Fold-maps and the space of base point preserving maps of spheres

Ando, Yoshifumi

doi:10.1215/kjm/1250517595

Cited by 24 publications

(25 citation statements)

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“…We proved the h-principle in (h-P) for fold-maps in [3] and [6]. Recently we introduced a very effective condition for the h-principle in (h-P) in [10].…”

Section: Classifying Spacementioning

confidence: 90%

Cobordisms of maps with singularities of given class

Ando

2008

Algebr. Geom. Topol.

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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 I -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 I -regular maps of n-dimensional closed manifolds into P in terms of certain stable homotopy groups. As an application we will study the relationship among the stable homotopy groups of spheres, the above cobordism group and higher singularities.

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“…We proved the h-principle in (h-P) for fold-maps in [3] and [6]. Recently we introduced a very effective condition for the h-principle in (h-P) in [10].…”

Section: Classifying Spacementioning

confidence: 90%

Cobordisms of maps with singularities of given class

Ando

2008

Algebr. Geom. Topol.

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“…A smooth map f: N•¨P is a fold map if and only if the image of j2 f is contained in ƒ ¶n-p+1,0(N,P) and transverse to ‡"n-p+1,0(N,P 2.4 and Proposition 5.7 have already been proved in [An3]. We refer to [B-R], [S-S] and its references in low dimensions (3, 2) and (4, 3) A jet of ‡"n-p+1,0(N,P) will be called a fold jet.…”

mentioning

confidence: 86%

Existence theorems of fold-maps

Ando

2004

Jpn. j. math

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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 T N ⊕ θN → T P covering f , then there exists a fold-map homotopic to f , where θN is the trivial line bundle. We also give an additional condition for finding a fold-map which folds only on a finite number of spheres of dimension p − 1.

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“…Suppose, conversely, that an embedded surface F = F 0 ∪ F 1 satisfying the above properties (1)- (4) is given. Note that, for the moment, we do not assume condition (5).…”

Section: Fold Maps and Embedded Surfacesmentioning

confidence: 99%

“…Ando [4,5,6] has shown the following theorem by using the homotopy principle for fold maps in the 2-jet level (see [3,Theorem 1] It has been known that the stable span of 4-manifolds satisfies the following (for details, see [40,19]). …”

Section: Stable Span and Fold Mapsmentioning

confidence: 99%