Numerical-valued index theories for G-spaces and G-pairs, when G is a compact Lie group, play an important role in the Ljusternik-Schnirelmann method and in theorems of the Bourgin- Yang type. The investigators have begun a study of an ideal- valued theory which can handle situations where the numerical- valued indices fail. A striking application is a Borsuk-Ulam theorem for maps on a Stiefel manifold with orthogonal group action where the numerical index is inadequate. This ideal- valued theory can be carried over to the case when the symmetry group G is not compact, within the context of infinitesimal deRham G-cohomology, and can be applied to obtain Borsuk-Ulam theorems where G is not compact. A deeper study of the ideal- valued theory as it applies to the critical point theory of invariant functionals, to Borsuk-Ulam theorems and to the equivariant J-homomorphism is a primary objective. The investigators also plan to continue their research in fixed point theory with special emphasis on the Nielsen fixed point theory of G-maps and local Reidemeister trace theory. They have made successful application of their prior work along these lines to establish the existence of equilibria for certain economic models.
