General Position of Equivariant Maps

Abstract

A natural generic notion of general position for smooth maps which are equivariant with respect to the action of a compact Lie group is introduced. If G is a compact Lie group, and M, N are smooth G-manifolds, then the set of smooth equivariant maps $F:M \to N$ which are in general position with respect to a closed invariant submanifold P of N, is open and dense in the Whitney topology. The inverse image of P, by an equivariant map in general position, is Whitney stratified. The inverse images, by nearby equivariant maps in general position, are topologically ambient isotopic. In the local context, let V, W be linear G-spaces, and $F:V \to W$ a smooth equivariant map. Let ${F_1}, \ldots ,{F_k}$ be a finite set of homogeneous polynomial generators for the module of smooth equivariant maps, over the ring of smooth invariant functions on V. There are invariant functions ${h_1}, \ldots ,{h_k}$ such that $F = U \circ$ graph h, where graph h is the graph of $h(x) = ({h_1}(x), \ldots ,{h_k}(x))$, and $U(x,h) = \Sigma _{i = 1}^k{h_i}{F_i}(x)$. The isomorphism class of the real affine algebraic subvariety $(U = 0)$ of $V \times {{\mathbf {R}}^k}$ is uniquely determined (up to product with an affine space) by V, W. F is said to be in general position with respect to $0 \in W$ at $0 \in V$ if graph $h:V \to V \times {{\mathbf {R}}^k}$ is transverse to the minimum Whitney stratification of $(U = 0)$, at $x \in V$.