On the Homotopy Types of the Groups of Equivariant Diffeomorphisms

The purpose of this paper is to study the homotopy type of the group of the equivariant diffeomorphisms of a closed connected smooth G-manifold M, when G is a compact Lie group and the orbit space M/G is homeomorphic to a unit interval [0, 1]. Let Diffg (M)0 denote the group of equivariant C°° diffeomorphisms of the G-manifold M which are G-isotopic to the identity, endowed with C°° topology. If M/G is homeomorphic to [0, 1], then M has two or three orbit types G/H, G/K0 and GjKl. We can choose the isotropy subgroups H, K0, K^ satisfying HdK0nK1. Moreover the G-manifold structure of M is determined by an element r\ of a factor group N(H)/H9 where N(H) is the normalizer of H in G (see §1). Let Q(N(H)/H; (N(H) fl N(K0))IH, (N(H) n N(rjK1ri~ y)IH)0 denote the connected component of the identity of the space of paths a: [0, l]-+N(H)IH satisfying a(0)e(N(H) n N(X0))/H and a(l)e(N(H) n