The Classification of Immersions of Spheres in Euclidean Spaces

Smale, Stephen T.

doi:10.2307/1970186

Cited by 213 publications

(116 citation statements)

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“…The analogous conclusion holds if dim X = 2 and X admits a Morse exhaustion function ρ : X → R without critical points of index >1. Theorem 1.4 is a holomorphic analogue of the Smale-Hirsch h-principle for smooth immersions [36,39,54] and of the Gromov-Phillips h-principle for smooth submersions [34,51]. The conclusion holds with a fixed Stein structure on X provided that Y satisfies a certain flexibility condition introduced (for submersions) in [18].…”

Section: Theorem 11 Let X Be a Stein Manifold With The Complex Strucmentioning

confidence: 95%

Stein structures and holomorphic mappings

Forstnerič

Slapar

2007

Math. Z.

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We prove that every continuous map from a Stein manifold X to a complex manifold Y can be made holomorphic by a homotopic deformation of both the map and the Stein structure on X. In the absence of topological obstructions the holomorphic map may be chosen to have pointwise maximal rank. The analogous result holds for any compact Hausdorff family of maps, but it fails in general for a noncompact family. Our main results are actually proved for smooth almost complex source manifolds (X,J) with the correct handlebody structure. The paper contains another proof of Eliashberg's (Int J Math 1:29--46, 1990) homotopy characterization of Stein manifolds and a slightly different explanation of the construction of exotic Stein surfaces due to Gompf (Ann Math 148 (2):619--693, 1998; J Symplectic Geom 3:565--587, 2005). (See also the related preprint math/0509419).Comment: The original publication is available at http://www.springerlink.co

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Section: Theorem 11 Let X Be a Stein Manifold With The Complex Strucmentioning

confidence: 95%

Stein structures and holomorphic mappings

Forstnerič

Slapar

2007

Math. Z.

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“…where ψ is an orientation-preserving diffeomorphism (see [20,Theorem 5.2] In our previous paper [24], we related the Hopf invariant for F to the normal Euler class of F; however, there was a sign error. This subsection is devoted to correcting the sign error and reviewing a formula for the Haefliger invariant with the corrected sign (see also [2,3,8] …”

Section: Definition 21mentioning

confidence: 99%

“…Ekholm and Szűcs [6, Theorem 1.1 and Remark 3.1] give a formula for the Smale invariant [20] ω : Imm[S 3 , R 5 ] → Z, which provides a group isomorphism between the group Imm[S 3 , R 5 ] of regular homotopy classes of immersions of S 3 in R 5 and the integers Z. …”

Section: Ekholm and Szűcs' Formulamentioning

confidence: 99%

The Hopf invariant of a Haefliger knot

Takase

2006

Math. Z.

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We show that Haefliger's differentiable (6,3)-knot bounds, in 6-space, a 4-manifold (a Seifert surface) of arbitrarily prescribed signature. This implies, according to our previous paper, that the Seifert surface has been prolonged in a prescribed direction near its boundary. This aspect enables us to understand a resemblance between Ekholm-Szucs' formula for the Smale invariant and Boechat-Haefliger's formula for Haefliger knots. As a consequence, we show that an immersion of the 3-sphere in 5-space can be regularly homotoped to the projection of an embedding in 6-space if and only if its Smale invariant is even. We also correct a sign error in our previous paper: "A geometric formula for Haefliger knots" [Topology 43 (2004) 1425-1447]

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“…Smale in [1] has surprised geometers when showing that any two immersions of S 2 in R 3 are regularly homotopic. This work has been extended to a general theory of immersions of manifolds by Smale and Hirsch in [2,3]. Implications of the Smale-Hirsch theory to immersions of surfaces appear in [4,5].…”

Section: Introductionmentioning

confidence: 98%