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The order of the commutator on SU(3) and an application to gauge groups.

1 Introduction

Let G be an H-group, defined as a homotopy associative H-space with a homotopy inverse. Let [bar.c]: G x G [right arrow] G be the map defined pointwise by [bar.c] (x,y) = [xyx.sup.-1] [y.sup.-1]. Consider the cofibration sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Observe that [bar.c] is null homotopic when restricted to G [disjunction] G, implying that [bar.c] factors through a map c: G [conjunction] G [right arrow] G. Since [summation](G x G) [equivalent] ([summation]G [disjunction] [summation] G) [disjunction] ([summation]G [conjunction] G), the cofibration connecting map [delta] is null homotopic. Thus the homotopy class of c is uniquely determined by the homotopy class of [bar.c]. Call the map c the commutator of G.

If G is finite then, rationally, it is homotopy equivalent to a product of Eilenberg-MacLane spaces as an H-space, implying that it is homotopy commutative. So, rationally, c is trivial, which implies that the order of c is finite. A fundamental problem is to determine the order of c. However, this is known only in extremely simple cases. For example, consider the case of SU(n). If n = 2 then SU(2) [equivalent] [S.sup.3] and the order of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], a consequence of Toda's calculations [To]. On the other hand, if n > 2 then the order of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is unknown. It is not even clear what an upper bound should be. In this paper we consider the case of SU(3) and show the following.

Theorem 1.1. The commutator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has order 120 = [2.sup.3] x 3 x 5.

Theorem 1.1 can be used to help determine the homotopy types of certain gauge groups. In general, if G is a topological group, X is a space, and P [right arrow] X is a principal G-bundle, then the gauge group G (P) of the bundle is the group of G-equivariant automorphisms of P which fix X. Crabb and Sutherland [CS] showed that if G is a compact, connected Lie group and X is a connected, finite CW-complex, then while there may be infinitely many distinct principal G-bundles over X, their corresponding gauge groups have only finitely many distinct homotopy types. However, Crabb and Sutherland gave no upper bound on the number of distinct gauge groups, and precise numbers have been worked out only in a few cases of very low rank [K, HK, KKKT, Th]. We consider instead the p-local homotopy types for a prime p, and produce explicit upper bounds for the number of distinct p-local homotopy types of gauge groups in a more restricted setting. The possible existence of a kind of Zabrodsky mixing of homotopy types means that it is not really clear how our p-local result relates to Crabb and Sutherland's integral result. For a prime p and an integer m, let [V.sub.p] (m) be the largest integer r such that [p.sup.r] divides m but [p.sup.r+1] does not divide m.

Theorem 1.2. Let G be a compact, connected Lie group and let Y be a space. Fix a homotopy class [f] [member of] [[summation]Y, BG]. For an integer k, let [P.sub.k] [right arrow] [summation]Y be the principal G-bundle induced by kf, and let [G.sub.k] be its gauge group. If the order of the commutator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is m, then the number of distinct p-local homotopy types for the gauge groups {[G.sub.k]} is at most [v.sub.p](m) + 1.

When G = SU(3) then for any homotopy class [f] [member of] [[summation]Y, BSU(3)], Theorems 1.1 and 1.2 imply that the number of distinct p-local homotopy types for the gauge groups {[G.sub.k]} is at most 4 if p = 2, 2 if p = 3 or p = 5, and 1 if p > 5. These general upper bounds closely match known lower bounds. If Y = [S.sup.3] and [f] [member of] [[S.sup.4], BSU(3)] represents an integral generator, then [HK] shows that there are exactly four 2-types, two 3-types and one p-type for p [greater than or equal to] 5. The advantage of Theorem 1.2 is that it works for any space Y, not just Y = [S.sup.3].

2 A lower bound on the order of c

Take homology with integer coefficients. Recall that [H.sup.*] (SU(3)) [congruent to] [LAMBDA](x, y) where [absolute value of x] = 3, [absolute value of y] = 5. Let i: [[summation]CP.sup.2] [right arrow] SU(3) be the canonical map which induces the projection onto the generating set in cohomology. Since i is the inclusion of the 7-skeleton of SU(3), whose next cell is in dimension 10, it follows that the map [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the inclusion of the 10-skeleton.

In this section we will show that the composite

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

has order 120. This implies that the order of c must be at least 120. To do this we use a method due to Hamanaka and the first author [HK].

We begin with a preliminary lemma. For any integer k [greater than or equal to] 0, let [W.sub.3,k] = SU(3 + k)/SU(3). Let [eta] be the stable generator of [[pi].sub.n+1]([S.sup.n]) [congruent to] Z/2Z.

