Cobordism invariants of fold maps

Abstract

This is a survey paper of author's results on cobordism groups and semigroups of fold maps and simple fold maps. The results include: establishing a relation between fold maps and immersions through geometrical invariants of cobordism classes of fold maps and simple fold maps in terms of immersions with prescribed normal bundles, detecting stable homotopy groups of spheres as direct summands of the cobordism semigroups of fold maps, Pontryagin-Thom type construction for -1 codimensional fold maps and estimations about the cobordism classes of manifolds which have fold maps into stably parallelizable manifolds. In the last section some of these results are extended and we show that our invariants also detect stable homotopy groups of the classifying spaces $BO(k)$ as direct summands of the cobordism semigroups of fold maps.

13 pages

References in corpus (5)

• Cobordisms of maps without prescribed singularities
• Fold cobordisms and stable homotopy groups
• Cobordisms of maps with singularities of a given class
• Cobordisms of fold maps and maps with prescribed number of cusps
• Fold maps and immersions from the viewpoint of cobordism

Cited by in corpus (8)

• Obstructions to the existence of fold maps
• Cobordism of fold maps, stably framed manifolds and immersions
• Cobordisms of maps with singularities of a given class
• Singular cobordism categories
• Invariants of singular sets of smooth maps
• Fold maps and immersions from the viewpoint of cobordism
• Cobordisms of fold maps of 4-manifolds into the space
• Fold maps, framed immersions and smooth structures