Nonlinear Similarity Begins in Dimension Six

Cappell, Sylvain E.; Shaneson, Julius L.; Steinberger, Mark; West, James E.

doi:10.2307/2374878

Cited by 20 publications

(24 citation statements)

References 17 publications

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“…According to [7,Th. 2], the element t − t 5 has order 2 r−2 in R Top (G) for G = C(2 r ), r > 3, and order 4 for G = C (8). We will verify this claim using our methods.…”

Section: ])mentioning

confidence: 68%

“…The 6-dimensional case was worked out in detail for cyclic 2-groups in [8], and general conditions A-D were announced in [7] for the classification of nonlinear similarities for cyclic 2-groups in any dimension. However, in Example 9.7 we give a counterexample to the necessity of [7,Cond.…”

Section: ])mentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…It is easy to see that the existence of a nonlinear similarity V ∼ t V implies dim V = dim V 5. Cappell, Shaneson, Steinberger and West [8] proved that 6-dimensional similarities exist for G = C(2 r ), r 4 and referred to the 1981 Cappell-Shaneson preprint (now published [6]) for the complete proof that 5-dimensional similarities do not exist for any finite group. See Corollary 9.3 for a direct argument using the criterion of Theorem A in the special case of cyclic 2-groups.…”

Section: Introductionmentioning

confidence: 99%

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Topological equivalence of linear representations for cyclic groups: II

Hambleton

Pedersen

2005

Forum Mathematicum

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“…According to [7,Th. 2], the element t − t 5 has order 2 r−2 in R Top (G) for G = C(2 r ), r > 3, and order 4 for G = C (8). We will verify this claim using our methods.…”

Section: ])mentioning

confidence: 68%

Section: ])mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Topological equivalence of linear representations for cyclic groups: II

Hambleton

Pedersen

2005

Forum Mathematicum

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“…For example, the concrete structure of these groups is relevant to the classification of linear representations of finite groups up to topological conjugacy ~5, 6,18] and the recent survey article I13] describes other applications. The purpose of the present paper is to provide a reference for further computations in this area.…”

Section: O Introductionmentioning

confidence: 99%

On the computation of the projective surgery obstruction groups

Hambleton¹,

Madsen²

1993

K-Theory

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Abstract. The computation of the projective surgery obstruction groups LP. (ZG), for G a hyperelementary finite group, is reduced to standard calculations in number theory, mostly involving class groups. Both the exponent of the torsion subgroup and the precise divisibility of the signatures are determined. For G a 2-hyperelementary group, the LP.(ZG) are detected by restriction to certain subquotients of G, and a complete set of invariants is given for oriented surgery obstructions.Key words. Surgery on manifolds, Hermitian K-theory. O. IntroductionThe projective surgery obstruction groups were first introduced by S. P. Novikov 1-30] in the context of Hermitian K-theory and the topology of infinite cycle covers of compact manifolds. These groups arise as the codimension 1 summands in a splitting theorem for the Wall surgery obstruction groups of a Laurent polynomial extension 1,39, 12], or more generally in the classification of noncompact manifolds [40,41,27,33]. The algebraic description of projective L-theory and the splitting theorem were given a systematic exposition by A. A. Ranicki 1-34, 36], including a definition of the lower L-groups by analogy with the lower K-groups of Bass.With the extensive development of 'bounded' or 'controlled' topology in the last decade, the role of projective and lower surgery obstruction groups has increased in importance. For example, the concrete structure of these groups is relevant to the classification of linear representations of finite groups up to topological conjugacy ~5, 6, 18] and the recent survey article I13] describes other applications. The purpose of the present paper is to provide a reference for further computations in this area.

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