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M. A. Kervaire showed in 1965 that every finitely presented group withdeficiency k, weight k and the free first homology of rank k is the fundamentalgroup of a smooth sphere-link of component k in a smooth homotopy 4-sphere. Usethe smooth unknotting conjecture and the smooth 4D Poincar{\'e} conjecture. Anysuch 2-link is shown to be equivalent to a sublink of a free ribbon sphere-linkin the 4-sphere, whose ribbon disk-link complement in the 4-disk is also shownto have a finite aspherical 2-complex as a spine. Every ribbon sphere-link inthe 4-sphere is also shown to be a sublink of a free ribbon sphere-link in the4-sphere, so that the complement of every ribbon disk-link in the 4-disk has afinite aspherical 2-complex as a spine. This implies that every subcomplex of afinite contractible 2-complex is aspherical (partially yes for J. H. C.Whitehead's conjecture).

Ribbonness of Kervaire's sphere-link in homotopy 4-sphere and its consequences to 2-complexes