Homotopy Type of

Abstract

The homotopy type of classifying spaces of gauge groups of principal SU(2) bundle over S 4 are classified by the absolute value of their instanton numbers. 1 Introduction Let P k be the principal SU(2) bundle over S 4 with c 2 (P k ) = k 2 Z, G k its gauge group. It is known ([1]) that BG k ' Map k (S 4 ; BSU(2)), where Map k (S 4 ; BSU(2)) is the connected component of Map(S 4 ; BSU(2)) including the map inducing P k . In this paper we will show the following result. Let BG k(p) denote the BG k localized at a prime p. Theorem 1.1. The following four conditions are equivalent. 1. jkj = jlj 2. G k ¸ = G l 3. BG k ' BG l 4. BG k(p) ' BG l(p) for any prime p. Remark 1.2. In the case of based gauge groups, G 3 k ¸ = G 3 0 and BG 3 k ' BG 3 0 . Remark 1.3. The conditions (3) and (4) are not always equivalent for general spaces. Remark 1.4. It is known that two compact Lie groups G and H are isomorphic as Lie groups if and only if BG and BH are homotopy equivale..