The Borsuk–Ulam Type Theorems for Finite-Dimensional Compact Group Actions

Clapp and Puppe (J. Reine Angew Math 418:1–29, 1991) proved that, if G is a torus or a p-torus, X is a path-connected G-space and Y is a finite-dimensional G-CW complex without fixed points, under certain cohomological conditions on X and Y, there is no equivariant map from X to Y. Also, Biasi and Mattos (Bull Braz Math Soc New Ser 37:127–137, 2006) proved that, again under certain cohomological conditions on X and Y, there is no equivariant map from X to Y provided that G is a compact Lie group and X, Y are path-connected, paracompact, free G-spaces. In this paper, our objective is to generalize these results for the actions of finite-dimensional pro-tori and compact groups, respectively.