Lemma 2.1. There are isomorphisms [[pi].sub.i] ([W.sub.3/[infinity]]) [congruent to] Z for i [member of] {7,9,11}.

Proof. Since there are fibrations [W.sub.3,k] [right arrow] [W.sub.3,k+1] [right arrow] [S.sup.2k+7] for each k [greater than or equal to] 0, by stability we obtain [[pi].sub.i]([W.sub.3,[infinity]]) [congruent to] [[pi].sub.i] ([W.sub.3,2]) for i [member of] {7,9} and [[pi].sub.11] ([W.sub.3,[infinity]]) = [[pi].sub.11] ([W.sub.3,3]). Now consider [W.sub.3,2]. By definition, [W.sub.3,2] = SU(5)/SU(3) so as a CW-complex we have [W.sub.3,2] = [S.sup.7] [[union].sub.[eta]] [e.sup.9] [union] [e.sup.16]. The Hurewicz homomorphism then implies that [[pi].sub.7]([W.sub.3,2]) [congruent to] Z. As well, by connectivity [[pi].sub.m]([W.sub.3,2]) [congruent to] [[pi].sub.m]([S.sup.7] [[union].sub.[eta]] [e.sup.9]) for 9 [less than or equal to] m [less than or equal to] 11. The cofibration [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] induces a long exact sequence of homotopy groups

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Observe that the first arrow is an isomorphism while the fourth arrow is an epimorphism. Thus [[pi].sub.9] ([S.sup.7] [[union].sub.[eta]] [e.sup.9]) [congruent to] Z. Similar exact sequence arguments show that [[pi].sub.10]([S.sup.7] [[union].sub.[eta]] [e.sup.9]) = [[pi].sub.11]([S.sup.7] [[union].sub.[eta]] [e.sup.9]) = 0. Hence [[pi].sub.9] ([W.sub.3,2]) = Z and [[pi].sub.10]([W.sub.3,2]) = [[pi].sub.11] ([W.sub.3,2]) = 0. Finally, the fibration [W.sub.3,2] [right arrow] [W.sub.3,3] [right arrow] [S.sup.11] induces a long exact sequence of homotopy groups

... [right arrow] [[pi].sub.11] ([W.sub.3,2]) [right arrow] [[pi].sub.11] ([W.sub.3,3]) [right arrow] [[pi].sub.11] ([S.sup.11]) [right arrow] [[pi].sub.10] ([W.sub.3,2]) [right arrow]....

Since [[pi].sub.10]([W.sub.3,2]) [congruent to] [[pi].sub.11]([W.sub.3,2]) [congruent to] 0, we immediately obtain [[pi].sub.11] ([W.sub.3,3]) [congruent to] [[pi].sub.11]([S.sup.11]) [congruent to] Z.

Consider the fibration sequence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since SU([infinity]) is an infinite loop space it is homotopy commutative. Thus the commutator map [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] lifts through [delta] to a map

[lambda]: SU(3) [conjunction] SU(3) [right arrow] [OMEGA][W.sub.3,[infinity]].

There may be many choices of such a lift. In [HK] a choice was made that satisfies the statement of Lemma 2.2. To describe this, recall that [H.sup.*] (SU([infinity])) [congruent to] [LAMBDA]([x.sub.3], [x.sub.5], ...). The generating set may be chosen so that [x.sub.2k+1] = [sigma]([c.sub.k+1]), where [c.sub.k+1] [member of] [H.sup.2k+2](BSU([infinity])) is the [(k + 1).sup.st] Chern class and [sigma] is the cohomology suspension. Then [H.sup.*] ([W.sub.3/[infinity]]) [congruent to] [LAMBDA]([[bar.x].sub.7], [[bar.x].sub.9], ...) where [[pi].sup.*] ([[bar.x].sub.2k+1]) = [x.sub.2k+1].

Lemma 2.2. The lift [gamma] may be chosen so that

[[lambda].sup.*] ([a.sub.2k]) = [[summation].sub.i+j=k+1][x.sub.i] [cross product] [x.sub.j]

where [a.sub.2k] = [sigma]([bar.x][2.sub.k+1]) [member of] [H.sup.2k]([OMEGA][W.sub.3,[infinity]]).

