Fixed point theory for homogeneous spaces, II}

Peter Wong¹

¹ Department of Mathematics Bates College Lewiston, ME 04240, U.S.A.

Fundamenta Mathematicae, Tome 186 (2005) no. 2, pp. 161-175

Résumé

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 $f:M\to M$.

Détail

DOI : 10.4064/fm186-2-4

Keywords: compact connected lie group closed subgroup homogeneous space right cosets suppose orientable selfmap rightarrow rightarrow where denote lefschetz nielsen reidemeister numbers respectively particular implies lefschetz number complete invariant deformable fixed point previously known under hypothesis * nontrivial where dim simple formula using equivariant degree given reidemeister trace selfmap