Differential Contracting Homotopy in the Linearized 3d Higher-Spin Theory

In this paper, the recently developed differential homotopy approach is applied to the problem of disentangling dynamical and topological fields of the $3d$ higher-spin gauge theory at the linear level. This formalism allows us to reproduce all known disentangling solutions in a unified form, including both the solutions obtained previously within the shifted homotopy approach in \cite{Korybut:2022kdx} and that derived by hand in \cite{Vasiliev:1992ix}, as well as other solutions including those associated with the cohomology of the background covariant derivative $D_0$. Also, within the differential homotopy framework an alternative way of derivation of disentangled equations through a non-conventional solution for the field $S_1$ is suggested. The obtained results are important for further analysis of nonlinear corrections to HS equations in $AdS_3$.