In the two parts of this paper we prove that the Reidemeister torsion invariants determine topological equivalence of G-representations, for G a ﬁnite cyclic group. 1. Introduction Let G be a ﬁnite group and V , V ﬁnite dimensional real orthogonal representations of G. Then V is said to be topologically equivalent to V (denoted V ∼t V ) if there exists a homeomorphism h : V → V which is G-equivariant. If V , V are topologically equivalent, but not linearly isomorphic, then such a homeomorphism is called a nonlinear similarity. .