The Equivariant Bundle Subtraction Theorem and its applications

In the theory of transformation groups, it is important to know what kind of isotropy subgroups of G do occur at points of the space upon which the given group G acts. In this article, for a finite group G, we prove the Equivariant Bundle Subtraction Theorem (Theorem 2.2) which allows us to construct smooth G-manifolds with prescribed isotropy subgroups around the G-fixed point sets. In Theorem 0.1, we restate Oliver’s result about manifolds M and G-vector bundles over M that occur, respectively, as the G-fixed point sets and their equivariant normal bundles for smooth G-actions on disks. In Theorems 0.2 and 0.3, we prove the corresponding results for smooth G-actions on disks with prescribed isotropy subgroups around M . In Theorems 0.4 and 0.5, for large classes of finite groups G, we explicitly describe manifolds M that occur as the G-fixed point sets for such actions on disks. These actions are expected to be useful for answering the question of which manifolds occur as the G-fixed points sets for smooth G-actions on spheres. 0. Introduction. In the theory of transformation groups, it often happens that a solution of a particular problem depends on the family of the isotropy subgroups that we allow to occur at points in the space upon which a given group G acts. In [O4], for a finite group G not of prime power order, Oliver describes necessary and sufficient conditions under which a smooth manifold M occurs as the G-fixed point set and a smooth G-vector ν over M stably occurs as the equivariant normal bundle of M in D (resp., E) for a 1991 Mathematics Subject Classification: Primary 57S17, 57S25; Secondary 55M35, 55R91.