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Principal bundle structure of the space of metric measure spaces
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 18 November 2024, pp. 1-31
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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 
$*$, as only one fixed-point. We prove that the 
${\mathbb{R}}_+$-action on 
$\mathcal{X}_* := \mathcal{X} \setminus \{*\}$ admits the structure of non-trivial and locally trivial principal 
${\mathbb{R}}_+$-bundle over the quotient space. Our bundle 
${\mathbb{R}}_+ \to \mathcal{X}_* \to \mathcal{X}_*/{\mathbb{R}}_+$ is a curious example of a non-trivial principal fibre bundle with contractible fibre. 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.
