Unstable K 1 -group [Σ 2n 2 CP 2, U(n)] and homotopy type of certain gauge groups

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1 Unstable K 1 -group [Σ n CP, U(n)] an homotopy type of certain gauge groups by Hiroai Hamanaa Department of Natural Science Hyogo University of Eucation Yashiro, Hyogo , JAPAN hammer@sci.hyogo-u.ac.jp an Aira Kono Department of Mathematics Kyoto University Kyoto , JAPAN ono@math.yoto-u.ac.jp October, 00 1 Introuction Let G be a compact connecte Lie group an let P be a principal G-bunle over a connecte finite complex B. We enote by G(P ), the group of G-equivariant self-maps of P covering the ientity map on B. G(P ) is calle the gauge group of P. For fixe B an G, M.C.Crabb an W.A.Sutherlan show as P ranges over all principal G-bunles with base space B, the number of homotopy types of G(P ) is finite ([]). Principal SU(n)-bunles over S are classifie by their secon Chern class. Denote by P n, the principal SU(n)-bunle over S with c (P n, ) =. The secon author shows in [6] G(P, ) is homotopy equivalent to G(P, ) if an only if (1, ) = (1, ). Therefore when B is S an G = SU(), there are precisely six homotopy types of G(P ). The purpose of this paper is to show the following: 1

2 Theorem 1. G(P 3, ) is homotopy equivalent to G(P 3, ) if an only if (, ) is equal to (, ). Theorem. If G(P n, ) is homotopy equivalent to G(P n, ), then (n(n 1), ) = (n(n 1), ). Remar 3. In [9] W.A.Sutherlan shows if G(P n, ) is homotopy equivalent to G(P n, ) then (n(n 1)/(, n + 1), ) is equal to (n(n 1)/(, n + 1), ). Therefore if n is even, then Theorem is the result of Sutherlan. If n is o, Theorem is an improvement of [9]. Denote by x 0 an y 0, the base points of S an BSU(n). Denote by ɛ a generator of π (BSU(n)). Note that ɛ is the classifying map of P n,. Put M n, = {f : S BSU(n) f ɛ } M n, = {f M n, f(x 0 ) = y 0 }. By Atiyah-Bott [1] M n, BG(P n, ). Note that M n, M n,0. Consier the fibre sequence () G(P n, ) SU(n) h n, Mn,0 M n, e BSU(n). In [SU(n), Mn,0] = [Σ SU(n), BSU(n)] = [Σ 3 SU(n), SU(n)], h n, is equal to γ (ɛ 1 SU(n) ) where γ is the commutator map an ɛ is a generator of π 3 (SU(n)) (see Lang [7]). Define ξ : [Σ n 5 CP, SU(n)] [Σ n CP, SU(n)] by ξ (α) = γ (ɛ α) for α [Σ n 5 CP, SU(n)]. In section, for o n, we etermine [Σ n CP, U(n)] an show Coerξ is finite an the orer of Coerξ is equal to the orer of Coerξ if an only if (n(n 1), ) is equal to (n(n 1), ). Applying the functor [Σ n 5 CP, ] to (), we get an exact sequence [Σ n 5 CP, SU(n)] [Σ n 5 CP, Mn,0] [Σ n 5 CP, M n, ] 0, = (h n, ) [Σ n CP, SU(n)] since [Σ n 5 CP, BSU(n)] = K 1 (Σ n 5 CP ) = 0. Note that (h n, ) is equal to ξ. If G(P n, ) G(P n, ) then (as a set) [Σ n 5 CP, M n, ] = [Σ n 5 CP, M n, ]. Therefore we get Theorem. Let ˆM 3,0 be the -connecte cover of M3,0. Since SU(3) is -connecte, [SU(3), ˆM 3,0] [SU(3), M3,0] is bijective. Note that ˆM 3,0 is a loop space, π j ( ˆM 3,0) is finite for any j an h 3, = h 3,1. In section 3 we show h 3,1 = 0. If (, ) = (, ), then we construct a homotopy equivalence ( ) : ˆM n,0 ˆM n,0 ( ) such that (h 3,1 ) h 3,1. Therefore Ω ˆM 3, Ω ˆM 3, if (, ) = (, ). Then we show G(P 3, ) S 1 Ω ˆM 3, an prove Theorem 1.