Each [a.sub.2k] [member of] [H.sup.2k]([OMEGA][W.sub.3,[infinity]]) is represented by a map [OMEGA][W.sub.3,[infinity]] [right arrow] K(Z,2k), which we also label as [a.sub.2k]. Taking the product of such maps for k [greater than or equal to] 3, we obtain a map

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Observe that a is both a loop map and a rational homotopy equivalence. Since, by Lemma 2.1, [[[summation]CP.sup.2] [conjunction] [[summation]CP.sup.2], [OMEGA][W.sub.3,[infinity]]] is a free abelian group, we therefore obtain the following.

Lemma 2.3. The induced map

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a monomorphism.

Applying the functor [[[summation]CP.sup.2] [conjunction] [[summation].sup.CP2]], to the fibration sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we obtain an exact sequence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [[summation]CP.sup.2] [conjunction] [[summation]CP.sup.2] is a CW-complex with cells only in even dimensions, we have [[??].sup.-1] ([[summation]CP.sup.2] [conjunction] [[summation]CP.sup.2]) = 0. Thus [[[summation]CP.sup.2] [conjunction] [[summation]CP.sup.2], SU(3)] is the cokernel of [([OMEGA][pi]).sub.*]. Let C be the cokernel of the composite [a.sub.*] x [([OMEGA][pi]).sub.*], where [a.sub.*] is the map in Lemma 2.3. Then we obtain a commutative diagram of exact sequences

where b is the induced map of cokernels. By Lemma 2.3, [a.sub.*] is a monomorphism. A diagram chase then implies that b is also a monomorphism. The composite [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] represents an element of [[[summation]CP.sup.2] [conjunction] [[summation]CP.sup.2], SU(3)]. Since c lifts through [delta] to the map [lambda], the composite [lambda] x (i [conjunction] i) represents a lift of c x (i [conjunction] i) through [[delta].sub.*]. The fact that b is a monomorphism then implies the following.

Lemma 2.4. The order of c x (i [conjunction] i) is the order of the image of [a.sub.*] ([lambda] x (i [conjunction] i)) in C.

Now we calculate the order of the image of [a.sub.*] ([lambda] x (i [conjunction] i)) in C. To understand the cokernel C of [a.sub.*] x [([OMEGA][pi]).sub.*], we first consider the image of [a.sub.*] x [([OMEGA][pi]).sub.*]. Observe that [a.sub.*] x [([OMEGA][pi]).sub.*] is induced by the composite [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The choice of the generating sets for [H.sup.*] (SU([infinity])) and [H.sup.*] ([OMEGA][W.sub.3,[infinity]]) implies that [([OMEGA][pi]).sub.*] ([a.sub.2k]) = [c.sub.2k] for k [greater than or equal to] 3. Thus a x [OMEGA][pi] corresponds to [[direct sum].sub.k [greater than or equal to] 3] k! [ch.sub.k] where [ch.sub.k] is the 2k-dimensional part of the Chern character. Hence for any [xi] [member of] [[??].sup.0] ([[summation]CP.sup.2] [conjunction] [[summation]CP.sup.2]) we have

[a.sub.*] x [([OMEGA][pi]).sub.*] ([xi]) = (3! [ch.sub.3]([xi]),4! [ch.sub.4]([xi]),5! [ch.sub.5]([xi])) [member of] [H.sup.6] [direct sum] [H.sup.8] [direct sun] [H.sup.10]. (1)

Let x be a generator of [[??].sup.0] ([CP.sup.2]) such that ch(x) = t + [t.sup.2]/2 for a generator t [member of] [H.sup.2]([CP.sup.2]). Then [[??].sup.0] ([[summation]CP.sub.2] [conjunction] [[summation]CP.sup.2]) is a free abelian group generated by [[summation].sup.2]([x.sup.i] [cross product] [x.sup.j]) for i,j = 1,2. Let [[epsilon].sub.i,j] = [a.sub.*] x [([OMEGA][pi]).sub.*] ([[summation].sup.E2] ([x.sup.i] [cross product] [x.sup.j])). Then (1) implies that the image of [a.sub.*] x [([OMEGA][pi]).sub.*] is as follows. In the group [Z[summation].sup.2] (t [cross product] t) [direct sum] [Z[summation].sup.2] (t [cross product] [t.sup.2]) [direct sum] [Z[summation].sup.2] ([t.sup.2] [cross product] t) [direct sun] [Z[summation].sup.2] ([t.sup.2] [cross product] [t.sup.2]) we have

[[epsilon].sub.1,1] = (3!,4!/2,4!/2,5!/4)

[[epsilon].sub.2.1] = (0,4!,0,5!/2)

[[epsilon].sub.1.2] = (0,0,4!,5!/2)

[[epsilon].sub.2,2] = (0,0,0,5!).

