Homotopy types of gauge groups of $$\mathrm {PU}(p)$$-bundles over spheres

Abstract

We examine the relation between the gauge groups of $$\mathrm {SU}(n)$$ SU ( n ) - and $$\mathrm {PU}(n)$$ PU ( n ) -bundles over $$S^{2i}$$ S 2 i , with $$2\le i\le n$$ 2 ≤ i ≤ n , particularly when n is a prime. As special cases, for $$\mathrm {PU}(5)$$ PU ( 5 ) -bundles over $$S^4$$ S 4 , we show that there is a rational or p-local equivalence $$\mathcal {G}_{2,k}\simeq _{(p)}\mathcal {G}_{2,l}$$ G 2 , k ≃ ( p ) G 2 , l for any prime p if, and only if, $$(120,k)=(120,l)$$ ( 120 , k ) = ( 120 , l ) , while for $$\mathrm {PU}(3)$$ PU ( 3 ) -bundles over $$S^6$$ S 6 there is an integral equivalence $$\mathcal {G}_{3,k}\simeq \mathcal {G}_{3,l}$$ G 3 , k ≃ G 3 , l if, and only if, $$(120,k)=(120,l)$$ ( 120 , k ) = ( 120 , l ) .

Homotopy types of gauge groups over Riemann surfaces

“…Subsequently, the precise number of homotopy types of gauge groups for specific G and X has been intensely studied. The study began with simply connected Lie groups by Cutler [7], Hamanaka, Hasui, Kishimoto, Kono, So, Theriault and Tsutaya [10; 12; 15; 16; 18; 20; 30; 31], and recently, nonsimply connected cases are also studied by Hasui, Kamiyama, Kishimoto, Kono, Membrillo-Solis, Sato, Theriault and Tsukuda [11; 14; 17] and Rea [26].…”

Homotopy types of gauge groups over Riemann surfaces

“…Then it has been intensely studied the precise number of homotopy types of gauge groups for specific G and X. The study began with simply-connected Lie groups [6,10,12,15,16,18,20,28,29], and recently, non-simply-connected cases are also studied as in [11,14,17,24].…”