An $n$-mean on a topological space $X$ is a symmetric continuous operation $p:X^n\to X$ satisfying $p(x,\dots,x)=x$ for every $x\in X$. The existence of continuous means is closely related to several classical questions in topology, particularly in connection with retract theory and extension properties.

In this talk, we discuss equivariant means associated with group actions on topological spaces and their connections with equivariant absolute extensors. Particular attention will be given to involutions (that is, $\mathbb Z_2$-actions) acting on spaces equipped with compatible lattice structures. We will present some existence results and applications in this setting, and explain how they relate to a classical open problem of Anderson.

This is a joint work with Ananda López Poo.