In this paper, we state and prove a $G$-equivariant version of the Dold Theorem mod $k$ for finite groups $G$. We then use this theorem to prove an equivariant version of the Adams Conjecture for $G$ cyclic, using the Becker-Gottlieb approach. The case for general $G$ and finite structure groups is also obtained by the methods of Quillen.

We would like to express our gratitude to Professor J. P. May for his encouragement and many useful suggestions, and to the referee for his critical reading of the manuscript, and for his improvements on several of our proofs.

In 1976 at the London Mathematical Symposium on Potential Theory at Durham, England, L. A. Rubel posed the problem of characterising weak Arakelian sets or at least showing whether or not there exists an Arakelian glove which is weak Arakelian. We shall give an example of an Arakelian glove which is weak Arakelian and we shall give a necessary condition for a closed set $F$ in the plane to be weak Arakelian. It seems likely that this necessary condition is also sufficient for $F$ to be weak Arakelian but at the present time we are not able to prove the sufficiency.