A geometric classification of the path components of the space of locally stable maps $S^3\to {\mathbb{R}}^4$

Abstract

Locally stable maps $S^3\to {\mathbb{R}}^4$ are classified up to homotopy through locally stable maps. The equivalence class of a map $f$ is determined by three invariants: the isotopy class $\sigma (f)$ of its framed singularity link, the generalized normal degree $\nu (f)$, and the algebraic number of cusps $\kappa (f)$ of any extension of $f$ to a locally stable map of the $4$-disk into ${\mathbb{R}}^5$. 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 $f:S^3\to {\mathbb{R}}^4$ with $\sigma (f)=\sigma$, $\nu (f)=\nu$, and $\kappa (f)=\kappa$. It follows in particular that every framed link in $S^3$ is the singularity set of some locally stable map into ${\mathbb{R}}^4$.

arXiv commentOld manuscript (2007) posted for archival purposes, 20 pages, 2 figures

References in corpus (3)

• A geometric classification of immersions of 3-manifolds into 5-space
• Regular homotopy classes of locally generic mappings
• Regular homotopy classes of singular maps