Cobordism invariants of fold maps

Abstract

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. IntroductionFold maps of (n + q)-dimensional manifolds into n-dimensional manifolds have the formulaas a local form around each singular point, and the subset of the singular points in the source manifold is a (q + 1)-codimensional submanifold (for results about fold maps, see, for example, [1,2,3,5,8,9,18,26,35,36]). 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. In [18] we showed that these invariants describe completely the cobordisms of simple fold maps of (n + 1)-dimensional manifolds into n-dimensional manifolds and in [17] we showed that these invariants detect direct summands of the cobordism group of fold maps, namely stable homotopy groups of spheres. In this paper we extend the results of [17] and show by constructing fibrations of Morse functions over immersed manifolds 2000 Mathematics Subject Classification. Primary 57R45; Secondary 57R75, 57R42, 55Q45.

Invariants of singular sets of smooth maps

“…Cobordism groups of various versions of Morin maps of non-negative dimension are studied in a series of papers by Ando [2], [3], [4], Kalmar [23], [24], [25], [26], Saeki [52], [51], Ikegami-Saeki [20], [21], Ikegami [19], and the author [48]. For a study in the case d < 0 we refer an interested reader to the paper of Szucz [54].…”

Compositions of equi-dimensional fold maps

“…The fold bordism groups have been studied by many authors (see e.g. [3][4][5][6][7][8]14]). In particular, Ando [3,5,6] has proven that SFold(n, 0) is isomorphic to the stable homotopy group π S n of spheres.…”