Fixed point theory for homogeneous spaces, II}

Volume 186 / 2005

Peter Wong Fundamenta Mathematicae 186 (2005), 161-175 MSC: Primary 55M20; Secondary 57S99. DOI: 10.4064/fm186-2-4

Abstract

Let $G$ be a compact connected Lie group, $K$ a closed subgroup and $M=G/K$ the homogeneous space of right cosets. Suppose that $M$ is orientable. We show that for any selfmap $f:M\to M$, $L(f)=0 \Rightarrow N(f)=0$ and $L(f)\ne 0 \Rightarrow N(f)=R(f)$ where $L(f)$, $N(f)$, and $R(f)$ denote the Lefschetz, Nielsen, and Reidemeister numbers of $f$, respectively. In particular, this implies that the Lefschetz number is a complete invariant, i.e., $L(f)=0$ iff $f$ is deformable to be fixed point free. This was previously known under the hypothesis that $p_*:H_n(G) \to H_n(M)$ is nontrivial where $n=\dim M$. A simple formula using equivariant degree is given for the Reidemeister trace of a selfmap $f:M\to M$.

Authors

  • Peter WongDepartment of Mathematics Bates College
    Lewiston, ME 04240, U.S.A.
    e-mail

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