homotopy equivalence G k (CP 2 ) G l (CP 2 ) similar to Theorem 1.3.
The author would like to thank his supervisor, Professor Stephen Theriault, for his guidance in writing this paper, and thank the Referee for his careful reading and useful comments.

Some facts about boundary map @ 1
Take M to be S 4 and CP 2 respectively in fibration (1) to obtain fibration sequences There is also a cofibration sequence where η is Hopf map and q is the quotient map. Due to the naturality of q * , we combine fibrations (2) and (3) to obtain a commutative diagram of fibration sequences It is known, [13], that ∂ k is triple adjoint to Samelson product where ı : S 3 → SU (n) is the inclusion of the bottom cell and 1, 1 is the Samelson product of the identity on G with itself. The order of ∂ k is its multiplicative order in the group [G, 3 0 G].
is not an H-space, so ∂ k has no order. In [18], Theriault defined the "order" of ∂ k to be the smallest number m such that the composition is null homotopic. In the following, we interpret the "order" of ∂ k as its multiplicative order in a group contained in [CP 2 ∧ G, BG]. Apply [− ∧ G, BG] to cofibration (4) to obtain an exact sequence of sets All terms except [CP 2 ∧ G, BG] are groups and ( η) * is a group homomorphism since η is a suspension. We want to refine this exact sequence so that the last term is replaced by a group. Observe that CP 2 is the cofiber of η and so there is a coaction ψ : CP 2 → CP 2 ∨ S 4 . We show that the coaction gives a group structure on I m(q * ).

be a cofibration sequence. If A is homotopy cocommutative, then I m(h * ) is an abelian group and
is an exact sequence of groups and group homomorphisms.
Proof Apply [−, Y ] to the cofibration to get an exact sequence of sets To check this is well-defined we need to show where σ : A → A ∨ A is the comultiplication and : Y ∨ Y → Y is the folding map. Since C is a cofiber, there is a coaction ψ : Then we obtain a string of equivalences The third line is due to the assumption h * α h * α . Therefore we have h * (α + β) This implies is well-defined. Due to the associativity of + in [ A, Y ], is associative since Clearly the trivial map * : C → Y is the identity of and h * (−α) is the inverse of h * α. Therefore is indeed a group multiplication. (6) to obtain a sequence of groups and group homomorphisms The exactness of (6) implies ker(h * ) = I m( f ) * , so the sequence is exact.
Applying Lemma 2.1 to cofibration 3 G → 2 G → CP 2 ∧ G and the space Y = BG, we obtain an exact sequence of abelian groups In the middle square of (5) ∂ k q * ∂ k , so ∂ k is in I m(q * ). For any number m, q * (m∂ k ) = mq * ∂ k , so the "order" of ∂ k defined in [18] coincides with the multiplicative order of ∂ k in I m(q * ). The exact sequence (7) allows us to compare the orders of ∂ 1 and ∂ 1 .

Lemma 2.2 Let m be the order of ∂ 1 and let m be the order of ∂ 1 . Then m is m or 2m .
Proof By exactness of (7), there is some f ∈ [ 3 G, BG] such that ( η) * f m ∂ 1 . Since η has order 2, 2m ∂ 1 is null homotopic. It follows that 2m is a multiple of m. Since m is greater than or equal to m , m is either m or 2m .
In the S 4 case, Theorem 1.1 gives a sufficient condition for G k (S 4 ) G l (S 4 ) when localized rationally or at any prime. In the CP 2 case, Theriault showed a similar counting statement, in which the sufficient condition depends on the order of ∂ 1 instead of ∂ 1 .

Theorem 2.3 (Theriault, [18]) Let m be the order of
is homotopy equivalent to G l (CP 2 ) when localized rationally or at any prime.
Lemma 2.2 can be used to improve the sufficient condition of Theorem 2.3.

