Connes’ calculus for the quantum double suspension

Given a spectral triple (A,H,D) Connes associated a canonical differential graded algebra ΩD•(A). However, so far this has been computed for very few special cases. We identify suitable hypotheses on a spectral triple that helps one to compute the associated Connes’ calculus for its quantum double suspension. This allows one to compute ΩD• for spectral triples obtained by iterated quantum double suspension of the spectral triple associated with a first order differential operator on a compact smooth manifold. This gives the first systematic computation of Connes’ calculus for a large family of spectral triples.