The homotopy types of PSp(n)-gauge groups over S2m
Introduction
Let G be a compact connected Lie group and let be a principal G-bundle over connected finite complex B. The gauge group 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:
-bundles over (see [4]); -bundles over when localized at any prime p or rationally (see [14]); -bundles over when localized at any prime p or rationally (see [15]); -bundles over when localised at an odd prime (see [2]); -bundles over (see [5]) and in the general case, -bundles over (see [10]); -bundles over (see [6]); -bundles over (see [9]).
In [7] Hasui, Kishimoto, Kono and Sato classify the homotopy types of the gauge groups for groups - and over . Also Simon Rea in [12] studies the homotopy types of the gauge groups of -bundles over spheres. The purpose of this article is to extend results in the direction of [7] by considering and and and , respectively. In each case let be the principal G-bundle classified by and where and are the generators of and , respectively, and let be the gauge group of . For integers a and b let be the greatest common divisor of and , we will prove the following theorems. Theorem 1.1 For , the following hold: (a) if is homotopy equivalent to then , (b) if then . Theorem 1.2 For , the following hold: (a) if is homotopy equivalent to then , (b) if then after localisation at any prime.
Section snippets
Preliminaries
Let M be a co-H-space that satisfies . Let be the component of the space of continuous unbased maps from M to BG which contains the map inducing P, similarly let 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 fibration where the map ev is evaluation map at the basepoint of M. Let be the classifying spaces of . By Atiyah-Bott [1], there is a homotopy
Samelson products in
Let A be the 7-skeleton of , that is . In this section we first compute and conclude that if then . We then prove part (b) of Theorem 1.1.
Recall that and that can be given the CW-structure of a three-cell complex . There is a homotopy cofibration sequence where g is the attaching map for A and is the pinch map to the top cell.
According to the projection homomorphism
Samelson products in
In this section we first show the order of the Samelson Product is either 168, 336 or 672. Next we show that if then , and in end we will prove Theorem 1.2.
Recall that and has the following CW-structure As in the case, in the projection homomorphism we have the following relation where 1 is the identity map on . The
Acknowledgement
The author thanks the reviewer for his helpful comments and suggestions.
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