Multiplicative properties of Morin maps

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 ${\mathrm{\Omega }}_{\ast }\otimes \text{Q}$, 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 ${\mathrm{\Sigma }}^2$ singularity type with $\text{Q}$ coefficients. Then we provide a product formula for the ${\mathrm{\Sigma }}^2$ and the ${\mathrm{\Sigma }}^{1,1}$ singularities.