Next, consider the image of [a.sub.*] ([lambda] x (i [conjunction] i)). By Lemma 2.2 and (1) we have

[a.sub.*] ([lambda] x (i [conjunction] i)) = (1,1,1,1).

Finally, observe that 20[[epsilon].sub.1,1] - 5([[epsilon].sub.1,2] + [[epsilon].sub.2,1]) + [[epsilon].sub.2,2] = 5!(1,1,1,1) and no other combination of [[epsilon].sub.i,j's] gives a smaller multiple of (1,1,1,1). That is, if y = [a.sub.*] ([lambda] x (i [conjunction] i)), then 5!y is in the image of [a.sub.*] x [([OMEGA][pi]).sub.*], and no smaller multiple of y is in the image. Thus y passes to an element in the cokernel C of order 5!. Lemma 2.4 therefore implies the following.

Proposition 2.5. The composite [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has order 51.

3 The odd primary components of the order of c

The adjoint of the identity map on SU(3) is the evaluation map ev: [summation]SU(3) [equivalent to] [summation][OMEGA]BSU(3) [right arrow] BSU(3). Let j be the composite j: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For the remainder of the section we localize all spaces and maps at an odd prime p. By [MNT], SU(3) [equivalent to] [S.sup.3] x [S.sup.5], so restricting to 5-skeleta we also have [[summation]CP.sup.2] [equivalent to] [S.sup.3] [conjunction] [S.sup.5]. The following lemma is a special case of an argument in [Mc].

Lemma 3.1. There is a homotopy commutative diagram

where t is a left homotopy inverse for [summation]i.

Proof. Since SU(3) [equivalent to] [S.sup.3] x [S.sup.5], we have [summation] SU(3) [equivalent to] [S.sup.4] [disjunction] [S.sup.6] [disjunction] [S.sup.9], so ev can be regarded as the wedge sum [summation]i + h for some map h: [S.sup.9] [right arrow] BSU(3)). By [G], the homotopy fibre of ev is the Hopf construction [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Observe that the restriction of [[mu].sup.*] to [S.sup.9] [equivalent to] [summation][S.sup.3] [conjunction] [S.sup.5] [right arrow] [summation] SU(3) [conjunction] SU(3) is onto in mod-p homology. Thus the equivalence for [summation]SU(3) can be chosen so that the [S.sup.9] summand of [summation]SU(3) factors through [[mu].sup.*]. Doing so, we obtain h [equivalent to] *. Hence ev factors through j and the lemma follows.

Note that the adjoint of the commutator SU(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and the adjoint of the composite [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is [j,j].

Lemma 3.2. The maps [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] have the same order.

Proof. Since c x (i [conjunction] i) factors through c, its order can be no larger than the order of c. To prove the converse, suppose that c x (i [conjunction] i) has order m. Then adjointing, the composite [ev, ev] x ([summation]i [conjunction] i) has order m. That is, [j, j] has order m. Observe that [j, j] is homotopic to the composite [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so [j, j] having order m implies that (m x j [disjunction] j) extends to a map [mu]: [[summation].sup.2] [CP.sup.2] x [[summation].sup.2] [CP.sup.2] [right arrow] BSU(3). Lemma 3.1 therefore implies that there is a map [[bar.[mu]: [summation]SU(3) x [summation]SU(3) [right arrow] BSU (3) which restricts to (m x ev [conjunction] ev) on [summation]SU (3) [conjunction] [summation]SU (3). Thus if m' = (m x j [conjunction] j), then the composite [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is null homotopic. But this composite is homotopic to m x [ev, ev]. Thus [ev, ev] has order m, implying that its adjoint c also has order m.

By Proposition 2.5 we know the order of c x (i [conjunction] i). Thus Lemma 3.2 immediately implies the following.

Proposition 3.3. The commutator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] satisfies:

(a) localized at 3, c has order 3;

(b) localized at 5, c has order 5;

(c) localized at p for p [greater than or equal to] 7, c is null homotopic.

4 The 2-component of the order of c

Throughout this section we localize all spaces and maps at 2. We will use a result of Mimura to reduce the calculation of the order of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to calculating the order of the composite [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Recall from Section 2 that the map [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the inclusion of the 10-skeleton. The following theorem incorporates this skeletal identification with Mimura's [M] description of the cell structure of SU(3) [conjunction] SU(3).

Recall that [eta]: [S.sup.n+1] [right arrow] [S.sup.n] represents the stable generator of [[pi].sub.n+1]([S.sup.n]) [congruent to] Z/2Z. For a space X, let [X.sub.n] be the n-skeleton of X and let [nabla]: X [disjunction] X [right arrow] X be the fold map.

