COBORDISMS OF FOLD MAPS AND MAPS WITH A PRESCRIBED NUMBER OF CUSPS

Abstract

A generic smooth map of a closed 2k-manifold into (3k - 1)-space has a finite number of cusps (Σ^1,1-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 (Σ^1,0-singularities). Two fold maps are left-right fold bordant if there are cobordisms between their source and 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. Cobordism classes of fold maps are known to form groups. We compute these groups for fold maps of (2k - 1)-manifolds into (3k - 2)-space. Analogous cobordism semigroups for arbitrary closed (3k - 2)-dimensional target manifolds are endowed with Abelian group structures and described. Left-right fold bordism groups in the same dimensions are also described.