Cobordism of maps on $Z_2$-Witt spaces

In this article we study the bordism groups of normally nonsingular maps $f: X \to Y$ defined on pseudomanifolds $X$ and $Y$. To characterize the bordism of such maps, inspired by the formula given by Stong, we give a general definition of Stiefel-Whitney numbers defined on $X$ and $Y$ using the Wu classes defined by Goresky and Pardon and we show that in several cases the cobordism class of a normally nonsingular map $f:X \to Y$ guarantees that these numbers are zero.