Let G be a simple, simply connected, compact Lie group, and let M be an orientable, smooth, connected, closed 4-manifold. In this paper, we calculate the homotopy type of the suspension of M and the homotopy types of the gauge groups of principal G-bundles over M when π1(M) is (1) ℤ*m, (2) ℤ/prℤ, or (3) ℤ*m*(*nj=1ℤ/prjjℤ), where p and the pj's are odd primes.
Let G be a simple, simply-connected, compact Lie group of low rank relative to a fixed prime p. After localization at p, there is a space A which "generates" G in a certain sense. Assuming G satisfies a homotopy nilpotency condition relative to p, we show that the Samelson product 1 G , 1 G of the identity of G equals the order of the Samelson product ı, ı of the inclusion ı : A → G. Applying this result, we calculate the orders of 1 G , 1 G for all p-regular Lie groups and give bounds of the orders of 1 G , 1 G for certain quasi-p-regular Lie groups.
Let G be a simple, simply-connected, compact Lie group and let M be an orientable, smooth, connected, closed 4-manifold. In this paper we calculate the homotopy type of the suspension of M and the homotopy types of the gauge groups of principal Gbundles over M when π 1 (M ) is (1) Z * m , (2) Z/p r Z, or (3) Z * m * ( * n j=1 Z/p rj j Z), where p and the p j 's are odd primes.
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