Theorem 4.1. There is a homotopy equivalence

SU(3) [conjunction] SU(3) [equivalent] B [conjunction] [[summation].sup.9] [CP.sup.2] [disjunction] [[summation].sup.9] [CP.sup.2]

where B satisfies the following:

(a) B = ([[summation]CP.sup.2] [conjunction] [CP.sup.2]) [union] [e.sup.16];

(b) B/[S.sup.6] [equivalent to] ([S.sup.8] x [S.sup.8]) [[union].sub.[bar.[eta]]] [e.sup.10], where [[bar.[eta]] = [eta] x [eta];

Now we begin the reduction procedure.

Lemma 4.2. The group [[[summation].sup.9][CP.sup.2], SU(3)] has order [less than or equal to] 8.

Proof. The cofibration [S.sup.11] [right arrow] [[summation].sup.9][CP.sup.2] [right arrow] [S.sup.13] induces an exact sequence [[S.sup.13],SU(3)] [right arrow] [[[summation].sup.9][CP.sup.2],SU(3)] [right arrow] [[S.sup.11],SU(3)]. By [MT], [[pi].sub.11] (SU(3)) [congruent to] Z/4Z and [[pi].sub.13] (SU(3)) [congruent to] Z/2Z. Thus, by exactness, [[[summation].sup.9][CP.sup.2], SU(3)] has order at most 8.

Corollary 4.3. If the composite [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has order [less than or equal to] 8 then c has order [less than or equal to] 8.

Proof. By Theorem 4.1, SU(3) [conjunction] SU(3) [equivalent to] B [disjunction] [[summation].sup.9][CP.sup.2] [disjunction] [[summation].sup.9][CP.sup.2]. By Lemma 4.2, the restriction of c to either copy of [[summation].sup.9][CP.sup.2] has order [less than or equal to] 8. Therefore, if the restriction of c to B also has order [less than or equal to] 8, then c has order [less than or equal to] 8.

Lemma 4.4. The map ([[summation]CP.sup.2] [conjunction] [[summation]CP.sup.2])/[S.sup.6] [right arrow] B/[S.sup.6] has a left homotopy inverse after suspending, implying there is a homotopy equivalence [summation]B/[S.sup.6] [equivalent to] ([[summation].sup.2][CP.sup.2] [conjunction] [CP.sup.2])/[S.sup.6] [disjunction] [S.sup.17].

Proof. By Theorem 4.1 (b), B/[S.sup.6] = ([S.sup.8] x [S.sup.8]) [[union].sub.[bar.[eta]]] [e.sup.10]. This implies that there is an inclusion [S.sup.8] x [S.sup.8] [right arrow] B/[S.sup.6] with the property that the pinch map B/[S.sup.6] [right arrow] [S.sup.16] to the top cell extends the pinch map [S.sup.8] x [S.sup.8] [right arrow] [S.sup.16]. After suspending, [summation] ([S.sup.8] x [S.sup.8]) [equivalent to] [S.sup.9] [disjunction] [S.sup.9] [disjunction] [S.sup.17]. Thus the top cell splits off [summation]B/[S.sup.6]. The lemma now follows since by Theorem 4.1 (a), [summation]B/[S.sup.6] = ([[summation].sup.2][CP.sup.2] [conjunction] [summation] [CP.sup.2])/[S.sup.6] [union] [e.sup.17].

In what follows we will have to distinguish between power maps and degree maps. In general, if X is an H-space let k: X [right arrow] X be the kth-power map and if Y is a co-H-space let k: Y [right arrow] Y be the map of degree k.

Lemma 4.5. If the composite [summation][CP.sup.2] [conjunction] [summation][CP.sup.2] [right arrow] B [right arrow] SU(3) [conjunction] SU(3) [??] SU(3) has order [less than or equal to] 8 then the composite B [right arrow] SU(3) [conjunction] SU(3) [??] SU(3) has order [less than or equal to] 8.

Proof. The proof of the lemma takes several steps.

Step 1. Let f be the composite B [right arrow] SU(3) [conjunction] SU(3) [??] SU(3). Let A = [summation][CP.sup.2] [conjunction] [summation][CP.sup.2] and let i: A [right arrow] B be the inclusion. Let i': A/[S.sup.6] [right arrow] B/[S.sup.6] be the map induced by pinching out the bottom cell common to both A and B. By [HK], the composite [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has order 2. Thus there is a homotopy commutative diagram

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

for some map g.

