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

2022/01/22 by Csépai, András
#55N22 #55P42 (Secondary) #57R45 #57R90 (Primary) 57R35 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2201.09014

Abstract

Classical results of Rohlin, Dold, Wall and Atiyah yield two exact sequences that connect the oriented and unoriented (abstract) cobordism groups Ω_n and \mathfrakN_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.

Related

• Cobordisms of singular maps
• Computations on cobordism groups of projected immersions
• Cobordism of singular maps
• Cobordisms of maps without prescribed singularities
• Cobordism maps in Khovanov homology and singular instanton homology II