Multiplicative properties of Morin maps

TL;DR

This paper develops a ring structure on cobordism classes of Morin maps, analyzes their singular strata, and computes Thom polynomials for specific singularities, advancing the algebraic understanding of these maps.

Contribution

It introduces a ring structure on cobordism classes of Morin maps and computes Thom polynomials for certain singularities, providing new algebraic tools.

Findings

Ring homomorphism from Morin maps to cobordism ring

Identification of the ring of Morin maps

Product formula for specific singularities

Abstract

In the first part of the paper we construct a ring structure on the rational cobordism classes of Morin maps (i. e. smooth generic maps of corank 1). We show that associating to a Morin map its singular strata defines a ring homomorphism to $\Omega_* \otimes \Q$, the rational oriented cobordism ring. This is proved by analyzing multiple-point sets of product immersion. Using these homomorphisms we are able to identify the ring of Morin maps. In the second part of the paper we compute the oriented Thom polynomial of the $\Sigma^2$ singularity type with $\Q$ coefficients. Then we provide a product formula for the $\Sigma^2$ and the $\Sigma^{1,1}$ singularities.