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 Zap. Nauchn. Sem. LOMI, 1982, Volume 122, Pages 135–136 (Mi znsl4092)

Nielsen numbers and fixed points of self-mappings of wedges of circles

V. G. Turaev

Abstract: It is well known that if $f$ is a self-mapping of a compact connected polyhedron then $f$ has at least $N(f)$ fixed points where $N(f)$ denotes the Hielsen number of $f$. The present paper shows that for some self-mappings of $S^1\vee S^1$ tnis estimate is far from being precise. Namely, the following theorem is proved:
If $\alpha$ and $\beta$ are the canonical generators of $\pi_1(S^1\vee S^1)$ and if $f$ is a mapping $S^1\vee S^1\to S^1\vee S^1$ such that $f_\sharp(\alpha)=1$ and $f_\sharp(\beta)$ is conjugate to $(\alpha\beta\alpha^{-1}\beta^{-1})^n\alpha\beta\alpha^{-1}$ with $n\geqslant1$ then $N(f)=0$ and any mapping homotopic to $f$ has at least $2n-1$ fixed points.

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Bibliographic databases:
UDC: 515.143

Citation: V. G. Turaev, “Nielsen numbers and fixed points of self-mappings of wedges of circles”, Investigations in topology. Part IV, Zap. Nauchn. Sem. LOMI, 122, "Nauka", Leningrad. Otdel., Leningrad, 1982, 135–136

Citation in format AMSBIB
\Bibitem{Tur82} \by V.~G.~Turaev \paper Nielsen numbers and fixed points of self-mappings of wedges of circles \inbook Investigations in topology. Part~IV \serial Zap. Nauchn. Sem. LOMI \yr 1982 \vol 122 \pages 135--136 \publ "Nauka", Leningrad. Otdel. \publaddr Leningrad \mathnet{http://mi.mathnet.ru/znsl4092} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=661472} \zmath{https://zbmath.org/?q=an:0492.55002} 

• http://mi.mathnet.ru/eng/znsl4092
• http://mi.mathnet.ru/eng/znsl/v122/p135

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This publication is cited in the following articles:
1. S. A. Bogatyi, D. L. Gonçalves, H. Zieschang, “Coincidence Theory: The Minimizing Problem”, Proc. Steklov Inst. Math., 225 (1999), 45–77