The homotopy types of PSp(n)-gauge groups over S2m

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Abstract

Let Gk(PSp(2)) and Gk(PSp(3)) be the gauge groups of principal PSp(2)-bundles over S8 and principal PSp(3)-bundles over S4 classified by kε1 and kε2, where ε1 and ε2 are generators of π8(BPSp(2))Z and π4(BPSp(3))Z, respectively. In this article we partially classify the homotopy types for these gauge groups.

Introduction

Let G be a compact connected Lie group and let PB be a principal G-bundle over connected finite complex B. The gauge group G(P) of P is the group of G-equivariant automorphisms of P which fix B. Crabb and Sutherland [3] have shown that if B and G are as above, then the number of homotopy types amongst all the gauge groups of principal G-bundles over B is finite. This is in spite of the fact that the number of isomorphism classes of principal G-bundles over B is often infinite.

It has been a subject of recent interest to determine the precise number of homotopy types in special cases. Precise enumerations of the homotopy types have been made in recent years. In the following, we mention some cases:

SU(3)-bundles over S4 (see [4]); SU(5)-bundles over S4 when localized at any prime p or rationally (see [14]); Sp(2)-bundles over S4 when localized at any prime p or rationally (see [15]); Sp(3)-bundles over S4 when localised at an odd prime (see [2]); SU(3)-bundles over S6 (see [5]) and in the general case, SU(n)-bundles over S6 (see [10]); Sp(2)-bundles over S8 (see [6]); SU(4)-bundles over S8 (see [9]).

In [7] Hasui, Kishimoto, Kono and Sato classify the homotopy types of the gauge groups for groups PU(3)- and PSp(2) over S4. Also Simon Rea in [12] studies the homotopy types of the gauge groups of PU(p)-bundles over spheres. The purpose of this article is to extend results in the direction of [7] by considering G=PSp(2) and B=S8 and G=PSp(3) and B=S4, respectively. In each case let Pk be the principal G-bundle classified by kε1 and kε2 where ε1 and ε2 are the generators of π8(BPSp(2))Z and π4(BPSp(3))Z, respectively, and let Gk=Gk(G) be the gauge group of Pk. For integers a and b let (a,b) be the greatest common divisor of |a| and |b|, we will prove the following theorems.

Theorem 1.1

For G=PSp(2), the following hold:

(a) if Gk(PSp(2)) is homotopy equivalent to Gk(PSp(2)) then (140,k)=(140,k),

(b) if (140,k)=(140,k) then ΩGk(PSp(2))ΩGk(PSp(2)).

Theorem 1.2

For G=PSp(3), the following hold:

(a) if Gk(PSp(3)) is homotopy equivalent to Gk(PSp(3)) then (84,k)=(84,k),

(b) if (672,k)=(672,k) then ΩGk(PSp(3))ΩGk(PSp(3)) after localisation at any prime.

Section snippets

Preliminaries

Let M be a co-H-space that satisfies [M,BG]Z. Let Mapk(M,BG) be the component of the space of continuous unbased maps from M to BG which contains the map inducing P, similarly let Mapk(M,BG) be the component of the space of pointed continuous maps from M and BG which contains the map inducing P. We know that there is a fibrationMapk(M,BG)Mapk(M,BG)evBG, where the map ev is evaluation map at the basepoint of M. Let BGk be the classifying spaces of Gk. By Atiyah-Bott [1], there is a homotopy

Samelson products in PSp(2)

Let A be the 7-skeleton of Sp(2), that is A=S3e7. In this section we first compute [A,BGk(PSp(2))] and conclude that if Gk(PSp(2))Gk(PSp(2)) then (140,k)=(140,k). We then prove part (b) of Theorem 1.1.

Recall that H(Sp(2);Z)Λ(x3,x7) and that Sp(2) can be given the CW-structure of a three-cell complex Sp(2)=S3e7e10. There is a homotopy cofibration sequenceS6gS3AπS7, where g is the attaching map for A and π is the pinch map to the top cell.

According to the projection homomorphism π:Sp(

Samelson products in PSp(3)

In this section we first show the order of the Samelson Product ε2,1PSp(3) is either 168, 336 or 672. Next we show that if Gk(PSp(3))Gk(PSp(3)) then (84,k)=(84,k), and in end we will prove Theorem 1.2.

Recall that H(Sp(3);Z)Λ(x3,x7,x11) and Sp(3) has the following CW-structureSp(3)S3v1e7e10e11e14e18e21. As in the PSp(2) case, in the projection homomorphism π:Sp(3)PSp(3) we have the following relationε2,1PSp(3)(1π)=πε2¯,1Sp(3), where 1 is the identity map on S3. The

Acknowledgement

The author thanks the reviewer for his helpful comments and suggestions.

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