Cobordisms of Fold Maps and Maps With a Prescribed Number of Cusps

Abstract

Abstract.A generic smooth map of a closed 2k-manifold into (3k − 1)-space has a finite number of cusps ( 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 ( 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.

Cobordism of singular maps

“…In [10] we computed the cobordism groups of fold maps Cob .n; k/ for all cases when n D 2k 1. For the case of fold maps of 2K C 2 manifolds in the Euclidean space R 3K C2 Terpai gave a complete computation in [42].…”

Cobordisms of fold maps of 2k+2-manifolds into R3k+2

“…Calculating π 3k+2 (X 1,0 ). The long exact sequence (2) gives a short exact sequence 0 → coker T → π 3k+2 (X 1,0 ) → ker T 1,0 3k+2 → 0 where ker T 1,0 3k+2 has been calculated in [3], so we need to determine coker T . First, we claim that Corollary 3 is applicable and the kernel of C is always trivial.…”

Cobordism invariants of fold maps

“…If we restrict a fold map to the set of its singular points, then we obtain a codimension one immersion into the target manifold of the fold map. This immersion together with more detailed informations about the neighbourhood of the set of singular points in the source manifold can be used as a geometrical invariant (see Section 2) of fold cobordism classes (see Definition 1.1) of fold maps (for results about cobordisms of singular maps with completely different approach from our present paper, see, for example, [7,11,12,21,25,34,48] and the works of Ando, Sadykov, Szűcs and the author in References). In this way we obtain a geometrical relation between fold maps and immersions with prescribed normal bundles via cobordisms.…”