Second Cohomology Classes of the Group of C-flat Diffeomorphisms of the Line

We study the cohomology of the group consisting of all C∞diffeomorphisms of the line, which are C1-flat to the identity at the origin. We construct non-trivial two second real cohomology classes and uncountably many second integral homology classes of this group. 1. Notations and main results We denote by a1 the Lie algebra of all formal vector fields on R with the Krull topology. For k ≥ 0, we denote by a1 the Lie subalgebra of a1 consisting of formal vector fields which are C-flat at the origin. Let Diff0 (R) be the group of orientation-preserving C∞-diffeomorphisms of R which fix the origin. Let G(1) be the group of germs of local C∞-diffeomorphisms at the origin of R. Let G∞(1) be the group of ∞-jets of local C∞-diffeomorphisms at the origin of R. For k ≥ 1, we denote by Diffk (R), Gk(1) and Gk (1) the subgroup of Diff ∞ 0 (R), G(1) and G∞(1) respectively, consisting of elements which are C-flat to the identity at the origin. The groups G∞(1) and Gk (1) can be considered as infinite-dimensional Lie groups, whose Lie algebras are a1 and a k 1 , respectively. We define the Gel’fand-Fuks cohomology of a1 in §2. It is known to be 2dimensional for each degree([2], [6]). Moreover, Millionschikov proved its generators in degree greater than 1 can be described by the Massey products([4]). We carried out the calculation of the Massey products on Diff1 (R), and we give two 2-cocycles of Diff1 (R) in §3. For l ≥ k and i ≥ 2, let αl and α l be the 1-cochains of Diff ∞ k (R) defined by αl(f) = d dxl f(0) for f ∈ Diffk (R), and α l(f) = αl(f) i for f ∈ Diffk (R), respectively. Then the following proposition holds. 2000 Mathematics Subject Classification. Primary 58D05, 57S05.