Step 2. The lemma asserts that if 8 [omicron] f [omicron] i is null homotopic then so is 8 [omicron] f. We claim that it suffices to show that if 4 [omicron] g [omicron] i' is null homotopic then so is 4 [omicron] g. To see this, suppose that 8 [omicron] f [omicron] i is null homotopic. Consider the cofibration sequence [S.sup.6] [right arrow] A [??] A/[S.sup.6] [right arrow] S7. Since 8 [omicron] f [omicron] i [equivalent] *, the homotopy commutativity of the outer rectangle in (2) implies that 4 [omicron] g [omicron] i' [omicron] [pi] is null homotopic. Thus 4 [omicron] g [omicron] i' extends through the cofibre of [pi] to a map [S.sup.7] [right arrow] SU(3). But by [MT], [[pi].sub.7](SU(3)) [congruent to] 0. Thus 4 [omicron] g [omicron] i' is null homotopic. We assume that this condition implies that 4 [omicron] g is null homotopic. But then the homotopy commutativity of the right square in (2) implies that 8 [omicron] f is null homotopic.

Step 3. It remains to show that if 4 [omicron] g [omicron] i' is null homotopic then so is 4 [omicron] g. In general, for a space X, let E: X [right arrow] [OMEGA][SIGMA]X be the suspension map. Applying the James construction [J] to the map g, we obtain an H-map [bar.g]: [OMEGA][SIGMA](B/[S.sup.6]) [right arrow] SU(3) such that [bar.g] [omicron] E [equivalent] g. Let A' = A/[S.sup.6]. By Lemma 4.4 there is a homotopy equivalence e: [SIGMA]A' [disjunction] [S.sup.17] [right arrow] [SIGMA]B/[S.sup.6] where the restriction of e to [SIGMA]A' is [SIGMA]i'. Consider the diagram

where [i.sub.1] is the inclusion of the left wedge summand. The rectangle homotopy commutes since the restriction of e to [SIGMA]A' is [SIGMA]i' and E commutes with suspensions. Since [bar.g] [omicron] E [equivalent] g, the upper direction around the diagram is homotopic to 4 [omicron] g [omicron] i', which we are assuming is null homotopic. Thus the lower direction around the diagram is also null homotopic. In addition, by [MT], [[pi].sub.16](SU(3)) [congruent to] Z/4Z [direct sum] Z/2Z, so in fact the entire bottom row of the diagram is null homotopic. On the other hand, since A' and [S.sup.16] are suspensions, the bottom row is homotopic to the composite

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore we have a string of homotopies

* [equivalent] 4 [omicron] [bar.g] [omicron] [OMEGA]e [omicron] E [equivalent] [bar.g] [omicron] [OMEGA]e [omicron] E [omicron] [4.bar] [equivalent] [bar.g] [omicron] [OMEGA]e [omicron] [OMEGA][SIGMA][4.bar] [omicron] E

where the last homotopy is due to the naturality of E. A consequence of the James construction is that the homotopy class of an H-map [OMEGA][SIGMA]X [??] Y is determined by the homotopy class of f [omicron] E. In our case, the null homotopy for [bar.g] [omicron] [OMEGA]e [omicron] [OMEGA][SIGMA][4.bar] [omicron] E implies that [bar.g] [omicron] [OMEGA]e [omicron] [OMEGA][SIGMA][4.bar] is null homotopic.

Step 4. By the Hilton-Milnor Theorem, there is a homotopy equivalence

[OMEGA]([SIGMA]A' [disjunction] [S.sup.17]) [equivalent] [OMEGA][SIGMA]A' x [OMEGA][S.sup.17] x [OMEGA]([SIGMA][OMEGA][SIGMA]A' [conjunction] [OMEGA][S.sup.17]).

Observe that as A' is 7-connected, [OMEGA]([SIGMA][OMEGA][SIGMA]A' [conjunction] [OMEGA][S.sup.17]) is 23- connected. The distributivity formula (see [C, [section]4], for example), therefore implies that the 4th-power map on [OMEGA]([SIGMA]A' [disjunction] [S.sup.17]) is homotopic to [OMEGA][SIGMA][4.bar] through dimension 23. Thus in dimensions [less than or equal to] 23, there is a string of homotopies

* [equivalent] [bar.g] [omicron] [OMEGA]e [omicron] [OMEGA][SIGMA][4.bar] [equivalent] [bar.g] [omicron] [OMEGA]e [omicron] 4 [equivalent] 4 [omicron] [bar.g] [omicron] [OMEGA]e

where the first homotopy is by Step 3 and last is due to the 4th-power map commuting with H-maps.

