Analogues of the Atiyah--Wall exact sequences for cobordism groups of singular maps
Andr\'as Cs\'epai

TL;DR
This paper extends classical cobordism exact sequences to maps with prescribed singularities, answering a long-standing question and revealing new insights in cobordism theory.
Contribution
It introduces analogues of Atiyah--Wall sequences for singular maps, bridging oriented and unoriented cobordism groups in a novel way.
Findings
Established new exact sequences for cobordism groups of singular maps
Provided a positive answer to Sz ext{"u}cs's fifteen-year-old question
Uncovered implications for cobordisms of immersions
Abstract
Classical results of Rohlin, Dold, Wall and Atiyah yield two exact sequences that connect the oriented and unoriented (abstract) cobordism groups and . 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 Sz\H{u}cs and has interesting consequences even in the case of cobordisms of immersions.
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Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Geometric Analysis and Curvature Flows
