TAILIEUCHUNG - Đề tài " Topological equivalence of linear representations for cyclic groups: I "

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. . | Annals of Mathematics Topological equivalence of linear representations for cyclic groups I By Ian Hambleton and Erik K. Pedersen Annals of Mathematics 161 2005 61 104 Topological equivalence of linear representations for cyclic groups I By Ian Hambleton and Erik K. Pedersen Abstract In the two parts of this paper we prove that the Reidemeister torsion invariants determine topological equivalence of -representations for G a finite cyclic group. 1. Introduction Let G be a finite group and V V1 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. These notions were introduced and studied by de Rham 31 32 and developed extensively in 3 4 22 23 and 8 . In the two parts of this paper referred to as I and II we complete de Rham s program by showing that Reidemeister torsion invariants and number theory determine nonlinear similarity for finite cyclic groups. A G-representation is called free if each element 1 g E G fixes only the zero vector. Every representation of a finite cyclic group has a unique maximal free subrepresentation. Theorem. Let G be a finite cyclic group and V-Ị- V2 be free G-represen-tations. For any G-representation W the existence of a nonlinear similarity VI W t V2 W is entirely determined by explicit congruences in the weights of the free summands Vi- V2 and the ratio A V1 A V2 of their Reidemeister torsions up to an algebraically described indeterminacy. Partially supported by NSERC grant A4000 and NSF grant DMS-9104026. The authors also wish to thank the Max Planck Institut fur Mathematik Bonn for its hospitality and support. 62 IAN HAMBLETON AND ERIK K. PEDERSEN The notation and the indeterminacy are given in Section 2 and a detailed statement of results in Theorems A-E. For cyclic groups of .

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