Step 5. We now have 4 [omicron] [bar.g] [omicron] [OMEGA]e [equivalent] * in dimensions [less than or equal to] 23. Since e is a homotopy equivalence, we can compose on the right with [OMEGA][e.sup.-1] to obtain 4 [omicron] [bar.g] [equivalent] * in dimensions [less than or equal to] 23. As B/[S.sup.6] is 16-dimensional, we therefore obtain 4 [omicron] [bar.g] [omicron] E [equivalent] * without any dimensional restriction. But [bar.g] was defined so that [bar.g] [omicron] E [equivalent] g. Hence 4 [omicron] g is null homotopic, as required.

Combining Lemma 4.5 and Corollary 4.3 immediately implies the following.

Proposition 4.6. If the composite [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has order [less than or equal to] 8, then c has order [less than or equal to] 8.

Now we can combine the results of the previous three sections to prove Theorem 1.1.

Proof of Theorem 1.1. Let m be the order of the commutator SU(3) [conjunction] SU (3) [??] SU(3). By Proposition 4.6, the 2-component of the order of c equals the 2-component of the order of c [omicron] (t [conjunction] t), which by Proposition 2.5 is 8. By Proposition 3.3, the 3-component of m is 3, the 5-component of m is 5, and the p component of m for p [greater than or equal to] 7 is 1. Thus m = [2.sup.3] x 3 x 5 = 120.

5 Counting gauge groups

In this section we revert to assuming that spaces and maps have not yet been localized. We begin by stating a general criterion proved in [Th] for determining when certain fibres are homotopy equivalent.

Lemma 5.1. Let Xbea space and Y be an H-space with a homotopy inverse. Let X [??] Y be a map of order m, where m is finite. Let [F.sub.k] be the homotopy fibre of the composite [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If (m, k) = (m, k') then [F.sub.k] and [F.sub.k'] are homotopy equivalent when localized rationally or at any prime.

In order to use this to help count gauge groups, we need a context in which gauge groups arise as homotopy fibres. Let G be a topological group, let X be a space, and let P [right arrow] X be a principal G-bundle with gauge group G(P). Let BG and BG(P) be the classifying spaces of G and G(P) respectively. In [AB] it was shown that there is a homotopy equivalence BG(P) [equivalent] [Map.sub.P](X, BG), where [Map.sub.P](X, BG) is the component of the space of continuous maps from X to BG which are freely homotopic to the map inducing P. Moreover, there is a fibration

[Map.sup.*.sub.P](X, BG) [right arrow] [Map.sub.P](X, BG) [??] BG

where [Map.sup.*.sub.P](X, BG) is the component of the space of continuous maps from X to BG which are based homotopic to the map inducing P, and ev evaluates a map at the basepoint.

Now specialize to X = [SIGMA]Y. Observe that the components of Map([SIGMA]Y, BG) and [Map.sup.*] ([SIGMA]Y, BG) are in one-to-one correspondence with the homotopy classes of maps [[SIGMA]Y, BG]. Fix a homotopy class [f] [member of] [[SIGMA]Y, BG]. For an integer k, let [P.sub.k] [right arrow] [SIGMA]Y be the principal G-bundle classified by the homotopy class of kf. Note that if [f] has infinite order then the bundles [P.sub.k] [right arrow] [SIGMA]Y are distinct, but if [f] has order m then there are bundle equivalences between [P.sub.ms+k] [right arrow] [SIGMA]Y and [P.sub.k] [right arrow] [SIGMA]Y for every integer s. Let [G.sub.k] be the gauge group of the principal G-bundle [P.sub.k] [right arrow] [SIGMA]Y. Then there is a homotopy equivalence B[G.sub.k] = [Map.sub.kf]([SIGMA]Y, BG). In the pointed case, the pointed exponential law implies that [Map.sup.*]([SIGMA]Y, BG) is homotopy equivalent to the loop space [OMEGA][Map.sup.*](Y, BG), and in general the components of a homotopy-associative H-space are homotopy equivalent. Explicitly in our case, the existence of a pointed wedge product [SIGMA]Y [right arrow] [SIGMA]Y [disjunction] [SIGMA]Y lets us define a map [bar.-kf]: [Map.sup.*.sub.kf]([SIGMA]Y, BG) [right arrow] [Map.sup.*.sub.0]([SIGMA]Y, BG) by sending g [member of] [Map.sup.*.sub.kf]([SIGMA]Y, BG) to the composite [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [nabla] is the fold map. Since the wedge product on [SIGMA]Y is associative, it follows that [bar.kf] [omicron] [bar.-kf] takes a map g [member of] [Map.sup.*.sub.kf]([SIGMA]Y, BG) to itself, implying that [Map.sup.*.sub.kf]([SIGMA]Y, BG) retracts off [Map.sup.*.sub.0]([SIGMA]Y, BG). A similar argument shows that [Map.sup.*.sub.0]([SIGMA]Y, BG) retracts off [Map.sup.*.sub.kf]([SIGMA]Y, BG), so in fact the two are homotopy equivalent. Therefore the evaluation fibration determines a homotopy fibration sequence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

