Homotopy Properties and Lower-Order Vertices in Higher-Spin Equations

New homotopy approach to the analysis of nonlinear higher-spin equations is developed. It is shown to directly reproduce the previously obtained local vertices. Simplest cubic (quartic in Lagrangian nomenclature) higher-spin interaction vertices in four dimensional theory are examined from locality perspective by the new approach and shown to be local. The results are obtained in a background independent fashion.