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

Locally stable maps $S^3\to R^4$ are classified up to homotopy through locally stable maps. The equivalence class of a map $f$ is determined by three invariants: the isotopy class $\sigma \left(f\right)$ of its framed singularity link, the generalized normal degree $\nu \left(f\right)$, and the algebraic number of cusps $\kappa \left(f\right)$ 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 $\sigma$, $\nu$, and $\kappa$ which satisfy these relations, there exists a map $f:S^3\to R^4$ with $\sigma \left(f\right)=\sigma$, $\nu \left(f\right)=\nu$, and $\kappa \left(f\right)=\kappa$. It follows in particular that every framed link in $S^3$ is the singularity set of some locally stable map into $R^4$.