Filament local product structures in homogeneous continua

This is a classifying study of homogeneous continua focused on decoding the structure of their neighborhoods. All non-locally-connected homogeneous continua have closed neighborhoods whose quotient space of components is homeomorphic to the Cantor set. Yet there are homogeneous non-locally-connected continua without neighborhoods homeomorphic to the product of a continuum and the Cantor set. The main result of this paper provides a useful criterion for identifying such neighborhoods. We show a number of applications of this result.