Principal bundle structure of the space of metric measure spaces

Abstract

We study the topological structure of the space $\mathcal{X}$ of isomorphism classes of metric measure spaces equipped with the box or concentration topologies. We consider the scale-change action of the multiplicative group ${\mathbb{R}}_{+}$ of positive real numbers on $\mathcal{X}$, which has a one-point metric measure space, say $\ast$, as only one fixed-point. We prove that the ${\mathbb{R}}_{+}$-action on ${\mathcal{X}}_{\ast }:=\mathcal{X}\setminus \{\ast \}$ admits the structure of nontrivial and locally trivial principal ${\mathbb{R}}_{+}$-bundle over the quotient space. Our bundle ${\mathbb{R}}_{+}\to {\mathcal{X}}_{\ast }\to {\mathcal{X}}_{\ast }/{\mathbb{R}}_{+}$ is a curious example of a nontrivial principal fiber bundle with contractible fiber. A similar statement is obtained for the pyramidal compactification of $\mathcal{X}$, where we completely determine the structure of the fixed-point set of the ${\mathbb{R}}_{+}$-action on the compactification.

18 pages