SECOND COHOMOLOGY CLASSES OF THE GROUP OF C-FLAT DIFFEOMORPHISMS OF THE LINE
SECOND COHOMOLOGY CLASSES OF THE GROUP OF C-FLAT DIFFEOMORPHISMS OF THE LINE
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直线C-平微分态群的第二上同调类
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发表时间：
2010
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通讯作者：
Tomohiko Ishida
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作者：
Tomohiko Ishida;Tomohiko Ishida
我们研究了直线上所有C∞微分同态组成的群的上同态，这些群在原点上是c1平的。构造了这个群的非平凡二秒实上同调类和无数个二阶整上同调类。1. 我们用a1表示R上具有Krull拓扑的所有形式向量场的李代数。当k≥0时，我们用a1表示a1的李子代数，它由在原点是c平的形式向量场组成。设Diff0 (R)为R的一组保持方向且固定原点的C∞-微分同态。设G(1)为R原点处局部C∞-微分同态的芽群。设G∞(1)为R原点处局部C∞-微分同态的∞-喷流群。对于k≥1，分别用Diffk (R)、Gk(1)和Gk(1)表示Diff∞0 (R)、G(1)和G∞(1)的子群，它们由原点处C平于恒等的元素组成。群G∞(1)和群Gk(1)可以看作无穷维李群，它们的李代数分别为a1和a1。我们在§2中定义了a1的Gel 'fand-Fuks上同。已知每个度（[2],[6]）都是二维的。此外，Millionschikov还证明了其产生器的次数大于1可以用Massey积（[4]）来描述。我们计算了Diff1 (R)上的Massey积，并在§3中给出了Diff1 (R)的两个2-环。当l≥k， i≥2时，设αl和αl为Diff∞k (R)的1-余链，分别定义为对于f∈Diffk (R) αl(f) = d dxl f(0)，对于f∈Diffk (R) αl(f) = αl(f) i。那么下面的命题成立。2000数学学科分类。主58D05, 57S05。
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.