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 paper 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 Szűcs and has interesting consequences even in the case of cobordisms of immersions.

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We will present proofs for two conjectures stated in [2]. The first one is that for an arbitrary manifold W , the homotopy classes of proper maps W × R n → R k+n stabilise as n → ∞, and the second one is that in a stable range there is a Pontryagin-Thom type bijection for proper maps W × R n → R k+n . The second one actually implies the first one and we shall prove the second one by giving an explicit construction.

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