Obstructions to Transversality for Compact Lie Groups

Throughout G is a compact Lie group which is topologically cyclic with dense generator g. Let N and M be smooth G manifolds without boundary and Fez M a closed invariant submanifold. All manifolds are oriented and G preserves orientation. Let f :N-^M be a proper G map. When is ƒ properly G homotopic to a map y which is transverse regular to Fc iM, written y(\\Y! We introduce obstructions which show that transversality is a global phenomena in contrast to the case G = l where everything is local and trivial. Without loss of generality, we may assume that ƒ : N->M is transverse to Y and set X = (f)~(Y). For each oriented real G vector bundle v over X such that the G representation on each fiber of v has no trivial factor and g preserves orientation on each fiber, let A±(v) be the ± eigenbundles of the canonical involution r on A(t;®C)=2 W(v®C) constructed from the orientation and an inner product on v. Let A__1(t;®C)= 2 (l)U ( t)0C), I e KG(TX ) be the index class of X9 i.e. the symbol of the operator D+. See [1, p. 575]. Let 0>^R(G) be the prime ideal of characters {X e R(G)\X(g)=Ö} and