Fixed point sets and the fundamental group II: Euler characteristics

For a finite group $G$ of not prime power order, Oliver showed that the obstruction for a finite CW-complex $F$ to be the fixed point set of a contractible finite $G$ -CW-complex is determined by the Euler characteristic $\chi (F)$ . (He also has similar results for compact Lie group actions.) We show that the analogous problem for $F$ to be the fixed point set of a finite $G$ -CW-complex of some given homotopy type is still determined by the Euler characteristic. Using trace maps on $K_0$ [2, 7, 18], we also see that there are interesting roles for the fundamental group and the component structure of the fixed point set.