Disentanglement of Topological and Dynamical Fields in 3d Higher-Spin Theory within Shifted Homotopy Approach

The first-order correction to the one-form sector of equations of the $3d$ higher-spin theory is derived from the generating nonlinear HS system by virtue of the shifted homotopy approach. The family of solutions to the generating system that disentangles equations for dynamical and topological fields in the first order of perturbation theory is found. This family is shown to belong to the different cohomology class compared to the solution found earlier by the direct methods. The related cohomology is shown to be the same as that underlying the mass deformation in the matter sector of $3d$ higher-spin equations.