3 The group [Σ n CP, U(n)] In this section we assume n is o. Put W n = U( )/U(n) an consier the fibre sequence ΩU( ) Ωπ ΩW n U(n) U( ) π W n. Since (W n ) (n+3) S n+1 η e n+3, where η is the generator of π n+ (S n+1 ). Denote a generator of H n+ (K(Z, n + 1)) by ɛ an consier the -stage Postniov system E K(Z, n + 1) ɛ K(Z, n + ). Let x be the generator of H n+1 (W n ) = Z. Since H n+ (W n ) = 0, x has a lift x : W n E, such that x is a (n + 3)-equivalence. Therefore if im X n +, then Ω x : [X, ΩW n ] [X, ΩE] is an isomorphism. Define a homomorphism λ : [X, ΩW n ] H n (X) H n+ (X) by λ(α) = (α (a n ), α (a n+ )), where a n an a n+ are generators of H n (ΩW n ) = H n+ (ΩW n ) = Z an α [X, ΩW n ]. If X = Σ n CP, we have the following: Lemma.1. If X = Σ n CP, λ : [X, ΩW n ] H n (X) H n+ (X) is monic an Imλ = {(a, b) a b mo }. Lemma.. If X = Σ n CP, Imλ (Ωπ) is generate by (n!, 1 (n + 1)!) an (0, (n + 1)!). Proof. First recall that (Ωπ) (a n ) = n!ch n an (Ωπ) (a n+ ) = (n + 1)!ch n+1. Denote by ξ n a generator of K(S n ). K(CP ) = Z[x]/(x 3 ), H (CP ) = Z[t]/(t 3 ) an chx = t + t. Therefore ch(ξ n 1 ˆ x) = σ n t + (σ n t /), ch(ξ n 1 ˆ x ) = σ n t. Note that [X, U( )] = K(X). Then λ (Ωπ) (ξ n 1 ˆ x) = (n!, 1 (n + 1)!) λ (Ωπ) (ξ n 1 ˆ x ) = (0, (n + 1)!). Since K 1 (Σ n CP ) = 0, we get Lemma.3. [Σ n CP, U(n)] = Coer(Ωπ). 3

4 Put u = (1, 1), v = (0, ) an l = 1 n!. Then (n!, 1 ( ) n 1 (n + 1)!) = (l, (n + 1)l) = lu + lv If n 3 mo, an therefore Note that (0, (n + 1)!) = (0, (n + 1)l) = (n + 1)lv. l(u + n 3 v) = lu + n 3 lv lv mo Im(Ωπ) l(n + 1)(u + n 3 v) = l(n + 1)v 0 mo Im(Ωπ). l(n + n 1 v) lu + Since 1 n 3 n 1 ( n 1 = n 1 ) lv 0 mo Im(Ωπ). n 3 = 1, u = (u+ n 3 v) an v = (u+ n 1 v) are generators of Imλ. Note that the orer of Coer(Ωπ) = (n + 1)l an we have Coer(Ωπ) = Z/(n + 1)! Z/( 1 n!). If l 1 mo l(u + n 3 ( ) n 3 v) = lu + lv lv mo Im(Ωπ) an therefore Note that (n + 1)l(u + n 3 v) (n + 1)lv 0 mo Im(Ωπ). Since n 3 1 n 1 l(u + n 1 v) 0 mo Im(Ωπ). = n 1 n 3 = 1, u = u+ n 3 v an v = u+ n 1 v are generators of Imλ. Note that the orer of Coer(Ωπ) = (n + 1)l an we have Coer(Ωπ) = Z/ ( 1 (n + 1)!) Z/(n!). { Theorem.. [Σ n CP Z/(n + 1)! Z/( 1, U(n)] = n!) if n 3 mo Z/ ( 1 (n + 1)!) Z/n! if n 1 mo.

5 Consier the Samelson prouct ξ : [Σ n 5 CP, U(n)] [Σ n CP, U(n)] given by ξ (α) = γ (ɛ α). Note that [Σ n 5 CP, U(n)] = K(Σ n CP ). The basis of K(Σ n CP ) = Z Z is ξ n ˆ x, ξ n ˆ x. Put L = (n )!. c n 1 (ξ n ˆ x) = Lσ n t, c n (ξ n ˆ x) = 1 (n 1)Lσn t c n 1 (ξ n ˆ x ) = 0, c n (ξ n ˆ x ) = (n 1)Lσ n t Let α j be the ajoint of ξ n ˆ x j (j = 1, ). For α : Σ n 5 CP U(n), put α = γ (ɛ α), where γ is the lift of γ constructe in []. Note that H (U(n)) = (σ(c 1 ),, σ(c n )). an γ (a n ) = Using the metho in [], we have i+j=n 1 γ (a n+ ) = i+j=n σ(c i ) σ(c j ). σ(c i ) σ(c j ). Therefore λ( α 1 ) = (L, 1 (n 1)L) an λ( α ) = (0, (n 1)L). If n 3 mo then v = v u. Put α = α +(n 1)α 1 then α 1 = Lu an α = 1 (n 1)Lv. If n 1 mo then v = v u. Put α = α + (n 1)α 1 then α 1 = 1 Lu an α = (n 1)Lv. Therefore we have Lemma.5. The orer of Coerξ is equal to 1 (n )!(n(n 1), ) (n 1)!(n, ) Corollary.6. The orer of Coerξ is equal to the orer of Coerξ only if (n(n 1), ) = (n(n 1), ). if an 3 Proof of Theorem 1 Consier the cofibering S 11 Σ 5 CP Σ SU(3). Since Σ 5 CP is 6-connecte, the suspension map Σ : [S 11, Σ 5 CP ] { S 11, Σ 5 CP } 5

