Odd primary homotopy types of SU(n)–gauge groups

Abstract

Abstract. Let G k (SU (n)) be the gauge group of the principal SU (n)-bundle with second Chern class k. If p is an odd prime and n ≤ (p − 1) 2 + 1, we classify the p-local homotopy types of G k (SU (n)).

Homotopy Types of Gauge Groups Over Non-Simplyconnected Closed 4-Manifolds

“…Theorem 1.2 shows that the homotopy type of G t (M) is related to that of G t (S 4 ) or G t (CP 2 ) in these three cases. Combining Theorem 1.2 and the known results in [3,11,10,19,22], we have the following classification. • when G = SU(n), there is a p-local homotopy equivalence G t (M) ≃ G s (M) if and only if (n(n 2 − 1), t) = (n(n 2 − 1), s) for any odd prime p such that n ≤ (p − 1) 2 + 1;…”

“…Let (a, b) be the greatest common divisor of a and b. Then [3,8,10,11,19,21,22] • when G = SU(2), there is a homotopy equivalence G t (S 4 ) ≃ G s (S 4 ) if and only if (12, t) = (12, s);…”

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

“…Theriault showed in [22] that, after localisation at an odd prime p and provided that n < (p − 1) 2 + 1, the order of the Samelson product δ 2 , 1 : S 3 ∧ SU(n) → SU(n) is the p-primary component of the integer n(n 2 − 1).…”

“…In [21], Theriault showed that the order of δ 2 , 1 : S 3 ∧ SU(5) → SU( 5) is 120. Hence, by Theorem 1, the order of 2 , 1 : S 3 ∧ PU(5) → PU( 5) is also 120.…”

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

“…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)…”