Multiplicative properties of Morin maps

Abstract

In the first part of the paper we construct a ring structure on the rational cobordism classes of Morin maps (ie smooth generic maps of corank 1). We show that associating to a Morin map its ${\Sigma }^{1_r}$ (or $A_r$) singular strata defines a ring homomorphism to ${\Omega }_{\ast }\otimes \mathbb{Q}$, the rational oriented cobordism ring. This is proved by analyzing the multiple-point sets of a product immersion. Using these homomorphisms we compute the ring of Morin maps.

In the second part of the paper we give a new method to find the oriented Thom polynomial of the ${\Sigma }^2$ singularity type with $\mathbb{Q}$ coefficients. Then we provide a product formula for the ${\Sigma }^2$ singularity in $\mathbb{Q}$ and the ${\Sigma }^{1,1}$ singularity in ${\mathbb{Z}}_2$ coefficients.