Local index formula for the quantum double suspension

Our understanding of local index formula in noncommutative geometry is stalled for a while because we do not have more than one explicit computation, namely that of Connes for quantum SU(2) and do not understand the meaning of the various multilinear functionals involved in the formula. In such a situation further progress in understanding necessitates more explicit computations and here we execute the second explicit computation for the quantum double suspension, a construction inspired by the Toeplitz extension. More specifically we compute local index formula for the quantum double suspensions of $C(S^2)$ and the noncommutative $2$-torus.