The homotopy of Гq and classifying spaces

Abstract

Let γ≥1 and Гq ^ be the topological groupoid of germs or orientation preserving local C(^v) diffeomorphisms of/R(^z). Then a E(^2) spectral sequence is constructed with the E(^2) terms computed from the homological properties of Г(_2), and E(^oo) is the bigraded module associated to the filtration of the homology of the classifying space, BГ given by Haefliger in [HA3]. Let S≥1 and Δ - rr [1,2,.....S+1] be the objects of the category C(_Δ)s with morphisms ≤ and Ѓ(_2)(Δ(^s) the space of functors from C(_Δ)-s to Г(_2) with the usual topology on Ѓ(_q). We prove that a) H(_t)(Ѓ(_q)(Δ(^s))(A')) = 0 for t > q. b) If Gl(_2) is the group of linear transformations of R(^2) with positive determinant then if V: Ѓ(_2)(Δ(^s))→(Gl(_2))(^s) is the map obtained by taking derivatives of ɤ (1 ≤ i) for 2 ≤ i ≤ s +1, ɤϵЃ(_2) is an isomorphism for t <q. These calculations go a long way towards calculating the E(^2) terms of the above spectral sequence. The spectral sequence is constructed for a large class of topological groupoids referred to as well formed topological groupoids, and the corresponding theorem on the high dimensional homologies of topological groupoids is proved for a special class of well formed topological groupoids which include the known topological groupoids associated with foliations.