The Topological Structure of Polish Groups and Groupoids of Measure Space Transformations

Abstract

It is proved that the groupoid of nonsingular partial isomorphisms of a Lebesgue space $(X,\mu )$ is weakly contractible in a “strong” sense: we present a contraction path which preserves invariant the subgroupoid of $\mu$-preserving partial isomorphisms as well as the group of nonsingular transformations of $X$. Moreover, let $\mathcal{R}$ be an ergodic measured discrete equivalence relation on $X$. The full group $[\mathcal{R}]$ endowed with the uniform topology is shown to be contractible. For an approximately finite $\mathcal{R}$ of type $II$ or ${III}_{\lambda }$, $0\le \lambda <1$, the normalizer $N[\mathcal{R}]$ of $\mathcal{R}$ furnished with the natural Polish topology is established to be homotopically equivalent to the centralizer of the associated Poincaré flow. These are the measure theoretical analogues of the resent results of S. Popa and M. Takesaki on the topological structure of the unitary and the automorphism group of a factor.