Invariants of singular sets of smooth maps

TL;DR

This paper introduces new invariants for the singular sets of smooth maps, especially when classical invariants are undefined, and applies them to classify cobordism groups of maps with Morin singularities.

Contribution

It develops novel invariants for singular sets lacking classical fundamental classes and applies these to classify cobordism groups of maps with Morin singularities.

Findings

New invariants are well-defined for certain singularity types.

These invariants, combined with characteristic classes, fully determine cobordism groups.

The invariants form a graded algebra of characteristic classes.

Abstract

A singular point of a smooth map F: M -> N of manifolds is a point in M at which the rank of the differential dF is less than the minimum of dimensions of M and N. The classical invariant of the set S of singular points of F of a given type is defined by taking the fundamental class [\bar{S}]\in H_*(M) of the closure of S. We introduce and study new invariants of singular sets for which the classical invariants may not be defined, i.e., for which \bar{S} may not possess the fundamental class. The simplest new invariant is defined by carefully choosing the fundamental class of the intersection of \bar{S} and its slight perturbation in M. Surprisingly, for certain singularity types such an invariant is well-define (and not trivial) despite the fact that \bar{S} does not possess the fundamental class. We determine new invariants for maps with Morin singularities---i.e., singularities of types A_k for k>0 in the ADE-classification of simple singularities by Dynkin diagrams---and, as an application, show that these invariants together with generalized Miller-Morita-Mumford classes form a commutative graded algebra of characteristic classes that completely determine the cobordism groups of maps with at most A_k-singularities for each k>0.