Thom polynomials relative to prescribed maps around the boundary

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 their extension maps, known as singular Seifert surfaces. To place these results in a unified framework, we aim in this paper to establish a relative version of Thom polynomial theory. Our results consist of four parts. (1) We introduce Thom polynomials relative to prescribed maps around the boundary (or a closed codimension-zero submanifold) that avoid singularities of given types. (2) We show a structure theorem for Thom polynomials relative to framable immersions. It expresses them as the sum of the term obtained by substituting Kervaire's relative characteristic classes into the absolute Thom polynomial and a universal correction term. (3) We determine correction terms in several cases, not only reinterpreting earlier works as instances of relative Thom polynomials but also recovering some of them. Most earlier formulas are summarized as the vanishing of correction terms. (4) We give suggestive evidence for the relative Thom polynomials of multi-singularity types $A_0^k$, with an application.