An ideal-valued cohomological index theory with applications to Borsuk—Ulam and Bourgin—Yang theorems

Abstract

AbstractNumerical-valued cohomological index theories for G-pairs (X, A) over B, where G is a compact Lie group, have proved useful in critical point theory and in proving Borsuk—Ulam and Bourgin—Yang theorems. More information (which is lost in taking numerical values) is obtained using an ideal-valued theory, and this theory is applied to estimating the size of the zero set of a G-map from certain G-manifolds to a G-module. Parametrized versions of these theorems are also obtained by a principle which applies quite generally.