Borsuk-ulam Type Theorems for Compact Lie Group Actions

Borsuk-Ulam type theorems for arbitrary compact Lie group actions are proven. The transfer plays a major role in this approach. We present Borsuk-Ulam type theorems for arbitrary compact Lie group actions. The essence of our approach is a generalization of the ideal-valued index of FadellHusseini [FH88] using transfer [Boa66], [BG75], [Dol76], [KP72], [Rou71]. Once an appropriate concept (Definition 0.3) is established in terms of transfer, then the proofs of our main results (Theorem 0.5, Theorem 0.6, and Corollary 0.9) are straightforward and simple. We start with some standard definitions of the transfer and Gysin homomorphism. Definition 0.1. (i) (cf. [BG75], [LMS86]) When G is a compact Lie group, any compact G-manifold F admits a G-embedding F ⊂ W in a finite-dimensional real G-representation W equipped with an invariant G-metric, by a theorem of Mostow [Mos57]. Then the Pontryagin-Thom construction gives us a map t : S → F ν , where ν is the G-equivariant normal bundle of F. However, we prefer to rewrite this in the equivariant stable homotopy category [LMS86] as t : S → F−τ(F , where τ(F ) is the tangent G-equivariant bundle of G satisfying τ(F )⊕ ν ∼= F ×W. Then for any honest G-equivariant vector bundle ξ on F, we have the composite t(ξ) : S γ −→ F−τ(F ) → F−τ(F )⊕ξ, where the second map between Thom spectra is induced by the inclusion of Gequivariant virtual bundles −τ(F ) ⊂ −τ(F ) ⊕ ξ. Given a G-CW complex X, smashing t(ξ) with the identity of the suspension spectrum ΣX+ yields another morphism, which is also denoted by t(ξ): t(ξ) : ΣX+ → F−τ(F )⊕ξ ∧ ΣX+. Received by the editors March 22, 2002 and, in revised form, August 12, 2002 and October 25, 2002. 2000 Mathematics Subject Classification. Primary 58E40, 55R12, 55N20; Secondary 55R35.