Cobordisms of Fold Maps and Maps with Prescribed Number of Cusps

A generic smooth map of a closed 2k-manifold into (3k− 1)-space has a finite number of cusps (Σ-singularities). We determine the possible numbers of cusps of such maps. A fold map is a map with singular set consisting of only fold singularities (Σsingularities). Two fold maps are fold bordant if there are cobordisms between their sourceand target manifolds with a fold map extending the two maps between the boundaries, if the two targets agree and the target cobordism can be taken as a product with a unit interval then the maps are fold cobordant. We compute the cobordism groups of fold maps of (2k − 1)-manifolds into (3k − 2)-space. Analogous cobordism semi-groups for arbitrary closed (3k − 2)-dimensional target manifolds are endowed with Abelian group structures and described. Fold bordism groups in the same dimensions are described as well.