Theorem 2.4 Let m be the order of
when localized rationally or at any prime.
Proof By Lemma 2.2, m is either m or 2m . If m = m , then the statement is same as Theorem 2.3. Assume m = 2m . Localize at an odd prime p. Let p r be the p- G l (CP 2 ) follows by Theorem 2.3. A similar argument works for rational localization. Now it remains to consider the case where m = 2m when localized at 2.
Assume m = 2 n and m = 2 n−1 . For any k, has the same homotopy type for both (2 n If (2 n , k) = 2 n , then k = 2 n t for some number t. By linearity of Samelson products, ∂ k k∂ 1 . Since ∂ k q * k∂ 1 q * 2 n t∂ 1 and ∂ 1 has order 2 n , ∂ k is null homotopic and we have If (2 n , k) = 2 n−1 , then k = 2 n−1 t for some odd number t. Writing t = 2s + 1 gives k = 2 n s + 2 n−1 . Since ∂ k q * k∂ 1 q * (2 n s + 2 n−1 )∂ 1 q * 2 n−1 ∂ 1 and ∂ 1 has order 2 n−1 , ∂ k is null homotopic and we have The same is true for G l (CP 2 ) and hence G k (CP 2 ) G l (CP 2 ).

Plan for the proofs of Theorems 1.5 and 1.6
From this section onward, we will focus on SU (n)-gauge groups over CP 2 . There is a fibration where p : SU (∞) → W n is the projection and W n is the symmetric space SU (∞)/SU (n). Then we havẽ where x 2n+1 has degree 2n +1, c i is the ith universal Chern class and where a 2i is the transgression of x 2i+1 .
The (2n + 4)-skeleton of W n is 2n−1 CP 2 for n odd, and is S 2n+3 ∨ S 2n+1 for n even, so its homotopy groups are as follows: The canonical map : of the generating set. Let C be the quotient CP n−1 /CP n−3 and letq : CP n−1 → C be the quotient map. Then there is a diagram where (∂ k ) * sends f to ∂ k • f and the rows are induced by fibration (3). In particular, in the second row the map : In Sect. 4, we use unstable K -theory to calculate the order of ∂ 1 • , giving a lower bound on the order of ∂ 1 . Furthermore, in [5] Hamanaka and Kono considered an exact sequence similar to the first row to give a necessary condition for G k (S 4 ) G l (S 4 ). In Sect. 5 we follow the same approach and use the first row to give a necessary condition for We remark that it is difficult to use only one of the two rows to prove both Theorems 1.5 and 1.6. On the one hand, ∂ 1 • factors through a map∂ : C → Map * (CP 2 , B SU (n)). There is no obvious method to show that∂ and ∂ 1 • have the same orders except direct calculation. Therefore we cannot compare the orders of ∂ and ∂ 1 to prove Theorem 1.5 without calculating the order of ∂ 1 • . On the other hand, applying the method used in Sect. 5 to the second row gives a much weaker conclusion than Theorem 1.6. This is because [ C, BG k (CP 2 )] is a much smaller group than [ CP n−1 , BG k (CP 2 )] and much information is lost by the mapq * .

A lower bound on the order of @ 1
The restriction of ∂ 1 to CP n−1 is ∂ 1 • , which is the triple adjoint of the composition Since SU (n) B SU (n), we can further take its adjoint and get ρ : where [ev, ev] is the Whitehead product of the evaluation map with itself. Similarly, the restriction ∂ 1 • is adjoint to the composition Since we will frequently refer to the facts established in [4,5], it is easier to follow their setting and consider its adjoint where T : CP 2 ∧ CP n−1 → CP 2 ∧ CP n−1 is the swapping map and τ : [ is the adjunction. By adjunction, the orders of ∂ 1 • , ρ and γ are the same. We will calculate the order of γ using unstable K -theory to prove Theorem 1.5. Apply [CP 2 ∧ CP n−1 , −] to fibration (8) to obtain the exact sequencẽ Since CP 2 ∧ CP n−1 is a CW-complex with even dimensional cells, the last item We refer to Table (9) freely for the homotopy groups of W n . Since π 2n−1 (W n ) and π 2n (W n ) are zero, Since π 2n (W n ) and π 2n+2 (W n ) are zero for n odd, so is Third, apply [ 2n−2 −, W n ] to (4) to obtain where η 1 and η 2 are induced by Hopf maps 2n η : S 2n+3 → S 2n+2 and 2n−1 η : S 2n+2 → S 2n+1 , and j is induced by the inclusion S 2n+1 → 2n−2 CP 2 of the bottom cell. When n is odd, π 2n+2 (W n ) is zero and π 2n+1 (W n ) and π 2n+3 (W n ) are Z, so [ 2n−2 CP n−1 , W n ] is Z ⊕ Z. When n is even, the (2n + 4)-skeleton of W n is S 2n+1 ∨ S 2n+3 . The inclusions generate π 2n+1 (W n ) and the Z-summand of π 2n+3 (W n ), and the compositions generate π 2n+2 (W n ) and the Z/2Z-summand of π 2n+3 (W n ) respectively. Since η 1 sends j 1 to j 2 , the cokernel of η 1 is Z. Similarly, η 2 sends i 1 to j 1 , so η 2 : Z → Z/2Z is surjective. This implies the preimage of j is a Z-summand. Therefore

