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Tubular neighbourhoods of embedded submanifolds
Definition
Let be a smooth embedding, and suppose its normal bundle (Normal and conormal bundles of an embedded submanifold) has been equipped with its standard smooth-vector-bundle structure. Under the existence of this structure is supplied by Assuming countable choice, normal and conormal bundles are smooth vector bundles.
A tubular neighbourhood of in consists of:
- an open neighbourhood of the zero section, and
- a smooth embedding
such that:
- for every , and
- is an open neighbourhood of in .
Equivalently, is a diffeomorphism from onto an ambient open neighbourhood of , and its restriction to the zero section is the original inclusion.
Depends on
Used by
- Compatible tubular neighbourhoods agree near compact sets up to ambient isotopy Corollary
- Dual surgery sphere Definition
- Framed embedded surgery sphere Definition
- The normal addition map for a Euclidean submanifold Definition
- The self-intersection number of a complementary-dimensional oriented submanifold Definition
- A fine Euclidean approximation lands in a prescribed tubular neighbourhood Lemma
- Embedded bands joining two framed spheres exist Lemma
- Moving a sphere off a lower-dimensional submanifold Lemma
- Normal push-off zeros are the self-intersection points Lemma
- One transverse intersection gives the standard local cancelling model Lemma
- Positive-codimension thickening reduces closed sources to the open case Lemma
- Pullback of the Thom class along a transverse section computes the Euler class Lemma
- Restriction of formal-immersion data has the parametric lifting property Lemma
- Spheres of adjacent critical levels have product neighbourhoods Lemma
- The complement of an oriented link is a connected smooth three-manifold Lemma
- The framing obstruction lives in the normal bundle of the surgery sphere Lemma
- The normal Thom class realizes the Poincare dual of a closed submanifold Lemma
- The velocity field of an isotopy extends to a neighbourhood Lemma
- Two tubular neighbourhood germs are isomorphic near the zero section Proposition
- The geometric intersection number is the Poincare-dual cup pairing Theorem
- The tubular neighbourhood theorem in a smooth ambient manifold Theorem
Dependency tree · two levels
16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Marco Gualtieri, Topology I: Smooth Manifolds, Part 11, Definition 3.53 (standard reference, not scraped)
- John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Tubular Neighborhoods (standard reference, not scraped)