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Thom polynomials relative to prescribed maps between codimension-zero submanifolds

Masato Tanabe

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.

Thom polynomials relative to prescribed maps between codimension-zero submanifolds

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 . We observe that these terms vanish or consist of classical invariants and their variants.
Paper Structure (36 sections, 31 theorems, 153 equations, 2 tables)

This paper contains 36 sections, 31 theorems, 153 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Main result
  4. Definition of relative Thom polynomials
  5. Structure of relative Thom polynomials
  6. Determining correction terms
  7. Absolute Thom polynomials
  8. Classification of singularities
  9. Thom polynomials
  10. Definition of relative Thom polynomials
  11. General situation
  12. For singularities of maps
  13. Relative Thom polynomials as obstruction classes
  14. Relative characteristic classes
  15. Definitions
  16. ...and 21 more sections

Key Result

Theorem 1

Let $\eta$ be a singularity type of codimension $q$. Then $\mathrm{Tp}(\eta) \in \mathbb Z_2[w_i, w'_j]$ is a unique homogeneous polynomial of degree $q$ satisfying the following: for any compact $m$-manifold $M$, $n$-manifold $N$, and map $f \colon M \to N$ generic with respect to $\eta$, where the left-hand side is the Poincaré dual to $\overline{\eta(f)} \subset M$.

Theorems & Definitions (91)

  • Definition 2.1
  • Example 1
  • Example 2: first order Thom--Boardman singularities
  • Remark 1
  • Definition 2.2
  • Theorem 1: Tho55HK56
  • proof
  • Remark 2
  • Remark 3: real $C^\infty$ oriented version
  • Example 3: And82, Rim00
  • ...and 81 more