Fibre techniques in Nielsen periodic point theory on nil and solvmanifolds II

Heath, Philip R.; Keppelmann, Edward C.

doi:10.1016/s0166-8641(99)00079-6

Cited by 20 publications

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“…Note that IEO (1) (f ) = N (f ), and IEO (n) (f ) = (1/n)N P n (f ) (as defined in [6, p. 69]). The proof of Theorem 2.4 also leads to the following consequence, which is comparable to [3,Theorem 3.4].…”

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confidence: 55%

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Reidemeister orbit sets

2004

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Abstract. The Reidemeister orbit set plays a crucial role in the Nielsen type theory of periodic orbits, much as the Reidemeister set does in Nielsen fixed point theory. Extending Ferrario's work on Reidemeister sets, we obtain algebraic results such as addition formulae for Reidemeister orbit sets. Similar formulae for Nielsen type essential orbit numbers are also proved for fibre preserving maps. Introduction. Nielsen fixed point theory has been extended to aNielsen type theory of periodic orbits [6, Section III.3]. In fixed point theory, the computation of the Nielsen number often relies on the knowledge of the Reidemeister set, that is, the set of Reidemeister conjugacy classes in the fundamental group. Ferrario [2] made an algebraic study of the Reidemeister set in relation to an invariant normal subgroup. He obtained addition formulae for Reidemeister numbers, and applied them to the Nielsen number of fibre preserving maps. Our aim in this paper is similar, but in the more complicated setting of periodic orbits: to study the Reidemeister orbit set of a group endomorphism in relation to an invariant normal subgroup, to obtain addition formulae for Reidemeister orbit numbers, and as application, to find addition formulae for Nielsen type essential orbit numbers of fibre preserving maps.Given a group endomorphism f : G → G, the Reidemeister set of f , denoted by R(f ), is the set of orbits of the left action of G on G via γ g → 2000 Mathematics Subject Classification: Primary 55M20.

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confidence: 55%

“…When f is the homomorphism induced by a map X → X on the fundamental group π 1 (X), R(f ) is an upper bound for the Nielsen number N (f ); N (f ) is usually the minimal number of fixed points in the homotopy class of the map. For many spaces, such as nil and solvmanifolds, the computation of the Nielsen number reduces to that of R(f ) (see [3]). …”

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confidence: 99%

Reidemeister orbit sets

2004

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“…Heath and Keppelmann [1,2] proved that the two Nielsen type numbers N P n (f ) and N Φ n (f ) can be computed according to the Nielsen numbers for most maps on nilmanifolds. Since there are already formulae for the Nielsen numbers of continuous maps on the 3-nilmanifolds (Proposition 3.5), N P n (f ) and N Φ n (f ) are known provided N (f n ) = 0.…”

Section: Nielsen Type Numbers Np N (F ) and N φ N (F )mentioning

confidence: 99%

“…However using fiber techniques on nilmanifolds and some solvmanifolds, Heath and Keppelmann [1,2] succeeded in showing that the Nielsen numbers and the two Nielsen type numbers are related to each other under certain condition.…”

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confidence: 99%

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Nielsen type numbers and homotopy minimal periods for maps on the 3-nilmanifolds

Lee

Zhao

2008

Sci. China Ser. A-Math.

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For all continuous maps on the 3-nilmanifolds, we give explicit formulae for a complete computation of the Nielsen type numbers NPn(f ) and N Φn(f ). The most general cases were explored by Heath and Keppelmann and the complementary part is studied in this paper. While studying the homotopy minimal periods of all maps on the 3-nilmanifolds, we also give a complete description of the homotopy minimal periods for all maps on the 3-nilmanifolds, including a correction to Jezierski and Marzantowicz's result.

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Fibre Techniques in Nielsen Theory Calculations

Heath¹

Handbook of Topological Fixed Point Theory

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Fibre techniques in Nielsen periodic point theory on nil and solvmanifolds II

Cited by 20 publications

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Reidemeister orbit sets

Reidemeister orbit sets

Nielsen type numbers and homotopy minimal periods for maps on the 3-nilmanifolds

Fibre Techniques in Nielsen Theory Calculations

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