which defines the map [[partial derivative].sub.k]. In [L] it was shown that the adjoint [SIGMA]Y [conjunction] G [right arrow] BG of [[partial derivative].sub.k] is homotopic to the Whitehead product [kf, ev]. As the Whitehead product is linear, we have [kf, ev] [equivalent] k[f, ev], implying that [[partial derivative].sub.k] [equivalent] k [omicron] [[partial derivative].sub.1]. Hence the fibration sequence (3) implies that [G.sub.k] is the homotopy fibre of the map k [omicron] [[partial derivative].sub.1].

Observe that the classifying space BG is rationally homotopy equivalent to a product of Eilenberg-MacLane spaces. That is, BG is rationally homotopy equivalent to an H-space. Therefore the adjoint of [[partial derivative].sub.1]--the Whitehead product [f, ev]--is rationally trivial. This implies that [[partial derivative].sub.1] has order [m.sub.f], where [m.sub.f] is finite. Now we can apply Lemma 5.1 to obtain the following.

Proposition 5.2. If ([m.sub.f], k) = ([m.sub.f], k') then there is a homotopy equivalence [G.sub.k] [equivalent] [G.sub.k'] after localizing rationally or at any prime.

Now we relate the order of [f, ev] to that of the commutator c to prove Theorem 1.2.

Proof of Theorem 1.2. We are given that the order of the commutator G [conjunction] G [??] G is m. The adjoint of c is the Whitehead product [ev, ev], so [ev, ev] has order m. By definition, the order of [f, ev] is [m.sub.f]. Since [f, ev] factors through [ev, ev], we must have [m.sub.f] dividing m. Thus ([m.sub.f], k) divides (m, k) for each k. So if (m, k) = (m, k') then ([m.sub.f], k) = ([m.sub.f], k') for each k. Proposition 5.2 therefore implies that there is a p-local homotopy equivalence [G.sub.k] [equivalent] [G.sub.k'].

Localized at p, we only need be concerned with the p-component of the integers (m, k). These p-components range from 0 to [v.sub.p](m). Thus the number of distinct p-local homotopy types of the gauge groups {[G.sub.k]} is bounded above by [v.sub.p](m) + 1.

References

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[B] R. Bott, A note on the Samelson product in the classical Lie groups, Comment. Math. Helv. 34 (1960), 245-256.

[C] F.R. Cohen, A short course in some aspects of classical homotopy theory, Lecture Notes in Math. 1286, Springer-Verlag (1987), 1-92.

[CS] M.C. Crabb and W.A. Sutherland, Counting homotopy types of gauge groups, Proc. London Math. Soc. 83 (2000), 747-768.

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[K] A. Kono, A note on the homotopy type of certain gauge groups, Proc. Roy. Soc. Edinburgh Sect. A 117 (1991), 295-297.

[L] G.E. Lang, The evaluation map and EHP sequences, Pacific J. Math. 44 (1973), 201-210.

[Mc] C.A. McGibbon, Homotopy commutativity in localized groups, Amer. J. Math 106 (1984), 665-687.

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and its applications, J. Math. Soc. Japan 23 (1971), 593-621.

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[Th] S.D. Theriault, The homotopy types of Sp(2)-gauge groups, J. Math. Kyoto Univ. 50 (2010), 591-605.

[To] H. Toda, Composition methods in homotopy groups of spheres, Annals of Math. Studies 49, Princeton University Press, 1962.

Received by the editors March 2012.

Communicated by Y. Felix.

2010 Mathematics Subject Classification: Primary 55P15, 57T99, Secondary 54C35.

Faculty of Science and Engineering, Doshisha University, Kyoto 610-0321, Japan

email:akono@mail.doshisha.ac.jp

School of Mathematics, University of Southampton, Southampton SO17 1QH, United Kingdom

email: S.D.Theriault@soton.ac.uk
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