Fixed Points of $n$-Valued Multimaps of the Circle

Tom 54 / 2006

Robert F. Brown Bulletin Polish Acad. Sci. Math. 54 (2006), 153-162 MSC: Primary 55M20; Secondary 54C60, 55M25. DOI: 10.4064/ba54-2-7


A multifunction $\phi \colon X \multimap Y$ is $n$-valued if $\phi(x)$ is an unordered subset of $n$ points of $Y$ for each $x \in X$. The (continuous) $n$-valued multimaps $\phi \colon S^1 \multimap S^1$ are classified up to homotopy by an integer-valued degree. In the Nielsen fixed point theory of such multimaps, due to Schirmer, the Nielsen number $N(\phi)$ of an $n$-valued $\phi \colon S^1 \multimap S^1$ of degree $d$ equals $|n - d|$ and $\phi$ is homotopic to an $n$-valued power map that has exactly $|n - d|$ fixed points. Thus the Wecken property, that Schirmer established for manifolds of dimension at least three, also holds for the circle. An $n$-valued multimap $\phi \colon S^1 \multimap S^1$ of degree $d$ splits into $n$ selfmaps of $S^1$ if and only if $d$ is a multiple of $n$.


  • Robert F. BrownDepartment of Mathematics
    University of California
    Los Angeles, CA 90095-1555, U.S.A.

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