6 is an isomorphism (see [10]). Note that Σ () = 0 (see [5]). Therefore = 0, Σ SU(3) Σ 5 CP S 1 an [Σ SU(3), BSU(3)] = [Σ 5 CP, BSU(3)] π 1 (BSU(3)) (1) = [Σ CP, SU(3)] π 11 (SU(3)) () = Z/ Z/3 Z/, (3) where π 11 (SU(3)) = Z/ (see [8]). Therefore we have: Lemma 3.1. (γ (ɛ 1 SU(3) )) = 0. Let X be a connecte loop space, its base point, µ : X X X its loop multiplication an ι : X X its homotopy inverse. For an integer n efine self map n : X X as follows: 0 =, 1 = 1 X, n = µ ((n 1) 1 X ) for a positive integer n. If n < 0 then n = ι ( n). Let Y be a finite complex an α : Y X. In the group [Y, X], nα is represente by n α. For a prime p, consier the localization l (p) : X X (p). For a map f : X X, there exists a map f (p) : X (p) X (p) satisfying f (p) l (p) l (p) f. f (p) is unique up to homotopy. X (p) is a loop space an l (p) is a loop map. In X (p), n ( (p) n. If (n, p) = 1, n : X (p) X (p) is a homotopy equivalence. For a ) b Z(p) ((a, b) = 1, (b, p) = 1)), b : X (p) X (p) is a homotopy equivalence, an efine ( ) a b = a b 1. If (a, p) = 1, ( ) a b is a homotopy equivalence. Lemma 3.. Let, an be non zero integers satisfying (, ) = (, ). If π j (X) is finite for any j an α = 0, then there exists a homotopy equivalence ( ) : X X ( ) satisfying α α. Proof. For an integer n 0 an n = p r q s is the factorization into prime powers, we efine ν p (n) = r. Define h p : X (p) X (p) by h p = {( ) if ν p () > ν p () 1 X(p) if ν p () ν p () an h = p h p. Since ν p ((, )) = min(ν p (), ν p ()), if ν p () > ν p (), then ν p () > ν p ( ) an ν p ( ) = ν p (). The orer of l (p) α is a power of p. If ν p () ν p () then ν p () ν p ( ) an therefore l (p) α l (p) α 0. Consier l = ( ( ) l (p) ) : X ΠX (p). l is a homotopy equivalence. Put = l 1 h l. We nee only show h l α l α or h (p) l (p) α l (p) α. If ν p () > ν p () h (p) l (p) α h (p) l (p) α 1 l (p) α l (p) α l (p) α. 6

7 If ν p () ν p (), h (p) = 1 X (p) an we have ( Therefore h (p) l (p) α h (p) l (p) α 0. ( ) is clearly a homotopy equivalence. ) α α. Now we can prove Theorem 1. Since BSU(3) is 3-connecte M 3, M 3,0 M 3, is -equivalence. Note that π 1 (M 3,0) = π 5 (BSU(3)) = 0 an π (M 3,0) = π 6 (BSU(3)) = Z. Denote the generators of H (M 3, ) = H (M 3,0) = Z by b an b respectively. The homotopy fibre of b an b are enote by ˆM 3, an ˆM 3,0 respectively. Therefore Ω ˆM 3, is the universal covering of ΩM 3, G 3,. Since π 1 (G 3, ) = Z, G 3, S 1 Ω ˆM 3,. Note that p : [SU(3), ˆM 3,0] [SU(3), M 3,0] is a bijection where p : ˆM 3,0 M 3,0 is the projection. Therefore there exists ĥ 3, : SU(3) ˆM 3,0 such that p ĥ3, h 3, an Ω ˆM 3, is the homotopy fibre of ĥ3,. ĥ 3, is unique up to homotopy an ĥ3, ĥ3,1. Since h 3,1 = 0, ĥ3,1 = 0. ˆM 3,0 is a loop space an π j ( ˆM 3,0) is finite for any j. Now by Lemma 3. we get if (, ) = (, ) then Ω ˆM 3, Ω ˆM 3,, an Theorem 1 is proven. References [1] M.F.Atiyah an R.Bott, The Yang-Mills equations over Riemann surfaces, Phils. Trans. Ray. Soc. Lonon Ser A., 308 (198), [] M.C.Crabb an W.A.Sutherlan, Counting homotopy types of gauge groups, Proc. Lonon. Math. Soc., (3)81(000), [3] H.Hamanaa, On [X, U(n)] when im X = n + 1, to appear in J. Math. Kyoto Univ. [] H.Hamanaa an A.Kono, On [X, U(n)] when im X = n, J. Math. Kyoto Univ., (003). [5] R.Kane, The homotopy of Hopf spaces, North-Hollan Math. Library 0. [6] A.Kono, A note on the homotopy type of certain gauge groups, Proc. Royal Soc. Einburgh, 117A(1991), [7] G.E.Lang, The evaluation map an EHP sequence, Pacific J. Math., (1973), [8] H.Matsunaga, The homotopy groups π n+i (U(n)) for i = 3, an 5. Mem. Fac. Sci. Kyushu Univ., 15(1961), [9] W.A.Sutherlan, Function spaces relate to gauge groups, Proc. Roy. Soc. Einburgh, 11A(199), [10] H.Toa, A survey of homotopy theory, Av. in Math. 10(1973),