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

V. G. Turaev

Zapiski Nauchnykh Seminarov POMI, Investigations in topology. Part IV, Tome 122 (1982), pp. 135-136

Résumé

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.