Cobordisms of fold maps and maps with prescribed number of cusps

TL;DR

This paper investigates the possible numbers of cusps in generic smooth maps from closed 2k-manifolds to (3k-1)-space, and computes cobordism groups of fold maps and fold bordism groups in related dimensions.

Contribution

It determines the possible cusp counts for certain smooth maps and computes the cobordism and bordism groups of fold maps in specific dimensions, extending understanding of their classification.

Findings

Possible cusp numbers are characterized for generic maps.

Cobordism groups of fold maps are explicitly computed.

Fold bordism groups are described as Abelian groups.

Abstract

A generic smooth map of a closed $2k$-manifold into $(3k-1)$-space has a finite number of cusps ($\Sigma^{1,1}$-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 ($\Sigma^{1,0}$-singularities). Two fold maps are 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. 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.