Thom polynomials relative to prescribed maps between codimension-zero submanifolds

TL;DR

This paper develops a relative version of Thom polynomial theory to study obstructions for extending maps between submanifolds without singularities, providing a unified framework and new insights into classical invariants.

Contribution

It introduces relative Thom polynomials, establishes a structure theorem for framed immersions, and computes correction terms, extending and unifying previous results in singularity theory.

Findings

Relative Thom polynomials serve as obstructions for extensions of maps avoiding singularities.

A structure theorem expresses these polynomials as a sum involving Kervaire's characteristic classes and correction terms.

Correction terms are identified as classical invariants or vanish in specific cases.

Abstract

Thom polynomials are universal cohomological obstructions to the appearance of singularities of given types in differentiable maps. As an application, various invariants of immersions have been expressed in terms of singularities of extensions of immersions (known as singular Seifert surfaces). To place these results in a unified framework, we aim in this paper to establish the foundation of a relative version of Thom polynomial theory. Our result consists of three parts. (1) We introduce the notion of relative Thom polynomials, which are relative cohomological obstructions for extensions of prescribed maps between codimension-zero submanifolds that avoid singularities of given types. (2) We show a structure theorem for relative Thom polynomials when the prescribed map is a framed immersion. It expresses them as the sum of the naive substitution of Kervaire's relative characteristic classes into the absolute Thom polynomial and a correction term. As a consequence, the correction term forms a regular homotopy invariant of the prescribed map. (3) We determine correction terms in several cases, not only reinterpreting earlier works as instances of relative Thom polynomials but also applying our framework to the type $A_1$. We observe that these terms vanish or consist of classical invariants and their variants.