Cobordism of singular maps
  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. 
Andr\'as Sz\H{u}cs