Lemma 4.2 The group [CP
When n is odd, C is 2n−6 CP 2 . Apply [ 2n−6 CP 2 ∧ −, W n ] to cofibration (4) to obtain the exact sequence The cohomology classx 2n+1 represents a mapx 2n+1 : W n → K (Z, 2n + 1) and a 2n = σ (x 2n+1 ) represents its loop x 2n+1 : W n → K (Z, 2n + 1). Similarly a 2n+2 = σ (x 2n+3 ) represents a loop map. This implies a is a group homomorphism. Furthermore, a 2n and a 2n+2 induce isomorphisms between H i ( W n ) and H i (K (2n, Z) × K (2n + 2, Z)) for i = 2n and 2n + 2. Since [CP 2 ∧ CP n−1 , W n ] is a free Z-module by Lemma 4.2, a is a monomorphism. Consider the diagram In the left square, is defined to be a • p * . In the right square, ψ is the quotient map and b is defined as follows. Any f ∈ [CP 2 ∧ CP n−1 , SU (n)] has a preimagef and b( f ) is defined to be ψ(a(f )). An easy diagram chase shows that b is well-defined and injective. Since b is injective, the order of γ ∈ [CP 2 ∧ CP n−1 , SU (n)] equals the order of b(γ ) ∈ Coker( ). In [4], Hamanaka and Kono gave an explicit formula for .

Theorem 4.3 (Hamanaka and Kono [4]) Let Y be a CW-complex. For any f ∈K 0 (Y )
we have where ch 2i ( f ) is the 2ith part of ch( f ).

The difference
is even since each term ( j − 1) j n−2 is even and n ≥ 3. Therefore A i −B i 2 is an integer and (L i ) is a linear combination of (L 1 ) and (n(n − 1), 0, 0). Furthermore, For n ≥ 4, since 2 n−3 and 3 n−2 − 3 · 2 n−3 are coprime to each other, there exist integers s and t such that 2 n−3 s + (3 n−2 − 3 · 2 n−3 )t = 1 and Therefore Back to diagram (10). The map γ has a liftγ : CP 2 ∧ CP n−1 → W n . By exactness, the order of γ equals the minimum number m such that mγ is contained in I m( p * ). Since a and b are injective, the order of γ equals the minimum number m such that m a(γ ) is contained in I m( ).

