A geometric classification of the path components of the space of locally stable maps S3→R4S3→R4

Locally stable maps S3→R4S3→R4 are classified up to homotopy through locally stable maps. The equivalence class of a map ff is determined by three invariants: the isotopy class σ(f)σ(f) of its framed singularity link, the generalized normal degree ν(f)ν(f), and the algebraic number of cusps κ(f)κ(f) of any extension of ff to a locally stable map of the 4-disk into R5R5. Relations between the invariants are described, and it is proved that for any σσ, νν, and κκ which satisfy these relations, there exists a map f:S3→R4f:S3→R4 with σ(f)=σσ(f)=σ, ν(f)=νν(f)=ν, and κ(f)=κκ(f)=κ. It follows in particular that every framed link in S3S3 is the singularity set of some locally stable map into R4R4.