Path concordances as detectors of codimension-one manifold factors

A collapsible conductive material includes a generally mesh-configured, collapsible surface, that defines the intended reflective geometry of an antenna. A distribution of tensionable cords and ties form radial truss elements with a plurality of inflatable radially extending ribs and posts of a support structure. The antenna is fully deployed once the support structure is inflated to at least a minimum pressure necessary to place the ties and cords in tension so that the reflective surface acquires a prescribed (e.g., parabolic) geometry, which is stably maintained by the radial truss elements.