Smooth maps to the plane and Pontryagin classes, Part I: Local aspects

TL;DR

This paper classifies local forms of smooth maps from manifolds to the plane, focusing on germs and multigerms, and discusses their relation to Pontryagin classes, emphasizing local classification aspects.

Contribution

It provides a detailed classification of germs and multigerms of smooth maps to the plane, connecting local map forms with topological invariants like Pontryagin classes.

Findings

Classification of germs of smooth maps to the plane

Classification of multigerms of smooth maps

Insights into local map structures related to Pontryagin classes

Abstract

We classify the most common local forms of smooth maps from a smooth manifold L to the plane. The word "local" can refer to locations in the source L, but also to locations in the target. The first point of view leads us to a classification of certain germs of maps, which we review here although it is very well known. The second point of view leads us to a classification of certain multigerms of maps.