Automorphism group of a smooth G-manifold and its applications.

(1)设M是光滑G-流形。我们考虑M的微分同胚、Lipschtz同胚和保持G-作用的同胚，它们通过紧支撑的同态与恒等式同构。当M有余维1个轨道时，我们确定了这些群的第一同调。从这些结果可以看出，第一同调反映了这些范畴的性质。(2)我们考虑了光滑流形对的微分同胚群的单位分支群的完备性和一致完备性。我们确定了当子流形的维度为1时，群一致完美的条件。这一结果可应用于纽结理论。

(1) Let M be a smooth G-manifold. We consider the diffeomorphisms, Lipschtz homeomorphisms and homeomorphisms of M preserving the G-action which are isotopic to the identity through the isotopies with compact support. We determined the first homology of those groups when M has codimension one orbits. From those results we see that the first homology reflects the properties of those categories.(2) We consider the perfectness and uniformly perfectness for the group of the identity component of the diffeomorphisms of a smooth manifold pair. We have determined the condition for the group to be uniformly perfect when the dimension of the submanifold is one. The result can be applied to the knot theory.