Gottlieb groups of spherical orbit spaces and a fixed point theorem

Bull. Korean Math. Soc. 1996 Vol. 33, No. 2, 303-310

D. S. Chun, K. H. Choi, and J. Pak Chonbug National University, Wonkwang University and Wayne State University

Abstract : The Gottlieb group of a compact connected $ANR \;\; X, \; G(X)$, consists of all $\alpha \in \Pi_{1} (X)$ such that there is anassociated map $A : S^{1} \times X \rightarrow X$ and a homotopy commutative diagram $$\CD S^1 \times X @>A>> X\\ @A\text{incl}AA @AA\alpha \lor idA\\ @. S^1 \lor X @. \endCD$$ Gottlieb has shown that if $X$ is a finite $K(\Pi, 1)$, then $G(X) =Z (\Pi_{1} (X))$, the center of $\Pi_{1} (X)$. More recently, Oprea has shownthat if $H$ is a finite group which acts freely on an odd dimensionalsphere $S^{2n + 1}, n \geq 1$, then $G(S^{2n + 1}/H) = Z(H)$. To prove his theorem, Oprea used rather complicated algebro-topological arguments. When the action is linear Broughton came up with a rathersimple and geometrical proof of this theorem. The purpose of this paperis to give more insight of these problems and apply toa fixed point theorem on the spherical orbit spaces.

Keywords : Gottlieb group, induced group representation, Nielsen number