On Knaster’s problem

Dold?s theorem gives sufficient conditions for proving that there is no G-equivariant mapping between two spaces. We prove a generalization of Dold?s theorem, which requires triviality of homology with some coefficients, up to dimension n, instead of n-connectedness. Then we apply it to a special case of the famous Knaster?s problem, and obtain a new proof of a result of C.T. Yang, which is much shorter and simpler than previous proofs. Also, we obtain a positive answer to some other cases of Knaster?s problem, and improve a result of V.V. Makeev, by weakening the conditions.