Let $G$ be a split semisimple group over a global function field $K$. Given a cuspidal automorphic representation $\Pi$ of $G$ satisfying a technical hypothesis, we prove that for almost all primes $\ell$, there is a cyclic base change lifting of $\Pi$ along any $\mathbb {Z}/\ell \mathbb {Z}$-extension of $K$. Our proof does not rely on any trace formulas; instead it is based on using modularity lifting theorems, together with a Smith theory argument, to obtain base change for residual representations. As an application, we also prove that for any split semisimple group $G$ over a local function field $F$, and almost all primes $\ell$, any irreducible admissible representation of $G(F)$ admits a base change along any $\mathbb {Z}/\ell \mathbb {Z}$-extension of $F$. Finally, we characterize local base change more explicitly for a class of toral representations considered in work of Chan and Oi.

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.

Smith theory says that the fixed point set of a semi-free action of a group $G$ on a contractible space is ${\mathbb {Z}}_p$-acyclic for any prime factor $p$ of the order of $G$. Jones proved the converse of Smith theory for the case $G$ is a cyclic group acting semi-freely on contractible, finite CW-complexes. We extend the theory to semi-free group actions on finite CW-complexes of given homotopy types, in various settings. In particular, the converse of Smith theory holds if and only if a certain $K$-theoretical obstruction vanishes. We also give some examples that show the geometrical effects of different types of $K$-theoretical obstructions.