We solve the lifting problem in C*-algebras for many sets of relations that include the relations x_j^N_j=0 for all variables. The remaining relations must be of the form ∥p(x₁,...,x_n)∥ ≦ C for C a positive constant and p a noncommutative *-polynomial that is in some sense homogeneous. For example, we prove liftability for the set of relations x³=0,    y⁴=0,    z⁵=0,    xx*+yy*+zz*≦1. Thus we find more noncommutative semialgebraic sets that have the topology of noncommutative absolute retracts.

This work was partially supported by a grant from the Simons Foundation (208723 to Loring), by the Efroymson fund at UNM, and by the NordForsk Research Network "Operator Algebras and Dynamics'' (grant 11580).