Global classification of isolated singularities in dimensions $(4,3)$ and $(8,5)$

TL;DR

This paper classifies isolated singularities of smooth maps from certain high-dimensional manifolds to lower-dimensional manifolds, determining minimal critical points and characterizing manifolds based on their singularity structure.

Contribution

It provides a classification of isolated singularities in specific dimensions and computes minimal critical points for these maps, extending previous understanding of manifold mappings.

Findings

Characterization of manifolds admitting smooth maps with finitely many critical points

Calculation of minimal critical points for specific dimensions

Application of local singularity classification and low-dimensional topology techniques

Abstract

We characterize those closed $2k$-manifolds admitting smooth maps into $(k+1)$-manifolds with only finitely many critical points, for $k\in\{2,4\}$. We compute then the minimal number of critical points of such smooth maps for $k=2$ and, under some fundamental group restrictions, also for $k=4$. The main ingredients are King's local classification of isolated singularities, decomposition theory, low dimensional cobordisms of spherical fibrations and 3-manifolds topology.