Nonabelian cohomologies of cyclic groups with coefficients in Lie groups

Abstract. In this paper we prove some properties of nonabelian cohomologies of cyclic groups with coefficients in Lie groups. In the case that the cyclic group is Z, we show that if G is a connected compact Lie group which has a nonabelian Z-module structure, then the canonical map H 1 (Z, T) → H 1 (Z, G) is surjective with finite fibers if and only if the Z-action on G is 1-semisimple, where T is a maximal compact torus of G Z. In the case that the cyclic group is Zn, we show that for arbitrary connected Lie group G which has a nonabelian Zn-module structure, there always exists a maximal compact subgroup K of G which is also a Zn-submodule, and the canonical map H 1 (Zn, K) → H 1 (Zn, G) is bijective, generalizing a theorem in Serre [10]. This implies that for each maximal compact torus T of G Zn, the canonical map H 1 (Zn, T) → H 1 (Zn, G) is surjective. In this case we also show that H 1 (Zn, G) coincides with the set of connected components of Z 1 (Zn, G), and each connected component of Z 1 (Zn, G) is a closed submanifold of G. Other conclusions which are also proved include that, for example, H 1 (Zn, G) is always finite, and for each of the above two cases, every connected component of H 0 (A, G/T) is a closed homogeneous submanifold of G/T, and the restriction of the coboundary operator δ: H 0 (A, G/T) → H 1 (A, T) to each connected component is constant, where A = Z or Zn, respectively.