Fold maps on 4-manifolds

Abstract

Abstract. We give a complete characterization of those closed orientable 4-manifolds which admit smooth maps into R 3 with only fold singularities. We also clarify the relationship between the existence problem of fold maps and that of linearly independent vector fields on manifolds. Mathematics Subject Classification (2000). Primary 57R45; Secondary 57N13, 57R25.

Self-intersection class for singularities and its application to fold maps

“…Remark 4.3. In [27], the second author obtained the special case of Theorem 1.3 for (n, p) = (4, 3) by using a different method. Corollary 4.2 was also obtained there.…”

“…Corollary 4.2 was also obtained there. (In fact, in [27], it was proved that the sufficient condition for the nonexistence of fold maps mentioned in Corollary 4.2 is also necessary.) Note that Corollary 4.2 generalizes Theorem 1.1 and the main theorem of [30].…”

“…For the existence results, refer to the works of Levine [20],Èliašberg [9], [10] and Ando [2]. For the dimension pair (n, p) = (4, 3), see [27].…”

“…Surprisingly enough, for (n, p) = (4, 3), the conclusion of the above corollary does not hold in general. In fact, if a closed oriented 4-manifold M has signature σ(M ) = ±1, ±2, then M always admits a fold map into R 3 (see [27]). Note that, in this case, k = 1 and 4k = n. Remark 4.8.…”

Obstructions to the existence of fold maps

“…As far as the authors know, there are only a few results about higher-order obstructions to the elimination of singularities. For example, the first and the second authors clarified such a secondary obstruction to eliminating cusp singularities for maps of closed orientable 4-manifolds into 3-manifolds (see [34,37]), and Szűcs [44] discussed this problem from a viewpoint of cobordism of maps.…”