The group PSL2R acts
transitively on the circle S1= R ∪∞, by linear fractional transformations. A
homeomorphism g : U → V between open subsets of R is called C1, piecewiseprojective if g is C1, and if there is some locally finite subset S of U such that, on
each component of U − S, g agrees with some element of PSL2R. Let ΓR be the
pseudogroup of such homeomorphisms. We show that the Haefliger classifying
space BΓR is simply connected, and that there is a homology isomorphism
i : BPSL2R→ BΓR. (PSL2R is the universal cover of PSL2R, considered as a
discrete group.) As a consequence, the classifying space of the discrete group of
compactly supported, C1 piecewise projective homeomorphisms of R is a “homology
loop space” of BPSL2R.
