A pr 2 00 7 Smooth maps with singularities of bounded K - codimensions ∗ † Yoshifumi ANDO

ar X iv : 0 70 4 . 01 15 v 2 [ m at h . G T ] 9 A pr 2 00 7 Smooth maps with singularities of bounded K - codimensions ∗ † Yoshifumi ANDO

Cobordisms of Maps with Singularities of a given Class

Let P be a smooth manifold of dimension p. We will describe the group of all cobordism classes of smooth maps of n-dimensional closed manifolds into P with singularities of a given class (including all fold singularities if n ≧ p) in terms of certain stable homotopy groups by applying the homotopy principle on the existence level, which is assumed to hold for those smooth maps. We will also dea...

A Homotopy Principle for Maps with Prescribed Thom-boardman Singularities

Let N and P be smooth manifolds of dimensions n and p (n ≥ p ≥ 2) respectively. Let Ω (N, P ) denote an open subspace of J∞(N, P ) which consists of all Boardman submanifolds Σ (N, P ) of symbols J with J ≤ I. An Ω -regular map f : N → P refers to a smooth map having only singularities in Ω (N, P ) and satisfying transversality condition. We will prove what is called the homotopy principle for ...

Cobordisms of Maps without Prescribed Singularities

Let N and P be smooth closed manifolds of dimensions n and p respectively. Given a Thom-Boardman symbol I, a smooth map f : N → P is called an Ω -regular map if and only if the Thom-Boardman symbol of each singular point of f is not greater than I in the lexicographic order. We will represent the group of all cobordism classes of Ω -regular maps of n-dimensional closed manifolds into P in terms...

Existence Theorems of Fold-maps

Let N and P be smooth manifolds of dimensions n and p (n ≥ p ≥ 2) respectively. A smooth map having only fold singularities is called a fold-map. We will study conditions for a continuous map f : N → P to be homotopic to a fold-map from the viewpoint of the homotopy principle. By certain homotopy principles for fold-maps, we prove that if there exists a fiberwise epimorphism TN ⊕ θN → TP coveri...