Cobordisms of singular maps

TL;DR

This paper reviews and organizes the development of cobordism groups of singular maps, highlighting key results and computational methods introduced since the concept's inception around 1980.

Contribution

It consolidates existing results and methodologies on cobordism groups of singular maps, providing a comprehensive overview of the field's progress.

Findings

Compilation of key results by Szűcs, Terpai, and others.

Development of classifying space techniques for cobordism groups.

Partial computations of cobordism groups in various dimensions.

Abstract

The notion of cobordism of singular maps was introduced around 1980 by A. Sz\H{u}cs and U. Koschorke independently. As an application, Sz\H{u}cs used it to compute cobordism groups of immersions and embeddings in dimensions where the classical theory did not succeed. His method of studying cobordism groups of singular maps involves the investigation of classifying spaces that were constructed by him, first in a few special cases, then, with the help of the work of Rim\'anyi, in complete generality. Since then quite a few results and (partial) computations were performed in this theory. The present thesis collects and organises these results of Sz\H{u}cs, Terpai and other coauthors towards and on the computation of cobordism groups of singular maps.