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

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 $\Omega_n$ and $\mathfrak{N}_n$. 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 Sz\H{u}cs and has interesting consequences even in the case of cobordisms of immersions.