On the topology of ending lamination space

Abstract

We show that if $S$ is a finite-type orientable surface of genus $g$ and with $p$ punctures, where $3g+p\ge 5$, then $\mathcal{ℰ}\mathcal{ℒ}\left(S\right)$ is $\left(n-1\right)$–connected and $\left(n-1\right)$–locally connected, where $dim\left(\mathcal{P}\mathcal{ℳ}\mathcal{ℒ}\left(S\right)\right)=2n+1=6g+2p-7$. Furthermore, if $g=0$, then $\mathcal{ℰ}\mathcal{ℒ}\left(S\right)$ is homeomorphic to the $\left(p-4\right)$–dimensional Nöbeling space. Finally if $n\ne 0$, then $\mathcal{ℱ}\mathcal{P}\mathcal{ℳ}\mathcal{ℒ}\left(S\right)$ is connected.