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

(1)设M为光滑g流形。本文研究了具有紧支撑的同位素与恒等的保g作用的M的微分同胚、Lipschtz同胚和同胚。当M的余维为1时，我们确定了这些群的第一个同源性。从这些结果我们看到，第一个同调反映了这些范畴的性质。(2)考虑光滑流形对的微分同构的恒等分量群的完备性和一致完备性。确定了子流形维数为1时群一致完美的条件。所得结果可应用于结理论。

(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.