Cobordism of singular maps

Abstract

We prove a conjecture due to M. Kazarian, connecting two classifying spaces in singularity theory. These spaces are: - Kazarian's space (generalizing Vassiliev's algebraic complex and) showing which cohomology classes are represented by singularity strata. - Author's space $X_{\tau }$ giving homotopy representation of cobordisms of singular maps with a given list of allowed singularities \cite{R--Sz}. As a consequence we obtain the ranks of cobordism groups of singular maps with a given list of allowed singularities, and also their $p$-torsion parts for big primes $p.$ Further we give complete answer to the problem of elimination of singularities by cobordisms. Obtain very clear homotopical description of the classifying space $X_{\tau }.$ We reveal some connection of the torsion parts of these cobordism groups to the stable homotopy groups of spheres and values of Thom polynomials.

Revised version, some parts were excluded, some parts are explained in greater detail

References in corpus (1)

• Cobordisms of fold maps and maps with prescribed number of cusps

Cited by in corpus (1)

• Cobordisms of singular maps