Ribbonness of Kervaire’s Sphere-Link in Homotopy 4-Sphere and its Consequences to 2-Complexes

M. A. Kervaire showed that every group of deficiency d and weight d is the fundamental group of a smooth sphere-link of d components in a smooth homotopy 4-sphere. In the use of the smooth unknotting conjecture and the smooth 4D Poincar´e conjecture, any such sphere-link is shown to be a sublink of a free ribbon sphere-link in the 4-sphere. Since every ribbon sphere-link in the 4-sphere is also shown to be a sublink of a free ribbon sphere-link in the 4-sphere, Kervaire’s sphere-link and the ribbon sphere-link are equivalent con- cepts. By applying this result to a ribbon disklink in the 4-disk, it is shown that the compact complement of every ribbon disk-link in the 4-disk is aspherical. By this property, a ribbon disk-link presentation for every contractible finite 2-complex is introduced. By using this presentation, it is shown that every connected subcomplex of a contractible finite 2-complex is aspherical (meaning partially yes for Whitehead aspherical conjecture).