Analogues of the Atiyah-Wall exact sequences for cobordism groups of singular maps

Classical results of Rohlin, Dold, Wall and Atiyah yield two exact sequences that connect the oriented and unoriented (abstract) cobordism groups Ω_n and N_n. In this talk we present analogous exact sequences connecting the oriented and unoriented cobordism groups of maps with prescribed singularities. This gives positive answer to a fifteen-year-old question posed by Szūcs and has interesting consequences even in the case of cobordisms of immersions.