Equivariant Alexander–Spanier cohomology and P.A. Smith theory

In Part 1 of the paper we give a new construction of equivariant Alexander–Spanier cohomology for actions of a finite group G, which is simpler and more direct than the original one given in [2]. In Part 2 we prove some useful results concerning equivariant Alexander–Spanier cohomology theory. In Part 3 we use equivariant Alexander–Spanier cohomology and the results proved in Part 2, in the case where G=CpG=Cp is a cyclic group of prime order p, to develop P.A. Smith theory from scratch, under minimal, in fact necessary, assumptions.