Lemma 4.5
Let α : X → SU (n) be a map for some space X . If α : CP 2 ∧ X → SU (n) is the adjoint of the composition then there is a liftα of α such thatα * (a 2i ) = u 2 ⊗ −1 α * (x 2i−3 ), where is the cohomology suspension isomorphism.
Proof In [4,5], Hamanaka and Kono constructed a lift : Let˜ be the compositioñ where u 3 is the generator of H 3 (S 3 ). Let T : CP 2 ∧ X → CP 2 ∧ X be the swapping map and let τ : [ CP 2 ∧ X , W n ] → [CP 2 ∧ X , W n ] be the adjunction. Takeα : CP 2 ∧ X → W n to be the adjoint of˜ , that isα = τ (˜ • T ). Thenα is a lift of α . Since
Proof Recall that γ is the adjoint of the composition Now we use Lemma 4.5 and take α to be : CP n−1 → SU (n). Then γ has a liftγ such thatγ * (a 2i ) Now we can calculate the order of ∂ 1 • , which gives a lower bound on the order of ∂ 1 .
Since s = −2r is even, the smallest positive value of t satisfying s = t(n − 1) is 1 for n odd and 2 for n even. Therefore m is 1 2 n(n 2 − 1) for n odd and n(n 2 − 1) for n even.
For SU (n)-gauge groups over S 4 , the order m of ∂ 1 has the form m = n(n 2 − 1) for n = 3 and 5 [5,19]. If p is an odd prime and n < (p − 1) 2 + 1, then m and n(n 2 − 1) have the same p-components [9,20]. These facts suggest it may be the case that m = n(n 2 − 1) for any n > 2. In fact, one can follow the method Hamanaka and Kono used in [5] and calculate the order of ∂ • to obtain a lower bound n(n 2 − 1) for n odd. However, it does not work for the n even case since [S 4 ∧ CP n−1 , W n ] is not a free Z-module. An interesting corollary of Theorem 4.7 is to give a lower bound on the order of ∂ 1 for n even. Corollary 4.8 When n is even and greater than 2, the order of ∂ 1 is at least n(n 2 − 1).

Proof
The order of ∂ 1 • is a lower bound on the order of ∂ 1 , which is either the same as or half of the order of ∂ 1 by Lemma 2.2. The corollary follows from Theorem 4.7.

A necessary condition for G k (CP 2 ) G l (CP 2 )
In this section we follow the approach in [5] to prove Theorem 1.6. The techniques used are similar to that in Sect. 4, except we are working with the quotient C = CP n−1 / CP n−1 instead of CP n−1 . When n is odd, C is 2n−6 CP 2 , and when n is even, C is S 2n−2 ∨ S 2n−4 . Apply [ C, −] to fibration (3) to obtain the exact sequence Proof When n is odd, C is 2n−6 CP 2 and the (2n +3)-skeleton of W n is 2n−2 CP 2 . To say (x, y, z) ∈ I m(a) means there exists f ∈ [CP 2 ∧ C, W n ] such that f * (a 2n ) = xu 2 v 2n−4 + yuv 2n−2 and f * (a 2n+2 ) = zu 2 v 2n−2 .

Lemma 5.3 When n is odd, the order of I m(∂ k ) * is
.
Proof Suppose n = 4m + 3 for some integer m. Then Transform M into Smith normal form The matrix B represents a basis change in I m(a) and A represents a basis change in I m( ). Therefore [CP 2 ∧ C, SU (n)] is isomorphic to We need to find the representation of∂ k under the new basis represented by B. The new coordinates ofα 1,k andα 2,k are the row vectors of the matrix Apply row operations to get Let μ = (k L, 0, −k L) and ν = ((4m + 2)k L, (2m + 1)k L, 0). Theñ If xμ + yν and x μ + y ν are the same modulo I m( ), then we have

T. So
We represent the coordinates of (L i )'s by the matrix Then I m( ) is spanned by the row vectors of M .
We represent their coordinates by a matrix Then the preimage∂ k = span{α 1,k ,α 2,k } of I m(∂ k ) * is spanned by the row vectors of M ∂ . We calculate as in the proof of Lemma 5.3 to obtain the following lemma.

Proof of Theorem 1.6
Before comparing the orders of I m(∂ k ) * and I m(∂ k ) * , we prove a preliminary lemma.

Lemma 5.5 Let n be an even number and let p be a prime. Denote the p-component of t by ν p (t).
If there are integers k and l such that ν p 1 2 n, k · ν p (n, k) = ν p 1 2 n, l · ν p (n, l), then ν p (n, k) = ν p (n, l).
Since s ≥ r , min(r − 1, s) = r − 1 and the right hand side is 2r − 1 which is odd. However, the left hand side is even, leading to a contradiction. This implies that this case does not satisfy the hypothesis. Case (4) is similar. Therefore ν 2 (n, k) = ν 2 (n, l) and the asserted statement follows.