On the homotopy types of Sp(n) gauge groups

Abstract

Let G k,n be the gauge group of the principal Sp(n)-bundle over S 4 corresponding to k ∈ Z ∼ = π3(Sp(n)). We refine the result of Sutherland on the homotopy types of G k,n and relate it with the order of a certain Samelson product in Sp(n). Then we classify the p-local homotopy types of G k,n for (p − 1) 2 + 1 ≥ 2n.2010 Mathematics Subject Classification. 55P15, 54C35.

Homotopy of gauge groups over non-simply-connected five-dimensional manifolds

“…Theriault [48] and Kishimoto-Kono [29] Spin(2n) (n 3) 2(n − 1) (p−1) 2 +1, p 5 In [49], Theriault proved a very useful lemma in the study of the homotopy of gauge groups, which is a general statement and then should be applicable to other situations. We will mainly use mod p version of his lemma.…”

“…Theriault [51] Sp(n) (n 2) 2n (p − 1) 2 +1, p 3 n(2n + 1) Kishimoto-Kono [29] Spin(2n + 1) (n 2) 2n (p − 1) 2 +1, p 3 n(2n + 1)…”

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

“…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.…”

Odd primary homotopy types of the gauge groups of exceptional Lie groups

“…As we will see, the homotopy type of G k is closely related with a certain Samelson product in G. It is shown in [11] that if p is a prime large with respect to the rank of G, then the classification of the p-local homotopy types of G k reduces to determining the order of this Samelson product. So in [10,21], the p-local classification is obtained by calculating this Samelson product when G is a classical group.…”