Endpoint-homogeneous fans

Abstract

A fan $F$ is \emph{endpoint-homogeneous} if for any two endpoints $e,e^{\prime }$ of $F$, there is a homeomorphism $h:F\to F$ such that $h(e)=e^{\prime }$. We prove there are uncountably many distinct homeomorphism types of endpoint-homogeneous smooth fans. To do this, we associate to each such fan $F$ a topological invariant, in the form of a characteristic subset $EPG(F)\subseteq [0,1]$ describing how the endpoints of $F$ limit onto any given blade of $F$. We describe precisely all the uncountably many different $X\subseteq [0,1]$ that can arise as $EPG(F)$ for some endpoint-homogeneous smooth fan $F$. We also prove the existence of $\frac{1}{n}$-homogeneous smooth fans for all $n\ge 5$.