A geometric classification of the path components of the space of locally stable maps

Locally stable maps S 3 → R 4 are classified up to homotopy through locally stable maps. The equivalence class of a map f is determined by three invariants: the isotopy class σ ( f ) of its framed singularity link, the generalized normal degree ν ( f ) , and the algebraic number of cusps κ ( f ) of any extension of f to a locally stable map of the 4-disk into R 5 . Relations between the invariants are described, and it is proved that for any σ , ν , and κ which satisfy these relations, there exists a map f : S 3 → R 4 with σ ( f ) = σ , ν ( f ) = ν , and κ ( f ) = κ . It follows in particular that every framed link in S 3 is the singularity set of some locally stable map into R 4 .