• Title, Summary, Keyword: Nielsen classes

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A NOTE ON NIELSEN TYPE NUMBERS

  • Lee, Seoung-Ho
    • Communications of the Korean Mathematical Society
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    • v.25 no.2
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    • pp.263-271
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    • 2010
  • The Reidemeister orbit set plays a crucial role in the Nielsen type theory of periodic orbits, such as the Reidemeister set does in Nielsen fixed point theory. In this paper, using Heath and You's methods on Nielsen type numbers, we show that these numbers can be evaluated by the set of essential orbit classes under suitable hypotheses, and obtain some formulas in some special cases.

A NOTE ON THE SET OF ROOT CLASSES

  • Lee, Seoung-Ho
    • Communications of the Korean Mathematical Society
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    • v.24 no.3
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    • pp.451-458
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    • 2009
  • The set of root classes plays a crucial role in the Nielsen root theory. Extending Brown et al.'s work on the set of root classes of iterates of maps, we rearrange it into the reduced orbit set and show that under suitable hypotheses, any reduced orbit has the full depth property as in the Nielsen type theory of periodic orbits.

IRREDUCIBLE REIDEMEISTER ORBIT SETS

  • Lee, Seoung Ho
    • Journal of the Chungcheong Mathematical Society
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    • v.27 no.4
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    • pp.721-734
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    • 2014
  • 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 our work on Reidemeister orbit sets, we obtain algebraic results such as addition formulae for irreducible Reidemeister orbit sets. Similar formulae for Nielsen type irreducible essential orbit numbers are also proved for fibre preserving maps.

A NIELSEN TYPE NUMBER OF FIBRE PRESERVING MAPS

  • Lee, Seoung Ho
    • Communications of the Korean Mathematical Society
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    • v.28 no.2
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    • pp.361-369
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    • 2013
  • We introduce a Nielsen type number of a fibre preserving map, and show that it is a lower bound for the number of $n$-orbits in the homotopy class. Under suitable conditions we show that it is equal to the Nielsen type relative essential $n$-orbit number. We also give necessary and sufficient conditions for it and the essential $n$-orbit number to coincide.

MINIMAL SETS OF PERIODS FOR MAPS ON THE KLEIN BOTTLE

  • Kim, Ju-Young;Kim, Sung-Sook;Zhao, Xuezhi
    • Journal of the Korean Mathematical Society
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    • v.45 no.3
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    • pp.883-902
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    • 2008
  • The main results concern with the self maps on the Klein bottle. We obtain the Reidemeister numbers and the Nielsen numbers for all self maps on the Klein bottle. In terms of the Nielsen numbers of their iterates, we totally determine the minimal sets of periods for all homotopy classes of self maps on the Klein bottle.

REIDEMEISTER SETS OF ITERATES

  • Lee, Seoung Ho
    • Journal of the Chungcheong Mathematical Society
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    • v.16 no.1
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    • pp.15-23
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    • 2003
  • In order to compute the Nielsen number N(f) of a self-map $f:X{\rightarrow}X$, some Reidemeister classes in the fundamental group ${\pi}_1(X)$ need to be distinguished. D. Ferrario has some algebraic results which allow distinguishing Reidemeister classes. In this paper we generalize these results to the Reidemeister sets of iterates.

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THE NIELSEN THEOREM FOR SEIFERT FIBERED SPACES OVER LOCALLY SYMMETRIC SPACES

  • RAYMOND, FRANK
    • Journal of the Korean Mathematical Society
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    • v.16 no.1
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    • pp.87-93
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    • 1979
  • In this note the geometric realization of a finite group of homotopy classes of self homotopy equivalences by a finite group of diffeomorphisms is investigated. In order for this to be accomplished an algebraic condition, which guarantees a certain group extension exists, must be satisfied. It is shown for a geometrically interesting class of aspherical manifolds, called injective Seifert fiber spaces over a locally symmetric space, that this necessary algebraic condition is also sufficient for geometric realization.

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