Cobordisms of maps with singularities of given class

Abstract

Let $P$ be a smooth manifold of dimension $p$. We will describe the group of all cobordism classes of smooth maps of $n$–dimensional closed manifolds into $P$ with singularities of given class (including all fold singularities if $n\ge p$) in terms of certain stable homotopy groups by applying the homotopy principle on the existence level, which is assumed to hold for those smooth maps. It will enable us to construct an explicit classifying space for this cobordism group in the dimensions $n<p$ and $n\ge p\ge 2$.