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

A generic smooth map of a closed $ 2k $-manifold into $ (3k - 1) $-space has a finite number of cusps ($ {sum}^{1,1} $-singularities). We determine the possible numbers of cusps of such maps. A fold map is a map with singular set consisting of only fold singularities ($ {sum}^{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